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THREE DIMENSIONAL ANALYSIS OF FIBRE
REINFORCED POLYMER LAMINATED
COMPOSITES
A thesis submitted to
THE UNIVERSITY OF MANCHESTER
for the degree of
DOCTOR OF PHILOSOPHY (PhD)
in the Faculty of Engineering and Physical Sciences
2012
Haji Elmi Haji Kamis
School of Mechanical, Aerospace and Civil Engineering
Contents
2
Contents
CONTENTS….....................................................................................................2
LIST OF FIGURES .............................................................................................4
LIST OF TABLES..................................... ..........................................................8
ABSTRACT….......................................... .........................................................10
DECLARATION........................................ ........................................................11
COPYRIGHT STATEMENT..............................................................................12
ACKNOWLEDGEMENTS ................................... .............................................13
CHAPTER 1 .INTRODUCTION ....................................................................15
1.1 Introduction.........................................................................................15 1.2 Research Aim and Objectives ............................................................17 1.3 Outline of Thesis.................................................................................18
CHAPTER 2 .LITERATURE REVIEW .................................. ........................21
2.1 Plate Theories ....................................................................................21 2.1.1 Rectangular Kirchhoff-Love plates...............................................24 2.1.2 Development of plate theories.....................................................27
2.2 FRP ....................................................................................................31 2.2.1 Advantages of FRP .....................................................................32 2.2.2 Disadvantages of FRP.................................................................32 2.2.3 Constitutive Material Properties of FRP.......................................33
2.3 FRP strengthening RC structures.......................................................36 2.4 Conclusions ........................................................................................56
CHAPTER 3 .STATE SPACE METHOD OF 3D ELASTICITY................ .....58
3.1 The Concept of State Space Method of 3D Elasticity .........................58 3.2 Governing Equations of Elasticity Problems .......................................61 3.3 State Equations for Simply Supported Orthotropic Plate ....................63 3.4 Conclusions ........................................................................................80
CHAPTER 4 .STATE SPACE SOLUTION OF CLAMPED EDGES LAMINATED PLATE .................................... ....................................................82
4.1 State Space Solution of a Single Layer Plate .....................................83 4.2 State Space Solution of Laminated Plate ...........................................85 4.3 Application of State Space Method to Laminated Plate....................116 4.4 Conclusions ......................................................................................126
Contents
3
CHAPTER 5 .NUMERICAL ANALYSIS OF LAMINATED PLATES ..........128
5.1 Introduction.......................................................................................128 5.2 Modelling of Composite Plate Using FEM ........................................129
5.2.1 Element types............................................................................130 5.2.2 Assembling the model ...............................................................131 5.2.3 Boundary Conditions .................................................................133 5.2.4 Analysis Type ............................................................................134
5.3 Material Properties............................................................................134 5.4 Mesh.................................................................................................135 5.5 Partially Clamped Edges Composite Plate with Variable Thickness to
Width Ratio .......................................................................................136 5.6 Fully Clamped Edges Composite Plate ............................................141 5.7 Laminated Plate Subjected To Hydrostatic Loading .........................161 5.8 Laminated Plate with increasing number of sub-layers.....................163 5.9 Conclusions ......................................................................................167
CHAPTER 6 .FLEXURAL DEFORMATION OF RC SLAB WITH FRP ......170
6.1 Numerical Modelling .........................................................................170 6.2 Geometric Properties of the Model ...................................................175 6.3 Element Type ...................................................................................178
6.3.1 Concrete Slab............................................................................179 6.3.2 Steel reinforcement bars ...........................................................179 6.3.3 FRP sheets................................................................................180 6.3.4 Interaction of FRP and concrete slab ........................................180 6.3.5 Boundary condition....................................................................182
6.4 Tension Stiffening.............................................................................183 6.4.1 Non-linear tension stiffening stress-strain relation.....................185 6.4.2 Linear tension stiffening stress-strain relation ...........................186 6.4.3 Multi-linear tension stiffening stress-strain relation....................187
6.5 FEM Against Experimental Test Results ..........................................188 6.6 Concrete slab reinforced with FRP ...................................................191 6.7 Conclusions ......................................................................................200
CHAPTER 7 .CONCLUSIONS AND RECOMMENDATIONS .................... 201
7.1 Conclusions ......................................................................................201 7.2 Future Works Recommendations .....................................................203
REFERENCES.. .............................................................................................205
LIST OF PUBLICATIONS ............................... ...............................................209
APPENDIX A… ........................................ ......................................................210
Word count: 37,568 words
List of Figures
4
List of Figures Figure 1.1: Layout of a UD lamina ........................................................................16
Figure 1.2: View of unstacked laminate ................................................................16
Figure 2.1: Normal and shear stresses .................................................................22
Figure 2.2: A simply supported rectangular plate under sinusoidal loading ..........25
Figure 2.3: An Ashby property map for composites ..............................................36
Figure 2.4: A test specimen with end-anchorage..................................................38
Figure 2.5: Mechanically anchored RC slab tested to failure in the laboratory .....39
Figure 2.6: RC Slab Section Strengthened with NSM CFRP rods ........................40
Figure 2.7: Loading system and slab dimensions .................................................41
Figure 2.8: Two way CFRP strips being installed..................................................41
Figure 2.9: (a) Slab dimension ; (b) Test set-up....................................................43
Figure 2.10: Dimensions and test set-up scheme.................................................44
Figure 2.11: Stress distribution for the analysed slabs..........................................45
Figure 2.12: Flexural Test Setup of Concrete Slabs .............................................46
Figure 2.13: Typical crack pattern of the CFRP grid reinforced slabs ...................46
Figure 2.14: Typical theoretical and experimental load versus midspan
deflection curve.....................................................................................................47
Figure 2.15: Strengthened RC beam details.........................................................50
Figure 2.16: Test setup and instrumentation of the two-way slab specimens .......51
Figure 2.17: slab specimen details and loading ....................................................52
Figure 2.18: Details of beam specimens (a) Cross Section and reinforcement
details (b) Zone of FRP repair...............................................................................53
Figure 2.19: Details of FRP retrofitting of slab specimens ....................................53
Figure 2.20: slab specimens tested to failure........................................................54
Figure 3.1: Spring - damper - mass system .........................................................59
Figure 3.2: Coordinate system and plate dimension.............................................63
Figure 4.1: A single layer plate..............................................................................83
Figure 4.2: Geometry and coordinate systems of the laminate.............................86
Figure 4.3: View of a clamped edges laminated plate.........................................117
Figure 5.1: Table used to model composite solid and shell in Abaqus................132
Figure 5.2: Geometry of plate consists of three plies..........................................137
List of Figures
5
Figure 5.3: Mesh sensitivity test results for stresses at x=y=z=0 for h/a = 0.2 ....138
Figure 5.4: Displacement (W C11/qh) distribution through the thickness of the
plate at x = a/2 and y = b/2 when h/a = 0.2 .........................................................143
Figure 5.5: Stress (σx/q) x = a/2 and y = b/2 when h/a = 0.2...............................143
Figure 5.6: Stress (σy/q) x = a/2 and y = b/2 when h/a = 0.2...............................144
Figure 5.7: Stress (σx/q) x = 0 and y = b/2 when h/a = 0.2..................................144
Figure 5.8: Stress (σy/q) x = 0 and y = b/2 when h/a = 0.2..................................145
Figure 5.9: Stress (σx/q) x = a/2 and y = 0 when h/a = 0.2..................................145
Figure 5.10: Stress (σy/q) x = a/2 and y = 0 when h/a = 0.2................................146
Figure 5.11: Stress (τxz/q) at x = 0 and y = b/2 when h/a = 0.2 ..........................146
Figure 5.12: Stress (τyz/q) at x = a/2 and y = 0 when h/a = 0.2 ..........................147
Figure 5.13: Displacement (W C11/qh) distribution through the thickness of the
plate at x = a/2 and y = b/2 when h/a = 0.4 .........................................................149
Figure 5.14: Stress (σx/q) x = a/2 and y = b/2 when h/a = 0.4.............................149
Figure 5.15: Stress (σy/q) x = a/2 and y = b/2 when h/a = 0.4.............................150
Figure 5.16: Stress (σx/q) at x = 0 and y = b/2 when h/a = 0.4............................150
Figure 5.17: Stress (σy/q) at x = 0 and y = b/2 when h/a = 0.4............................151
Figure 5.18: Stress (τxz/q) at x = 0 and y = b/2 when h/a = 0.4 ..........................151
Figure 5.19: Stress (σx/q) at x = a/2 and y = 0 when h/a = 0.4............................152
Figure 5.20: Stress (σy/q) at x = a/2 and y = 0 when h/a = 0.4............................152
Figure 5.21: Stress (τyz/q) at x = a/2 and y = 0 when h/a = 0.4 ..........................153
Figure 5.22: Displacement (W C11/qh) distribution through the thickness of the
plate at x = a/2 and y = b/2 when h/a = 0.6 .........................................................155
Figure 5.23: Stress (σx/q) at x = a/2 and y = b/2 when h/a = 0.6.........................155
Figure 5.24: Stress (σy/q) at x = a/2 and y = b/2 when h/a = 0.6.........................156
Figure 5.25: Stress (σx/q) at x = 0 and y = b/2 when h/a = 0.6............................156
Figure 5.26: Stress (σy/q) at x = 0 and y = b/2 when h/a = 0.6............................157
Figure 5.27: Stress (σx/q) at x = a/2 and y = 0 when h/a = 0.6............................157
Figure 5.28: Stress (σy/q) at x = a/2 and y = 0 when h/a = 0.6............................158
Figure 5.29: Stress (τxz/q) at x = 0 and y = b/2 when h/a = 0.6 ..........................158
Figure 5.30: Stress (τyz/q) at x = a/2 and y = 0 when h/a = 0.6 ..........................159
Figure 5.31: Partially clamped plate subjected to hydrostatic loading for h/a=0.4161
List of Figures
6
Figure 5.32: Central deflection across the thickness of plate h/a=0.4.................166
Figure 5.33: Stress,sx of plate h/a = 0.4 at x = a/2 y = b/2 .................................166
Figure 5.34: Stress,sy of plate h/a = 0.4 at x = a/2 y = b/2 .................................167
Figure 6.1: Stress and strain relationship of (a) concrete (b) steel and (c) FRP .172
Figure 6.2: View of RC slab model (a) without FRP (b) with FRP.......................177
Figure 6.3: FEM of CFRP strengthened RC slab................................................178
Figure 6.4: Damage traction-separation response used in FEM.........................181
Figure 6.5: Concrete tensile response characterized by damaged plasticity ......183
Figure 6.6: Exponential tension stiffening (Hordijk).............................................185
Figure 6.7: Linear tension stiffening....................................................................186
Figure 6.8: Multi-linear tension stiffening ............................................................187
Figure 6.9: Relationship of load against central deflection of RC slab by variable
tension stiffening response .................................................................................189
Figure 6.10: Relationship of load against central deflection of RC slab by
variable element types ........................................................................................189
Figure 6.11: Load against central deflection of un-strengthened and
strengthened of RC slab with CFRP ...................................................................190
Figure 6.12: Geometry and coordinate systems of the layered slab ...................192
Figure 6.13: Deflection distribution (W C11/qh) at x=a/2, y = b/2 h/a = 0.1 ..........197
Figure 6.14: Deflection distribution (W C11/qh) at x=a/2, y = b/2 h/a = 0.2 ..........198
Figure 6.15: Deflection distribution (W C11/qh) at x=a/2, y = b/2 h/a = 0.3 ..........198
Figure 6.16: Deflection distribution (W C11/qh) at x=a/2, y = b/2 h/a = 0.4 ..........199
Figure 6.17: Deflection distribution (W C11/qh) at x=a/2, y = b/2 h/a = 0.6 ..........199
Figure A-1: Deflection (WC11 / qh) at x = a/2 , y = b/2 at z = 0............................221
Figure A-2: Stress (σx/q) at x = a/2 and y = b/2 Ply1 top.....................................221
Figure A-3: Stress (σx/q) at x = a/2 and y = b/2 Ply1 bottom...............................222
Figure A-4: Stress (σx/q) at x = a/2 and y = b/2 Ply2 top.....................................222
Figure A-5: Stress (σx/q) at x = a/2 and y = b/2 Ply2 bottom...............................223
Figure A-6: Stress (σx/q) at x = a/2 and y = b/2 Ply3 top.....................................223
Figure A-7: Stress (σx/q) at x = a/2 and y = b/2 Ply3 bottom...............................224
Figure A-8: Stress (σy/q) at x = a/2 and y = b/2 Ply1 top.....................................224
Figure A-9: Stress (σy/q) at x = a/2 and y = b/2 Ply1 bottom...............................225
Figure A-10: Stress (σy/q) at x = a/2 and y = b/2 Ply2 top...................................225
List of Figures
7
Figure A-11: Stress (σy/q) at x = a/2 and y = b/2 Ply2 bottom.............................226
Figure A-12: Stress (σy/q) at x = a/2 and y = b/2 Ply3 top...................................226
Figure A-13: Stress (σy/q) at x = a/2 and y = b/2 Ply3 bottom.............................227
Figure A-14: Stress (σx/q) at x = 0 and y = b/2 Ply1 top......................................227
Figure A-15: Stress (σx/q) at x = 0 and y = b/2 Ply1 bottom................................228
Figure A-16: Stress (σx/q) at x = 0 and y = b/2 Ply2 top......................................228
Figure A-17: Stress (σx/q) at x = 0 and y = b/2 Ply2 bottom................................229
Figure A-18: Stress (σx/q) at x = 0 and y = b/2 Ply3 top......................................229
Figure A-19: Stress (σx/q) at x = 0 and y = b/2 Ply3 bottom................................230
Figure A-20: Stress (σy/q) at x = 0 and y = b/2 Ply1 top......................................230
Figure A-21: Stress (σy/q) at x = 0 and y = b/2 Ply1 bottom................................231
Figure A-22: Stress (σy/q) at x = 0 and y = b/2 Ply2 top......................................231
Figure A-23: Stress (σy/q) at x = 0 and y = b/2 Ply2 bottom................................232
Figure A-24: Stress (σy/q) at x = 0 and y = b/2 Ply3 top......................................232
Figure A-25: Stress (σy/q) at x = 0 and y = b/2 Ply3 bottom................................233
Figure A-26: Stress (τxz/q) at x = 0 and y = b/2 Ply1 bottom ..............................233
Figure A-27: Stress (τxz/q) at x = 0 and y = b/2 Ply2 top ....................................234
Figure A-28: Stress (τxz/q) at x = 0 and y = b/2 Ply2 bottom ..............................234
Figure A-29: Stress (τxz/q) at x = 0 and y = b/2 Ply3 top ....................................235
List of Tables
8
List of Tables Table 2-1: Material properties ...............................................................................34
Table 5-1: Displacement and stresses distribution of fully clamped laminate
when h/a = 0.2 ....................................................................................................142
Table 5-2: Displacement and stresses distribution of fully clamped laminate
when h/a = 0.4 ....................................................................................................148
Table 5-3: Displacement and stresses distribution of fully clamped laminate
when h/a = 0.6 ....................................................................................................154
Table 5-4: Exact solution versus FEM for plate subjected to hydrostatic loading162
Table 5-5: The exact solutions and FEM of Case 1 ............................................163
Table 5-6: The exact solutions and FEM of Case 2 ............................................164
Table 5-7: The exact solutions and FEM of Case 3 ............................................165
Table 6-1: The evolution of the damage variable for compression and tension ..173
Table 6-2: Basic material properties used in FEM ..............................................175
Table 6-3: Deflection distribution (WC11/qh) for h/a = 0.1 ...................................195
Table 6-4: Deflection distribution (WC11/qh) for h/a = 0.2 ...................................195
Table 6-5: Deflection distribution (WC11/qh) for h/a = 0.3 ...................................196
Table 6-6: Deflection distribution (WC11/qh) for h/a = 0.4 ...................................196
Table 6-7: Deflection distribution (WC11/qh) for h/a = 0.6 ...................................197
Table A-1: Maximum deflection at x = a/2, y = b/2 and z = 0 (W C11/ qh) ...........210
Table A-2: Stress (σx/q) at x = a/2, y = b/2 at Ply1 top surface ...........................210
Table A-3: Stress (σx/q) at x = a/2, y = b/2 at Ply1 bottom surface .....................210
Table A-4: Stress (σx/q) at x = a/2, y = b/2 at Ply2 top surface ...........................211
Table A-5: Stress (σx/q) at x = a/2, y = b/2 at Ply2 bottom surface .....................211
Table A-6: Stress (σx/q) at x = a/2, y = b/2 at Ply3 top surface ...........................211
Table A-7: Stress (σx/q) at x = a/2, y = b/2 at Ply3 bottom surface .....................212
Table A-8: Stress (σy/q) at x = a/2, y = b/2 at Ply1 top surface ...........................212
Table A-9: Stress (σy/q) at x = a/2, y = b/2 at Ply1 bottom surface .....................212
Table A-10: Stress (σy/q) at x = a/2, y = b/2 at Ply2 top surface .........................213
Table A-11: Stress (σy/q) at x = a/2, y = b/2 at Ply2 bottom surface ..................213
Table A-12: Stress (σy/q) at x = a/2, y = b/2 at Ply3 top surface .........................213
List of Tables
9
Table A-13: Stress (σy/q) at x = a/2, y = b/2 at Ply3 bottom surface ...................214
Table A-14: Stress (σx/q) at x = 0, y = b/2 at Ply1 top surface ............................214
Table A-15: Stress (σx/q) at x = 0, y = b/2 at Ply1 bottom surface ......................214
Table A-16: Stress (σx/q) at x = 0, y = b/2 at Ply2 top surface ............................215
Table A-17: Stress (σx/q) at x = 0, y = b/2 at Ply2 bottom surface ......................215
Table A-18: Stress (σx/q) at x = 0, y = b/2 at Ply3 top surface ............................215
Table A-19: Stress (σx/q) at x = 0, y = b/2 at Ply3 bottom surface ......................216
Table A-20: Stress (σy/q) at x = 0, y = b/2 at Ply1 top surface ............................216
Table A-21: Stress (σy/q) at x = 0, y = b/2 at Ply1 bottom surface ......................216
Table A-22: Stress (σy/q) at x = 0, y = b/2 at Ply2 top surface ............................217
Table A-23: Stress (σy/q) at x = 0, y = b/2 at Ply2 bottom surface ......................217
Table A-24: Stress (σy/q) at x = 0, y = b/2 at Ply3 top surface ............................217
Table A-25: Stress (σy/q) at x = 0, y = b/2 at Ply3 bottom surface ......................218
Table A-26: Stress (τxz/q) at x = 0, y = b/2 at Ply1 top surface...........................218
Table A-27: Stress (τxz/q) at x = 0, y = b/2 at Ply1 bottom surface.....................218
Table A-28: Stress (τxz/q) at x = 0, y = b/2 at Ply2 top surface...........................219
Table A-29: Stress (τxz/q) at x = 0, y = b/2 at Ply2 bottom surface.....................219
Table A-30: Stress (τxz/q) at x = 0, y = b/2 at Ply3 top surface...........................219
Table A-31: Stress (τxz/q) at x = 0, y = b/2 at Ply3 bottom surface.....................220
Table A-32: Stress (τxy/q) at x = 0, y = b/2 across the thickness........................220
Abstract
10
Abstract
Name of the University: The University of Manchester Candidate’s full name: Haji Elmi Haji Kamis Degree Title: Doctor of Philosophy (PhD) Thesis Title: Three Dimensional Analysis of Fibre Reinforced Polymer Laminated Composites Date: 08th July 2012 The thesis presents the structural behaviour of fibre reinforced polymer (FRP) laminated composites based on 3D elasticity formulation and finite element modeling using Abaqus. This investigation into the performance of the laminate included subjecting it to various parameters i.e. different boundary conditions, material properties and loading conditions to examine the structural responses of deformation and stress. Both analytical and numerical investigations were performed to determine the stress and displacement distributions at any point of the laminates. Other investigative work undertaken in this study includes the numerical analysis of the effect of flexural deformation of the FRP strengthened RC slab. The formulation of 3D elasticity and enforced boundary conditions were applied to establish the state equation of the laminated composites. Transfer matrix and recursive solutions were then used to produce analytical solutions which satisfied all the boundary conditions throughout all the layers of the composites. These analytical solutions were then compared with numerical analysis through one of the commercial finite element analysis programs, Abaqus. Out of wide variety of element types available in the Abaqus element library, shells and solids elements are chosen to model the composites. From these FEM results, comparison can be made to the solution obtained from the analytical. The novel work and results presented in this thesis are the analysis of fully clamped laminated composite plates. The breakthrough results of fully clamped laminated composite plate can be used as a benchmark for further investigation. These analytical solutions were verified with FEM solutions which showed that only the solid element (C3D20) exhibited close results to the exact solutions. However, FEM gave poor results on the transverse shear stresses particularly at the boundary edges. As an application of the work above, it is noticed that the FEM results for the FRP strengthened RC slab, agreed well with the experimental work conducted in the laboratory. The flexural capacity of the RC slab showed significant increase, both at service and ultimate limit states, after FRP sheets were applied at the bottom surface of the slab. Given the established and developed programming codes, exact solutions of deflection and stresses can be determined for any reduced material properties, boundary and loading conditions, using Mathematica.
Declaration
11
Declaration
No portion of the work referred to in this thesis has been submitted in support of
an application for another degree of qualification of this or any other university,
or other institution of learning.
Copyright Statement
12
Copyright Statement
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thesis) owns certain copyright or related right in it (the “Copyright”) and s/he has
given The University of Manchester certain rights to use such Copyright,
including for administrative purposes.
Copies of this thesis, either in full or in extracts and whether in hard or
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intellectual property (the “Intellectual property”) and any reproductions of
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which may be described in this thesis, may not be owned by the author and
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cannot and must not be made available for use without the prior written
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Library, the University Library’s regulations (see
http:www.manchester.ac.uk/library/aboutus/regulations) and in The University’s
policy on presentation of Theses.
Acknowledgments
13
Acknowledgements
Firstly, I am grateful to Allah S.W.T. for His continuous blessings and devotions
such that I could not see, hear nor sense anything other than His sole presence
guiding me throughout my journey.
My first academic year had not been a smooth sailing start for me unlike for
most students. Due to unforeseen circumstances, I had to commute to Cardiff
three days a week after spending the day work at the University. Having to
travel in chilled evenings by train, I continued my research work on my laptop
during the three and a half hours journeys. Cardiff University was where my wife
pursued her Master degree and where my family lived. On the following
mornings while my wife attended her lectures I had to sent my two young
children to school and took care of my one year old son at home. Such three
days a week tasks became a routine throughout my wife’s one and a half year
Master Degree course.
From this difficult and challenging experience, I had learnt on how tough it was
to pursue PhD study especially for those who are with families. I had
understood that the ability to juggle with time between family and study surely
needed sheer patience. However, with strong will and determination, I managed
to complete my study after significant amount of time and efforts were sacrificed.
Knowing the limited time I had left to complete my study after my family
returned home during my third year of study, I had pushed myself to the limits,
going through ups and downs of mixed emotions, until at some points I had to
stop spending the night work at the University where it had jeopardised my
health condition.
Nevertheless, I would like to send my deepest gratitude to my family especially
my beloved wife, Suzan for her strengths and patience for taking care of all our
children while I was away in Manchester. To all my lovely sons, Md Naim Bazli,
Md Nabil Wafie, Md Muazzam and Khairul Harisin, letting them grown up
Acknowledgments
14
without my presence and missing the full opportunities to observe every single
of their actions was heart- breaking. I really miss and love you all. My
appreciation also goes to my parents whom continuously give support and pray
for my success.
Special thank also goes to The Brunei Government whom provided me the
great opportunities to pursue my study by awarding me the sponsorship for the
third time.
Last but certainly not the least, I wish to thank Dr Jack Wu, my supervisor for
his assistance throughout my study. I can not also forget to thank all my friends,
Dr Ahmad Abdullah, Dr Hisham, Dr Ashfaq Khan, Dr Rao Krishnamoorthy, Dr
Sabri Jamil, Dr Yakub, Dr Febian, Dr Maturose Suchatawat, Dr Ramadan
Eghlio, Amin, Ahmad Sabri, Abdul Rahman, Moustafa, Aftab, Israr, Kwan Sete,
Noorhafiza, Sutham, Salwan, Michael, Barbie, Chao Han, and Dominic for all
their help and support throughout my study.
One of the sweet memories that I gained in Manchester was when my team (My
B team comprises of Dr Sabri, Dr Nasrun, Aizul, Abdul Latip and myself) won
the Badminton competition organised by North West Tri-Badminton Series 2012
at Belle Vue, Longsight, Manchester last January.
After going through all the experiences and mixed emotions, I am pretty sure
that I will miss Manchester especially the life as a student once I return back
home and get back to work. I really hope one day I can have the chance to
come and visit Manchester again, the birth place of my youngest son, Khairul
Harisin.
Chapter 1 Introduction
15
Chapter 1 Introduction
1.1 Introduction
Fibre reinforced polymer (FRP) composite materials have been widely used in
construction for buildings and bridges since 1980’s [1]. They have been used in
structural engineering for new construction and for strengthening and repair
purposes of existing building and bridge structures. FRP materials come in a
variety of shapes and sizes, ranging from internal reinforcing bars for concrete
members and sheets or strips for external strengthening of concrete or other
structures. The use of FRP applications has also been beneficial in other
industries including aerospace, manufacturing, automotive, biomedical,
infrastructure, sports equipments and others [2].
Typically, the use of FRP materials in civil engineering are ready made and
supplied with relevant dimensions and specifications to meet the desired design
requirements. In other words, FRPs material can be tailored according to the
designer’s or engineer’s requirement.
One typical type of composite laminae which consists of a single layer of FRP is
known as uni-directional (UD) fibre lamina as shown in Figure 1.1. This type of
lamina is commonly flat shape with the fibres being arranged parallel along their
principal material axes. These fibres are the main load carrying medium and
they are hold together by matrix. By combining the fibres and matrix together,
they make the UD lamina become strong and stiff. This type of UD lamina is
used in this study as the main type of FRP.
Furthermore, if there is a number of lamina being stacked and bonded together
with various fibre orientations in each lamina, they become a laminate as shown
in Figure 1.2. By forming this laminate, the material properties of each lamina
can be varied and specified to the designer’s requirement.
Chapter 1 Introduction
16
Figure 1.1: Layout of a UD lamina [3]
Figure 1.2: View of unstacked laminate [3]
Chapter 1 Introduction
17
The major purpose of forming the lamination is to meet the required strength
and stiffness of the composite material. Due to such great significant
advantages of laminate’s inherent property over the conventional metal, it has
inspired the author to investigate the structural response of composite
laminated plate by subjecting it to various parameters including boundary,
loading conditions and material properties.
Reinforced concrete (RC) slab is also another type of composite material in
such a way that steel reinforcement bars are the fibres and concrete is the
matrix. These materials are also being investigated in this study as the main
material properties for application purposes. Using the available experimental
test results of FRP strengthened RC slab, numerical analysis of the same
specimens are also reviewed using one of the finite element method (FEM),
Abaqus. For the analytical part, composite laminated plates are investigated
using classical plate theory and compared with the three dimensional analysis
of elasticity with state space method.
1.2 Research Aim and Objectives
The aim of this study is to investigate the structural performance such as
stresses and displacements of composite laminated plates subjected to various
parameters including loading, boundary conditions and material properties. Both
analytical and numerical analyses were carried out to examine the responses of
the structures. The analytical study has focused on the application of three
dimensional elasticity and state space method whereas numerical method of
investigation was carried out based on FEM analysis.
In order to achieve this aim, the following objectives of this research studies
have been identified:
• To investigate the structural response of laminated composite plate by
analytical method based on the 3D elasticity and state space method.
• The use of several different FEM element types to find a suitable element
for comparison with the analytical method.
Chapter 1 Introduction
18
• To carry out further case studies of laminated plate by changing various
parameters including material properties, boundary and loading
conditions to reflect the true response of composite laminated plate.
• To review the effect of flexural deformation of RC slab with and without
FRP. This task is performed by modeling and simulating the specimen’s
deflection, numerically. The available experimental test results are used
and compared with the numerical solutions by Abaqus/CAE.
• To realise the analytical solutions for the structural response of any
reduced material properties, boundary and loading conditions by
developing program codes with Mathematica.
• To compare the results of slab’s deflection between classical plate theory
and the exact solution of three dimensional elasticity approach.
1.3 Outline of Thesis
Thesis content can be briefly summarized as follows,
Chapter 1 gives the introduction of the FRP composites material. The definition
and layout view of UD laminate has been outlined and shown, respectively. The
function of laminate has also been briefly described in this chapter. The other
main content on this chapter is to state the research’s aim and objectives in
which it describes how and what tasks will be involved during the investigation.
Chapter 2 briefs the literature review of plate theory and it’s development from
two to three dimensional analyses. Some theoretical formulations are also
shown in this chapter to calculate the deflection of a simply supported plate
subjected to various loading conditions. These fundamental formulation of
deflection are based on the assumption of the Kirchhoff-Love theory. The uses
of FRP towards strengthening of structures have also been reviewed. This
chapter also summaries on why the novelty of work presented in this thesis are
carried out in connection to the available literature.
Chapter 1 Introduction
19
Chapter 3 deals with the concept of state space method for plate structures and
the governing equations of elasticity. A classical (continuum) or analytical
method based on the application of the equations of equilibrium and kinematics,
together with the stress and strain relations for the material are applied to
produce governing equations which should be solved to obtain displacements
and stresses. From these governing equations in state space, further
development into various applications having different boundary and loading
conditions can be explored. It is importance to understand thoroughly on this
chapter since the concept is extended to the following chapter.
Chapter 4 shows the application of state space method for clamped plates with
various loading and boundary conditions. It is clearly showed that the concept of
this clamped edges plate is applied from the simply supported plate. The exact
solution approach consists of determining the displacements of rectangular
plate by setting a general expression of the displacement field according to the
available boundary and loading conditions. State space methods together with
state transfer matrix are presented with the aid of programming code, in this
chapter. To understand the process of transfer matrix involved, initially only a
single layer of plate considered, and then followed by a plate consisting of many
sub-layers. The derivations of expressions are also shown in this chapter for
clarification purpose.
Chapter 5 demonstrates the use of numerical solutions by means of finite
element method in order to compare with the analytical works based on the 3D
elasticity and state space method. The objective of this method is to simulate
the composite plate behavior in such a way that the model (a continuum) is
discretized into simple geometric shapes which can be used to perform some
other parametric studies. FEM is an alternative approach to solving the
governing equations of a complicated structural problem particularly beyond the
elastic phase. The other important results presented here are the numerical
solutions of FEM verified the exact solutions of laminated plate subjected to
different loading conditions and material properties. The novelty solutions and
comparison of fully clamped laminated plate are presented in this chapter.
Chapter 1 Introduction
20
Chapter 6 illustrates the work undertaken during the first and second year of
PhD study. The deflection of RC slab strengthened with FRP using FEM is
reviewed. Modelling of FRP strengthened RC slab by Abaqus is explained in
this chapter. Numerical flexural deformation performance can be compared
against the behaviour of composition materials of the structure. The accuracy of
the modelling of deflection of the structures is verified with the available
experimental test results obtained from the full scale FRP strengthened RC
slabs tested in the University of Manchester two years ago. The results of
plate’s deflection are also compared between classical plate theory and the
exact solution based on the material properties of concrete and FRP lamina.
Chapter 7 outlines the conclusion and future work recommendations. This
chapter also summaries the ideal method of analysis that is to be used in the
future investigation of laminated plate.
Chapter 2 Literature Review
21
Chapter 2
LITERATURE REVIEW
2.1 Plate Theories
For the analysis of plate, classical finite difference and finite element methods, a
starting point of historical development of the theory and analysis of laminated
plates, are primarily based on the theory of thin plate, known as Kirchhoff-Love
plate theory. It is a two-dimensional theory developed by Kirchhoff-Love [4] from
the extension of the hypothesis of Euler-Bernoulli beam. Such a theory is
established on a few assumptions which neglect some parameters for the
analysis, namely, transverse shear deformations and rotatory inertia. Because
of the assumptions, this theory is inaccurate and will be unable to give a
solution precisely. More errors for the analysis of thick plate are expected,
particularly for the analysis of anisotropic laminated plate structures.
Classical plate bending theory makes errors when the ratio of the elastic
modulus to shear modulus becomes large. For instance, graphite epoxy and
boron epoxy have these modulus ratios of about 25 and 45, respectively against
ratio of 2.6 for isotropic materials [5]. Composite plate can leads to a
complicated coupling effect and significant changes of stresses in magnitude at
the interface of the laminate. As a result, classical plate theory is not accurate
for the analysis of anisotropic composite plates.
The assumptions that are made in this theory [6]:
• In-plane deformations in the x and y directions at the mid surface are
zero.
• straight lines normal to the mid-surface remain normal to the mid-surface
after deformation
Chapter 2 Literature Review
22
• the stress σz is negligible compared with other stresses in the transverse
cross section, i.e. σx, σy and τxy as shown in Figure 2.1.
where σx,σy and σz are the in-plane and transverse stresses while τxy is the
shear stress as shown in Figure 2.1
These assumptions give approximate solutions and the following conditions.
Figure 2.1: Normal and shear stresses [6]
The first assumption implies that at the mid surface, the in-plane displacements
U(x,y,0) = 0 and V(x,y,0) = 0.
Second and third assumptions have lead transverse shear strains
yzxzγandγ and the normal strain,
zzε to be disappeared, i.e. [6]
0
0
0
=∂
∂=
=∂∂+
∂∂=
=∂∂+
∂∂=
z
Wε
z
V
y
Wγ
z
U
x
Wγ
zz
yz
xz
(2-1)
The remaining strains to be considered are xyyyxx
γandε,ε such that [6]
z(W)
x(U)
y(V)
h
σz
σx
σy
τxy
Chapter 2 Literature Review
23
yx
Wz
yx
Wz
xy
Wz
x
V
y
Uγ
y
Wz
y
Vε
x
Wz
x
Uε
xy
yy
xx
∂∂∂−=
∂∂∂−
∂∂∂−=
∂∂+
∂∂=
∂∂−=
∂∂=
∂∂−=
∂∂=
222
2
2
2
2
2
(2-2)
The relationship of stress and strain is presented as follows:
For isotropic [6]:
−−=
xy
yy
xx
xy
yy
xx E
γεε
µµ
µ
µτσσ
2
100
01
01
1 2 and
( )µ+=
12
EG (2-3)
where
E is the Elastic modulus
G is the shear modulus
µ is the Poisson ratio
The above formulation is said to be true assuming the plate thickness is
relatively thin compared to its other dimensions. The typical ratio of thickness to
the plate plane dimension is less than 0.1, i.e. 1.0<a
h. Further formulations can
be obtained from the above equations including:
Strains from eqn.(2.2),
∂∂∂−
∂∂−
∂∂−
=
yx
W
y
Wx
W
z
γ
ε
ε
xy
yy
xx
2
2
2
2
2
2
(2-4)
Chapter 2 Literature Review
24
Moments [7],
∂∂∂−
∂∂−
∂∂−
−=
=
∫−
yx
W
y
Wx
W
µ
µ
µ
Ddzz
τ
σ
σ
M
M
M h
h
xy
yy
xx
xy
yy
xx
2
2
2
2
2
2
2
22
100
01
01
where ( )2
3
112 µ−= Eh
D
(2-5)
D is also known as the flexural rigidity of the plate [7].
Shears [6],
∂∂∂+
∂∂−=
2
3
3
3
yx
W
x
WDQ
x and
∂∂∂+
∂∂−=
2
3
3
3
xy
W
y
WDQ
y (2-6)
Based on the Love-Kirchhoff assumptions and the equations of equilibrium at a
point for a plate, the following governing equations can be deduced [6]:
WD
q
y
W
yx
W
x
W 44
4
22
42
4
4∇==
∂
∂+
∂∂
∂+
∂
∂ (2-7)
The equation (2-7) is well known as bi-harmonic equation. For a specified
loading, q, and specified boundary conditions, the solution of the plate problem
is reduced to finding a deflection, W. Subsequently, the values of bending
moments and shears can be calculated. The following case is illustrated to
determine deflection of plate.
2.1.1 Rectangular Kirchhoff-Love plates
Plate bending refers to the small deflection of a plate out of its original plane
under the action of external transverse loading. The plate has relatively smaller
thickness than its other dimensions (less than 0.1 of it’s width). Classical plate
theory can be applied to the plate where shear strains are neglected across the
thickness. Only in-plane direct and shear stresses are considered.
Chapter 2 Literature Review
25
For a simply supported rectangular plate subjected to an applied loading as
shown in Figure 2.2, Navier [6] established the displacement and stress
expressed in terms of Fourier expansion.
Figure 2.2: A simply supported rectangular plate un der sinusoidal loading
For a single distributed lateral load of the form
( )b
yπnSin
a
xπmSinqy,xq
o= (2-8)
where qo is the constant (with the dimension of pressure), substitute
b
yπnSin
a
xπmSinWW
o= into equation (2-7) and verifying the boundary
conditions w = 0 and x = 0 and a; w = 0 at y = 0 and b, are satisfied, the
deflection can be determined as
b
yπnSin
a
xπmSin
b
n
a
mDπ
qW o
2
2
2
2
24
+
= , where 2
2
2
2
24
+
=
b
n
a
mDπ
qW o
o
z
x
y
a
b
h
q
Chapter 2 Literature Review
26
Therefore, the general expression of displacement can be deduced as follows
[6]:
For a plate subjected to uniform loading, the deflection is given by
( )∞=
+
= ∑∑∞
=
∞
=......,.........,,n,m
b
n
a
mmn
b
yπnSin
a
xπmSin
Dπ
qW
m n
o 53116
1 12
2
2
2
26 (2-9)
For the case of plate subjected to concentrated load at the centre, its deflection
is given by
b
yπnSin
a
xπmSin
b
n
a
m
b
πηnSin
a
πξmSin
abDπ
PW
m n∑∑∞
=
∞
=
+
=1 1
2
2
2
2
24
4. (2-10)
where ;2
;2
by
ax ==== ηξ if the load is applied at the middle.
Further expression of theoretical formulations for various loading and boundary
conditions can be referred to Timoshenko [6].
It is important to note that classical laminate theory provides a simple and direct
method to determine stresses and strains. However, it is not very accurate as it
does not satisfy the equations of elasticity at every point of laminated plate. It
also ignores shear deformations of layers because of the assumed bond
between two laminae which are non-shear deformable. During loading, shear
stresses are developed at the interfaces. The transverse stresses
yzxzzand ττσ , are negligible in the regions away from the plate edges.
Therefore, laminate theory is only sufficient in the regions away from the plate
boundary. For the case of regions near the boundary, however, a plane stress
is no longer true, a 3D stress state would become more appropriate.
Chapter 2 Literature Review
27
2.1.2 Development of plate theories
More rigorous studies have been performed to improve such theory without
neglecting the transverse shear deformations and rotatory inertia. The study
that relates governing equation of plates and incorporates the effect of shear
was first established by Reissner [8]. The assumptions made by Reissner gave
a linear bending stress and a parabolic shear stress distribution through the
thickness of the plate. Since Reissner’s work, there have been a number of
refinement and further generalization beyond the classical plate theory. This
includes further improvement by Mindlin [9] which allowed shear strains to occur.
His theory assumed that there was a linear variation of displacements (U and V)
across the plate thickness, however, the deflection through the thickness did not
change during the loading, i.e. W had no relation with the thickness direction (z-
coordinate). This leads to the normal stress through the thickness is being
ignored, which is similar situation to the plane stress condition. Both Mindlin and
Reissner theories however have some similarities in such ways that their works
are based on the extension of Kirchhoff-Love plate theory and taken into
account of shear deformations through the thickness of a plate where the
normal to the mid-surface remains straight but not necessarily perpendicular to
the mid surface and rotatory inertia. Both theories satisfy the three boundary
conditions on the edge but do not satisfy the differential governing equations of
three dimensional elasticity. The form of Mindlin-Reissner plate theory also
includes in-plane shear strains and is often known as a first-order shear
deformation plate theory with a linear displacement variation through the
thickness. More details about the relationship between Reissner’s and Mindlin’s
theories are presented by Wang et al. [10]
In reality, the transverse shear strains cannot be constant through the thickness
of the plate. They are frequently, but not always, zero at the top and bottom
surfaces of the plate and they are generally undergo in parabolic form across
the thickness. Therefore, Mindlin theory represents average transverse shears
through the thickness.
Chapter 2 Literature Review
28
Ambartsumyam’s theory [11] improved the mid-thick plate theories by
introducing a quadratic function to represent the variation of shear stresses
yzxzand ττ (or the corresponding deformations
yzxzand γγ ) across the
thickness.
Some related studies of interlaminar stress distributions of laminate which led to
important development on this field were reported by Pagano and Halpin [12],
Whitney and Leissa [13] and Whitney [14]. However, the first direct approach to
the problem was made by Puppo and Evenson [15] who derived an
approximate formulation of the laminate made of surface layers and an isotropic
shear layer. The anisotropic layers were assumed to carry only in-plane loads
hence resulting plane stress whereas the isotropic shear layers were assumed
to carry interlaminar shear stress.
The experimental work of Pipes and Daniels [16] confirmed the theoretical
results of Pipes and Pagano [17] by using the Moire technique. The Moire effect
used the arrays of images viewed when lights were transmitted to the specimen.
The observed images known as the Moire patterns, revealed the axial
displacement and strain fields of the laminate subjected to axial loading. The
results obtained from the test were accurately shown as compared to the
theoretical solutions.
Pagano and Pipes [18] extended their work on laminated plate to study the
effect of arranging the orientation direction of the laminate across the thickness
to provide optimum laminate strength. However, the results obtained were
merely approximate particularly at interlaminar stresses and at the boundary
edges.
Apart from classical solutions, numerical techniques had been developed for
comparison. This includes the work of Rybicki [19] on 3D FEM for the analysis
of finite width laminate which confirmed the sign changes of σz, as per Pagano
work.
Chapter 2 Literature Review
29
Many researchers have continued to develop the theories showing that the
deflection W has no relationship with z-coordinate. Some works however have
considered the relation of deflection w and z coordinate by higher order theories
such as Lo et al [20]. However, these refined theories are still insufficient and do
not meet all governing equations of three dimensional elasticity. These theories
produce some form of approximation but not exact solutions.
Further investigations have been performed to study the bending, vibration and
buckling analyses of laminated rectangular plates associated with the three
dimensional elasticity.
The work of Pagano [21] on a cylindrical bending of a simply supported
orthotropic strip under sinusoidal transverse load showed that classical
laminated plate theory based on Kirchhoff-Love theory underestimated the plate
deflection and gave a very poor result as the span to depth ratio of plate
reduced.
Pagano [22] extended his study of three-dimensional elasticity solutions for the
case of rectangular laminates composed of orthotropic layers with pinned edges
under static loading. He concluded that the accuracy of classical thin plate
theory for a particular problem depended on the material properties, lamination
orientation and span to depth ratios.
Srinivas and Rao [23] have also established an exact analysis of three
dimensional elasticity theory solutions for the analysis of simply supported thick
orthotropic rectangular laminates subjected to sinusoidal normal pressures.
From the result obtained, they showed that as the thickness of the plate
increased, the stress and displacement distributions across thickness became
more complex and great computation process was required.
Bahar [24] presented a fundamental formulation for associated classical 2-D
elasticity by use of the transfer matrix approach in state space.
Chapter 2 Literature Review
30
Based on the same approach, Iyengar and Pandya [25] combined it with the
method of initial functions (MIF) which was initially proposed by Vlasov [26] to
formulate the general solutions of orthotropic rectangular thick plates in three
dimensional elasticity. The result proved to be a kind of higher order theory.
They formed equations using a Taylor series expansion with respect to z-
coordinate. The main criticism, if any, concentrates mainly on the error in this
method by an expansion which cannot be avoided.
For orthotropic, transversely isotropic and isotropic 3-D simply supported elastic
plates, Wu [27] has developed the methods mentioned above and gave the
various close-form solutions explicitly, for the first time in this subject.
Fan and Ye [28-29] extended the work of Wu [27] and Srinivas and Rao [23] to
a three ply orthotropic thick plate. The subsequent equations were expressed in
terms of the initial value when z = 0 (top surface). A compatibility equation was
used at each interface of the plies, resulting from the continuity of the interface
displacements and stresses. A set of algebraic equations were then established
following the successive formulation from top layer until the bottom layer of the
laminated plate.
Fan and Sheng [30] have further presented the state equation for thick laminate
with clamped edges by introducing delta function to establish appropriate
boundary conditions along the edges of a plate. The results presented in their
study were incomplete, i.e. the exact solutions for a partially clamped laminated
plate.
Rogers et al. [31] presented general expressions for the deformation and stress
distribution for some particular cases such as an elliptical plate with moderate
thickness, a semi-infinite strip clamped along its two edges etc.
Agbossou and Mougin [32] investigated the static and dynamic analysis of
rectangular reinforced concrete slabs based on the laminated theory. Non linear
behaviour of simply supported slabs were designed and analysed from the
basis of laminated theory by changing the neutral axis position and steel and
Chapter 2 Literature Review
31
concrete effective modulus when concrete slab started to crack. The objective
of the work is to analyse and design the slab to resist impact loading from rock
falls. After reaching the ultimate tensile strength of concrete and when the steel
reinforcement begins to resist further tensile loading, bending stiffness matrix
has been modified to take into account of the effect of the damaged concrete
and plasticity of steel bars within the laminate. The results provide good
relationship between analytical, FEM and experimental tests.
Recent work related to the application of state space equation to clamped thick
laminated plates included that of Sheng et al. [33]. Analytical solution was
presented for the analysis of laminated piezoelectric plate with clamped and
electric open-circuited boundary conditions. These results were also verified
with numerical outcomes from FEM.
Furthermore, Li et al. [34] used finite integral transform method to establish an
exact bending solutions for fully clamped orthotropic rectangular thin plates
subjected to arbitrary loading. The mathematical method did not require any
displacement functions to satisfy the governing equations of 2D elasticity and
the boundary conditions. Further works need to be done are to validate the
formulations including the case of 3D orthotropic plate on various boundary
conditions.
2.2 FRP
FRP offers an effective, sustainable method of structural strengthening and
rehabilitation. FRP composites consist of fibres of high tensile strength and high
modulus within a polymer matrix. The two or more materials combine together
to produce desirable properties that cannot be achieved with any single
constituent acting alone. The bonding of these aligned fibres into the matrix
material results in a fibre reinforced composite material with superior properties
in the fibre direction. [35]
Due to the fact that fibres are highly directional, the resultant composite will
exhibit anisotropic behaviour. One common example of such behaviour is steel
Chapter 2 Literature Review
32
reinforced concrete members. The steel bars are the fibres, and the concrete
material is the matrix that keeps the fibres together. In this application, when
structural members are subjected to a load, fibres act as the principal load
carrying members and the concrete provides a load transfer medium between
fibres and protect fibres from being exposed to the environment such as
humidity.
Typical FRP composite material properties include low specific gravity, high
strength to weight ratio and a high modulus to weight ratio. Most FRP materials
require minimum protection and are very resistant to corrosion. Generally, FRP
materials behave in a linear elastic stress strain curve until failure under tension.
Brittle failure is the common mode of failure of FRP under excessive stress.
2.2.1 Advantages of FRP
In the case of repair of concrete structures, the use of fibre reinforced polymer
composites has the following advantages over conventional materials:
• Very light weight
• Superior toughness
• Low thermal conductivity (ability to conduct heat)
• Durability
• Ease of installation and transportation
• Can be wrapped over curved surfaces such as columns
The obvious advantages of using composite materials such as greater strength
and stiffness combined with lightness and durability are the reasons why they
are commonly used in a wide variety of applications.
2.2.2 Disadvantages of FRP
The significant disadvantage of externally strengthening structures with FRP is
the risk of fire, vandalism or accidental damage especially if they are not
protected. Poor workmanship during the installation of the FRP can lead to
unfavourable results. The performance of such a strengthening system can be
Chapter 2 Literature Review
33
reduced if the adhesive layer does not achieve the desired quality. Another
obvious disadvantage of using FRP for strengthening purposes is the relatively
expensive cost of the material. However, this is balanced out given the
significant advantages FRP has over other systems such as steel plate. For
instance, FRP can be installed speedily without delaying and disrupting to the
user of the structure.
2.2.3 Constitutive Material Properties of FRP
Composite materials are often both non homogenous and non isotropic
(orthotropic or anisotropic).
A non homogenous body has non uniform properties over the body, i.e. the
properties depend on position in the body.
An orthotropic body has material properties that are different in three mutually
perpendicular directions at a point in the body and has three mutually
perpendicular planes of material property symmetry. Thus, the properties
depend on orientation at a point in the body.
An anisotropic body has material properties that are different in all directions at
a point in the body. No plane’s symmetry of material property exists. The
properties depend on orientation at a point in the body.
Composite materials have many mechanical behaviour characteristics that are
different from other typical engineering materials such as steel, aluminum and
other metals [36]. The desired properties of FRPs are achieved by the favorable
characteristics of the two major constituents, namely the fibre and the matrix.
Properties of Fibre
A fibre is characterised generally by its very high length-to-diameter ratio having
high strength and stiffness properties with low density when compared to
common materials such as aluminium, titanium, steel and others. Recall that,
Chapter 2 Literature Review
34
fibres must have a surrounding matrix to achieve desired structural performance.
Typically, fibres comprise of 60 – 70% (by volume) of the composite.
Table 2-1 shows some materials properties arranged in increasing values of
ρS
and ρE
, where S is the tensile strength, E is the tensile stiffness and ρ is
the density. Fibres tend to have low transverse strength however they primarily
act in tension. Various form of fibres finish are manufactured and readily
available in the market including in ‘bundles’ called tows or rovings which can
be then further processed into tow sheets, fabrics or mats. In some types of
composite, the fibres are oriented randomly within a plane, while in others the
material is made up of a stack of differently-oriented “plies” to form a laminate,
each ply containing an aligned set of parallel fibres.
Table 2-1: Material properties [3]
FIBRE or
WIRE
Density, ρ
(kN/m3)
Tensile
Strength, S
(GPa)
ρS
(km)
Tensile
Stiffness,
E
(GPa)
ρE
(Mm or
x 106 m)
Aluminium 26.3 0.62 24 73 2.8
Titanium 46.1 1.9 41 115 2.5
Steel 76.6 0.5 * 6.5 * 207 2.7
E-glass 25.0 2.0 * 80 * 72 2.9
S-glass 24.4 4.8 197 86 3.5
Carbon 13.8 1.7 123 190 14
Beryllium 18.2 0.6 * 33 * 300 16
Boron 25.2 3.4 135 400 16
Graphite 13.8 1.7 123 250 18
(Note *: the typical values of tensile strength and specific tensile strength ( )ρ
S
of the material rather than as stated in the reference).
Chapter 2 Literature Review
35
For advanced FRP used in civil engineering, the fibres are generally made of
carbon or glass in a polymer matrix such as vinylester or epoxy. From the
above Table 2-1, it shows that Graphite and S-glass have the higher stiffness
and strength to density ratios, respectively. For these reasons, these materials
are prominent type of materials used in today’s composite structures.
Note that the meanings of some basic mechanical properties of any material
such as the following:
Stiffness - The ability of material to resist elastic deformation. It is characterized
by the Young's modulus.
Strength - A measure of a material's resistance to failure. It depends of details
on how it is measured, specimen geometry and, for brittle materials, on the
presence of flaws.
Toughness – The ability of a material to absorb energy and plastically deform
without fracturing.
Properties of Matrix
By nature, matrix materials are at least an order of magnitude weaker than the
reinforcing embedded fibres. All matrix materials exhibit significant magnitude of
creep and have large coefficient of thermal expansion compared to traditional
construction materials. However, composite laminates could not exist without
matrix materials. In fact, the roles of a matrix are to support and protect the
fibres and transfer stress between broken fibres through shear. The matrix of a
composite acts as a binder that bond fibres together.
The most common type of matrix used in structural strengthening purposes is
polymers which consist of rubbers, thermoplastics and thermosets.
Typical examples of thermoplastics are nylon, polyethylene and polysulfone.
Epoxies, phenolics and polymides are common examples of thermosets.
Chapter 2 Literature Review
36
The choices of composition and of the materials used as matrix and fibre are
dependent on the required properties. This can be deduced by deriving a merit
index for the performance required followed by the use of Ashby property maps
as shown in Figure 2.3.
Figure 2.3: An Ashby property map for composites [ 37]
2.3 FRP strengthening RC structures
In civil engineering, a growing level of activity in the repair and rehabilitation of
structures has been certainly acknowledged. Over many years, scientific
investigation of decay and deterioration of historical buildings in general and
concrete structures in particular has contributed significant results to gain
understanding of structural defects and finding ways to improve its durability. As
a consequence, service life assessment and rehabilitation are now widely
recognised as important issues.
Building or construction is considered to be among the earliest human activities.
To build and own a house has been essential to human needs and survival for
thousand of years. For any house constructed with building materials such as
concrete, timber, bricks, stones, mortars, plaster and steel, deterioration has
required the process of repairing and rebuilding to be an ongoing activity.
Chapter 2 Literature Review
37
The load bearing capacity and the durability of all structures, ranging from
bridges to buildings, are susceptible to deterioration can either locally or globally,
by various mechanisms. The strength of the structures that are severely
exposed to environmental factors such as weather, aggressive chemical attack,
changes of usage or imposed loading etc can be adversely impacted. In the
21st century, concern about the decay and deterioration of buildings has grown
in tandem with the impact of industrial activities on the environment. Increasing
air pollution is believed to be responsible for the acceleration of building defects
and deterioration, particularly reinforced concrete structures. Corrosion of
reinforcement is the most common structural defect, having significant impact
on the economic process of repair and maintenance. The structural integrity of
the reinforced concrete members such as beams, columns, slabs or masonry
walls, deteriorates when they experience prolonged contact with acid rain,
sulphate or chloride attack. Sometimes the function of a structural member is
altered by the owner of the building. The departure from the intended use of
floor slab of a building results in it being overloaded. Furthermore, the capacity
and performance of the obsolete heritage buildings, landmark structures, ageing
prominent bridges and others are often degraded over time.
Due to the above factors, the needs for strengthening and rehabilitation of the
building structures are inevitably higher as the number of structures in the world
continues to increase. Properly designed and constructed RC structures which
operate under normal conditions of exposure and use, normally require
minimum maintenance. However, it is wrong to suggest that they are
maintenance free. The maintenance work for restoration or for increasing the
capacity of the structures has been in high demand. It is statistically published
that about 60% of investments are concerned with the maintenance and repair
of existing structures while only about 40% relate to the building of new
structures [38]. In addition, the performance of the strengthened structural
members has to be evaluated and monitored constantly, as safety is not an
issue which can be compromised. It is also equally important that the lessons
which can be learnt from the remedial work and correct treatment of the
strengthened (existing) structures should be put to practical use in the design
and construction of new structures. Lines of communication and exchange of
Chapter 2 Literature Review
38
information are essential between repair parties involved and the design team
so that durability of structures can be enhanced.
In order to increase the loading capacity of the concrete structures, they have
been strengthened using bonding steel plates to the tension surface with
adhesives or bolts since 1960s. Towards the late of 1980s, the use of FRP has
been developing rapidly in the application of strengthening of concrete
structures [35].
Limited number of investigations on the behaviour of the strengthened RC slabs
with FRP has been carried out. The research activities have been increasing
with the application of modern fibre reinforced composite materials such as
CFRP and GRP [39].
The development of strengthening methods has evolved as there is strong
interest to the whole concrete repair community for the benefits of sustainable
life.
El Maaddawy and Soudki [40] investigated the use of mechanically anchored
un-bonded FRP (MA-UFRP) system to strengthen RC slabs as shown in Figure
2.4
Figure 2.4: A test specimen with end-anchorage [40]
A CFRP strengthened RC slab, with dimension 500mm width, 100mm thickness
and 1800mm length and reinforced with three 10mm diameter deformed steel
Chapter 2 Literature Review
39
bars at the tension side, was tested to failure in the laboratory as shown in
Figure 2.5.
One CFRP strip having a width of 50mm and a thickness of 1.2mm was bonded
to the tension face of the slab. This CFRP composite strip had an elasticity
modulus of 155 GPa, tensile strength of 3.1 GPa at failure, ultimate elongation
of 1.9%. These correspond to steel reinforcement ratio (ρs) and FRP
reinforcement ratio (ρf) of 0.8% and 0.12% respectively. The average
compressive strength of concrete was 28 MPa and the steel reinforcement of
Grade 400 were noted during the investigation. Test results showed that the
ultimate load capacity and the mid span deflection of FRP strengthened RC
slabs was increased by 46% and reduced by 45%, respectively relative to a
control slab.
Figure 2.5: Mechanically anchored RC slab tested to failure in the laboratory [40]
The work of Foret and Limam [41] has also contributed to the significant use of
composite materials for strengthening purpose. They carried out an
experimental investigation to examine RC two-way slabs strengthened with
near surface mounted (NSM) CFRP rods as shown in Figure 2.6.
Chapter 2 Literature Review
40
Figure 2.6: RC Slab Section Strengthened with NSM C FRP rods [41]
Two CFRP strengthened RC slabs, each measuring 1650mm length, 1150mm
width and 70mm thickness, reinforced with 6mm diameter of steel bars
positioned at two orthogonal directions were tested in the laboratory. The
concrete had compressive strength of about 40 MPa and the average elastic
modulus of steel was about 200GPa. CFRP strips had elastic modulus of
163GPa, 50mm width and a thickness of 1.4mm. These correspond to ρs and ρf
of 0.35% and 1.0% respectively. Results from the experimental tests had shown
the increase of flexural load capacity of CFRP strengthened slab up to 81% as
compared to the un-strengthened slab and the decrease of mid span deflection
of about 76% that of the control slab.
Michel et al [42] investigated the effective use of composite materials for
ultimate punching load of concrete slabs strengthened by CFRP. CFRP
strengthening RC slabs with 1200 x 1200 mm square geometry having ρs of
0.636% and ρf of 0.35% for one cross layer and 1.05% with three cross layers,
were loaded until failure as shown in Figure 2.7. The compressive strength of
concrete slab was 36 MPa and the steel yield strength was 500 MPa. Two-way
CFRP strips with elastic modulus of 240 GPa, ultimate tensile strength of 4 GPa
and ultimate elongation at break of 1.6%, were bonded at the bottom surface of
slab as shown in Figure 2.8. Experimental results show CFRP strips have
increased the slab ultimate punching load. The punching load for the slab
bonded with one cross layer CFRP has increased to about 15% and by 30% for
Chapter 2 Literature Review
41
three cross layers. Reduction of the mid span deflection up to 35% with one
cross layers and 45% on three cross layers were also recorded.
Figure 2.7: Loading system and slab dimensions [42]
Figure 2.8: Two way CFRP strips being installed [42 ]
Chapter 2 Literature Review
42
El-Sayed, El-Salakawy and Benmokrane [43] assessed the shear strength of
one-way concrete slabs reinforced with different types of FRP bars. All the eight
slabs of each size 3100mm long, 1000mm wide and 200mm deep were tested
under four point bending over a simply supported clear span of 2500mm and
shear span of 1000mm as shown in Figure 2.9. Five slabs were reinforced with
GRP bars and with FRP reinforcement ratio of 0.86%, 1.70%, 1.71%, 2.44%
and 2.63%. Three other slabs were reinforced with CFRP with reinforcement
ratio of 0.39%, 0.78% and 1.18%. The average concrete compressive strength
was 40 MPa, a modulus of elasticity of 30 GPa and average concrete tensile
strength of 3.5 MPa. The properties of reinforcing bars used include modulus of
elasticity of CFRP and GRP were 114 GPa and 40 GPa, respectively and
tensile strength of 1536 MPa and 597 MPa, respectively. From the tests, it
showed that the flexural stiffness of the slabs reinforced either with glass or
carbon FRP bars increased with an increase in the reinforcement ratio. The test
results indicate that the shear strength increases as the FRPs reinforcements
increases. With CFRP bars, the increase of ρf from 0.39 to 0.78% and 0.39 to
1.18% resulted shear capacity increased by 19% to 36%, respectively. For GRP,
an increase of 44% to 49% was obtained by increasing ρf from 0.86 to 1.71%
and 0.86 to 2.63% respectively. All tested slabs were failed in shear.
Chapter 2 Literature Review
43
Figure 2.9: (a) Slab dimension ; (b) Test set-up [4 3]
Rochdi et al [44] investigated an analytical and experimental evaluation of the
strength of the two-way concrete slabs externally bonded with CFRP composite
sheets.
Eight slabs, each of in-plane size 600mm by 600mm and 50mm in thickness
(Figure 2.10) were tested to failure. Six slabs were strengthened with externally
bonded CFRP with variable thickness and the remaining two slabs were just
ordinary RC slabs with no FRP installed and acted as control specimens.
Material properties of Uni-directional (UD) CFRP used were elastic modulus of
82.6 GPa and tensile strength of 1140 MPa. RC slabs consist of an average
concrete compressive strength of about 25 MPa and were reinforced with steel
bars having reinforcement ratio of 0.12%. The steel bars had an average tensile
strength of 770MPa. The test showed that central deflections of all strengthened
Chapter 2 Literature Review
44
slabs were considerably lower than the control slabs. The ultimate punching
shear strength of slab increases with increasing composite section in the range
of 67% to 177% over the control slabs. The average deflection at the ultimate
load was also reduced by 28% that of the corresponding reference specimens.
Shear failure was observed by the forming of the flexural cracks that appeared
at the slab top directly below the concentrated load. These cracks propagated
towards the sides of the slabs as the load increased. Finally, punching shear
failure occurred and the FRP reinforcement was also delaminated.
Figure 2.10: Dimensions and test set-up scheme [44]
These experimental results were also compared with the finite element
simulation. Using a 3D element, one quarter of the slab was modelled due to
the geometrical and loading symmetry as shown in Figure 2.11. The predicted
failure load presented was successfully showed a good comparison with the
experimental results.
The numerical study concluded that CFRP strengthening produced significant
improvements in punching shear strength.
Chapter 2 Literature Review
45
Figure 2.11: Stress distribution for the analysed s labs [44]
Zhang et al [45] studied the flexural behaviour of one way concrete slabs
reinforced with CFRP grid reinforcements. The results of this behaviour were
then compared with conventional steel reinforcing rebars. All three slabs were
reinforced with CFRP grid reinforcement and one slab with steel rebars had the
same measurement of 3300 x 1000 x 250mm but with different reinforcement
ratios. Each of them was simply supported and tested under both static and
cyclic loading conditions in the laboratory to evaluate their flexural and shear
limit states (Figure 2.12).
Two types of CFRP were used in the experiment, namely, New Fibre
Composite Material for Reinforcing Concrete, NEFMAC C16 and NEFMAC
C19-R2. These materials comprised of modulus of elasticity of 98.1 and 90 GPa
with tensile strength of 1180 MPa and 1400 MPa and 0.49% and 0.99% of
reinforcement ratio, respectively. The slab’s concrete average compressive
strength of 45 MPa and modulus of elasticity of 37 GPa. One of the slabs that
acts as the control specimen was only reinforced with steel bars of grade 400
with reinforcement ratio of 0.69%.
Chapter 2 Literature Review
46
Figure 2.12: Flexural Test Setup of Concrete Slabs [45]
A fairly regular crack pattern was observed at the location of the transverse bars
of the FRP grid as shown in Figure 2.13. It was also observed that the crack
width was larger, generally, in FRP reinforced slab than the steel reinforced slab
under the same applied loads. The deflection behaviour of steel reinforced slab
was generally characterised by plastic deformation as the steel yielded and non
plastic deformation behaviour was observed in FRP reinforced slabs followed
by brittle fracture due to crushing of the concrete.
Figure 2.13: Typical crack pattern of the CFRP grid reinforced slabs [45]
Chapter 2 Literature Review
47
From the research work investigated, the experimental ultimate moment
recorded for steel bars reinforced slab was 184 kNm, and after post-
strengthening with CFRP 0.49% reinforcement ratio, the ultimate moment was
197 kNm. FRP reinforced slab also characterised by bi-linearly elastic until
failure as shown in Figure 2.14. This showed that FRP reinforcement structures
had greater ultimate flexural capacity than steel reinforced.
Figure 2.14: Typical theoretical and experimental l oad versus midspan deflection curve [45]
Ebead and Marzouk [46] investigated the effect of FRP strengthening on the
tensile behaviour of concrete slabs using finite element analysis (FEA). The
available experimental results of the strengthened reinforced concrete slabs
were used to calibrate the finite element model based on the ultimate load
carrying capacity of the two-way slabs. An overall increase in the post-peak
stiffness based on the tensile stress-strain relationship was observed. The
comparison of study between the tension-stiffening model of FRP strengthened
and un-strengthened concrete was the main focus of the research.
Slab specimens were measured by 1900 x 1900 x 150mm thick. Column stubs
were cast at the centre of the slab and of size 250 x 250mm. Two un-
strengthened specimens with variable reinforcement ratio ρs of 0.35% and 0.5%
Chapter 2 Literature Review
48
were tested to failure and the ultimate load carrying capacity were 250kN and
330kN, respectively. Both GRP and CFRP strips were used at individual slab
and located at the tension side of the slab with reinforcement ratio ρs of 0.35%
and 0.5% for each type of FRPs. The modulus of elasticity of concrete, Ec, was
26.6 GPa and the compressive strength of concrete was 35 MPa. The tensile
strength of concrete was 2.8 MPa. The steel reinforcement’s yield stress and
the modulus of elasticity of 440 MPa and 210 GPa, respectively, were defined in
the simulation process.
The test results had been used to calibrate the FEA inputs data. The test
showed that GRP reinforced slabs exhibited an average gain in the load
carrying capacity of about 31% over that of the reference un-strengthened slabs.
Meanwhile, about an average of 40% increase with CFRP strips over the
control specimen was found.
From the finite analysis results, both CFRP and GRP strengthened slabs
showed the post behaviour of slabs was stiffened. The slope of the tensile
stress-tensile strain was decreased in the post-peak zone indicating the
contribution of the FRP strengthening materials in increasing the post-peak
stiffness of concrete in tension. The results of the load carrying capacity
between the FEA and the experimental test were compared and a good
agreement was found. FEA also exhibited a stiffer deformational behaviour
compared to the experimental results.
The study concluded that FRP strengthened concrete showed a stiffer post-
peak response than conventional reinforced concrete. Experimental test results
show that the use of FRP strengthened strips or laminates could lead to an
average gain in the load carrying capacity of about 36% over the un-
strengthened specimens. However, a decrease in ductility and energy
absorption was recorded due to the brittle nature of the FRP composites. It was
also observed that de-bonding between FRP composites and concrete was the
main cause of failure. None of the FRP composites experienced rupture. Slabs
were failed after exceeding flexural capacity.
Chapter 2 Literature Review
49
Pesic and Pilakoutas [47] developed a numerical method for the computation of
the bending moment capacity and deflection of FRP plate reinforced concrete
beams and prediction of the flexural failure modes. The numerical procedure
was validated with available experimental results.
Material properties of concrete beam were defined such that concrete
compressive strength was 35 MPa, steel reinforcement ratio and yield strength
of 1.8% and 456 MPa, respectively, GRP elastic modulus, tensile strength and
reinforcement ratio of 32.7 GPa, 400 MPa and 1.11% respectively. The
experimental tests result showed that an increase of 33.3% of loading capacity
was achieved after post strengthening of RC beam with GRP plate.
Ashour et al [48] tested 16 RC continuous beams with different arrangements of
internal steel bars and external CFRP laminates (Figure 2.15). All test
specimens had the same geometrical dimensions and were classified into three
groups according to the amount of internal steel reinforcement. Each group
included one un-strengthened control beam designed to fail in flexure. Different
parameters including the length, thickness, position and form of the CFRP
laminates were investigated. Material properties used in the tests consist of
concrete compressive strength ranging from 24.0 MPa to 47.8 MPa and
average Young’s modulus, yield strength and ultimate strength of steel bars
were 200 GPa, 512 MPa and 616 MPa, respectively. The uni-directional CFRP
plates and sheets were used in the experiment. According to the manufacturer’s
recommendations, tensile strengths of CFRP plates of 1.2 mm thickness and
100 mm width and CFRP sheets of 0.1117 mm thickness and 110 mm width
were 3900 MPa and 2500 MPa. They have Young modulus of 240 GPa and
150 GPa, respectively. Results from the tests showed that an increase of
ultimate load was recorded after post strengthening with CFRP sheets and
plates. With CFRP sheets, failure load was increased up to 57% and 37% after
post strengthening. Three failure modes of beams with external CFRP
laminates were observed, namely, laminate rupture, laminate separation and
peeling failure of the concrete cover attached to the laminate. The ductility of all
strengthened beams was reduced compared with that of the respective un-
strengthened control beam.
Chapter 2 Literature Review
50
Figure 2.15: Strengthened RC beam details [48]
The results from the experimental and the simplified methods were also
compared which showed that most beams were close to achieving their full
flexural capacity.
Mosallam and Mosalam [49] investigated the performance of two-way RC slabs
retrofitted with FRP composite laminates. CFRP strengthened RC slabs, each
measuring 2640mm width, 2640mm length and 76mm thick, were tested to the
ultimate load and compared with the control slab as shown in Figure 2.16.
Concrete slab’s average compressive strength was 32.87 MPa, steel bars of
grade 400 and CFRP having a thickness 0.58mm and 457mm width and elastic
modulus of 1.1GPa. These correspond to ρs and ρf of 0.64% and 0.79%
respectively. Results from the tests showed that CFRP strengthened RC slab
provided about 198% enhancement in flexural strength relative to the control
slab.
Chapter 2 Literature Review
51
Figure 2.16: Test setup and instrumentation of the two-way slab specimens [49]
Chapter 2 Literature Review
52
Sheikh [50] investigated the structural performance with the application of FRP
to strengthen and repair damaged slabs (Figure 2.17) and beams (Figure 2.18).
Figure 2.17: slab specimen details and loading [50]
Chapter 2 Literature Review
53
Figure 2.18: Details of beam specimens (a) Cross Se ction and reinforcement details (b) Zone of FRP repair [50]
For each reinforced slab specimen of 1200mm wide, 2100mm long and 250mm
thick to be repaired with FRP (Figure 2.19 and Figure 2.20), three strips of
CFRP of about 600mm wide were used for slab2 and GRP for slab3. Slab2 was
loaded to failure in shear at a load of 478kN while slab3 shear failure occurred
at an applied load of 442kN.
Compressive strength of concrete was 30 MPa and grade 60 and grade 400
steel bars were used in the test.
Figure 2.19: Details of FRP retrofitting of slab sp ecimens [50]
Chapter 2 Literature Review
54
Figure 2.20: slab specimens tested to failure [50]
Beam1 which considered as a control specimen formed crack widths exceeded
2.0mm at 1600kN loading. Beam1 failed at a load of 1700kN with a
corresponding deflection of 14mm. Beam2 was wrapped around the damaged
zone with three strips of 610mm wide carbon fabric. The maximum load applied
to this beam was 2528kN with corresponding deflection of 143mm.
Chapter 2 Literature Review
55
Sheikh found out that FRP resulted a substantial increase in the ultimate
capacity of the slabs. The tests showed that 119% increase for GRP and 148%
for CFRP. The load corresponding to the shear capacity was much lower than
that for the enhanced flexural capacity. The failure in both repaired slabs was
caused by shear.
Sheikh concluded his investigation from experimental works that retrofitting with
FRP provided a feasible rehabilitation technique for repair as well as
strengthening. FRP wrapping was very effective in enhancing flexural strength
of the damage slabs, shear resistance of the damaged beams. Both CFRP and
GRP provided approximately 150% enhancement in flexural strength. Installing
one layer of CFRP had increased the beam failure capacity from 1700kN to
2528kN (about 49% enhancement).
Based on one of the experimental work conducted recently in the laboratory
within the school of civil engineering, University of Manchester [51], CFRP
strengthened RC slabs of dimension 1800mm length, 1800mm width and
150mm thick were tested to failure. Two CFRP strips, each measuring 100mm
width and 1.2mm thickness, having an elastic modulus of 150GPa and average
tensile strength of about 3 GPa, were bonded to the tensile face of the slabs in
each direction. The average concrete compressive strength was about 41 MPa
and steel reinforcement bars grade 600 were observed for the tests. The
corresponding ρs and ρf were 0.75% and 0.2% respectively. Results from the
experimental tests showed the increase of ultimate loading capacity of FRP
strengthened RC slab to about 43% and the reduction of central deflection up to
58%. These deflection results obtained from experimental tests were verified
with FEM which will be explained in the next section.
From the literature review stated above, obviously, different results of flexural
loading capacity enhancement and reduction of central deflection of post-
strengthening RC slab with FRP from the experimental works had been noted.
These variable results were affected due to several parameters including
material properties, geometric characteristics, loading and boundary conditions.
Chapter 2 Literature Review
56
2.4 Conclusions
Plate theory remains one of the interesting topics to the present day particularly
for the applied mechanicians and structural engineers. It is interesting to note
that the subject of bending of plates was extended to shell where such element
was applied for finite element method. The thickness of the plate is very small
when compared to the other dimensions and full three dimensional numerical
analysis becomes not only costly but it often leads to serious numerical ill
conditioning problems.
In order to ease the solution, several classical assumptions with regard to the
behaviour of the structures were introduced which resulting a series of
approximations. Both Reissner and Mindlin have improved the classical thin
plate theory by incorporating the effect of transverse shear deformation,
however, the normal stress had been neglected in the Mindlin but it was
accounted for Reissner plate theory. From the literature, it clearly shows the
weakness of plate theory due to the assumptions made. The main problems on
the classical plate bending are due to the fact that the deflection is not a
constant across the thickness of the plate and it is only suitable for isotropic
plate having thickness to side length ratio of less than 1/10.
From the shortcomings of the plate theory, it brings interest and attention to
make comparison of classical plate theory over the exact three dimensional
elastic analysis. For this reason, the application of concrete slab with FRP as
laminated plate is investigated in this study to illustrate their differences on the
flexural deformation of the structure. The other application of the exact elasticity
analysis was conducted by Fan et. al. on partially clamped edges laminated
plate. Such problem with the given exact solutions is not complete, i.e. the plate
is not fully fixed at all edges. For this main reason, it attracts a new challenge
for the author to further investigate analytical exact analysis of a fully clamped
laminated plate. The novel solutions of such findings are presented in this thesis.
Chapter 2 Literature Review
57
From the numerous researches conducted which are available in the literature
with regard to FRP strengthened RC slab, numerical analysis of FEM is also
carried out in this study for review purpose. This study aims to complement and
review on the practical significant enhancement of structural loading capacity of
RC slab when FRP is used.
The numerical results of this modeling and simulation using Abaqus are then
compared to the existing available experimental test results of similar
specimens. The results showed that punching shear strength and the stiffness
of the RC slabs have increased after FRPs are bonded to the slab.
FEM results also agreed well with the experimental test that the stiffness and
deflection of RC slabs have significantly increased and reduced, respectively
with the application of FRP.
Chapter 3 State Space Method of 3D Elasticity
58
Chapter 3
STATE SPACE METHOD OF 3D
ELASTICITY
The objective of this chapter is to review the concept of state space method and
the governing equations of elasticity. A classical (continuum) or analytical
method based on the application of the equations of equilibrium and
compatibility, together with the stress and strain relations for the material are
applied to produce governing equations which must be solved to obtain
displacements and stresses. It also shows the fundamental equations for
orthotropic plates based on the theory of plates and elasticity with respect to a
rectangular Cartesian coordinate system, x,y and z only. It is important to note
that it can also be represented in other coordinate system such as polar
coordinates and cylindrical coordinate systems.
From these governing equations for state space method, further development
into various applications having different boundary and loading conditions can
be explored. In the next chapter, state space equations are solved for the case
of partially fixed edges laminated plates subjected to various loading conditions.
Based on the formulation applied to a simply supported rectangular laminated
plate, state space methods are extended to the case of clamped edges
laminated plates.
3.1 The Concept of State Space Method of 3D Elastic ity
The state space method of 3D elasticity was initiated from Vlasov’s method of
initial function (MIF) [26] and Bahar’s transfer matrix approach [24]. The term
‘state space’ deals with a linear control system between the relationships of
actions and the responses of the related system. This system can be
Chapter 3 State Space Method of 3D Elasticity
59
represented an electrical, hydraulic, mechanical, thermal systems or others, for
instance, a spring-damper-mass system when subjected to excitation as shown
in Figure 3.1. The concept involved solving linear time-invariant system of the
elastic mechanical system subjected to an external source of loading. By using
the boundary conditions as the initial state of the system, a solution can be
determined in terms of a set of arbitrary constants which satisfy these boundary
conditions. From this, all the remaining outputs can be found at any time
between initial stage and the subsequent stages.
As an example to illustrate state space variables, a governing second order
linear differential equation of motion of such system can be represented by [52]:
( ) ( ) ( ) ( )tFtkxtxctxm =++ &&& (3-1)
where
m is the mass of the object
x is the displacement
t is the time
F is the force applied
c is the damping coefficient
k is spring constant
Figure 3.1: Spring - damper - mass system [52]
The equation (3-1) can be converted into a first order linear differential equation
system in a matrix form of [52]:
( )( )
( )( ) ( )
+
−−=
mtFtx
tx
m
c
m
ktx
tx
dt
d
/
010
&& (3-2)
k
c
m F ( t )
x(t)
Chapter 3 State Space Method of 3D Elasticity
60
or
( ){ } [ ] ( ){ } [ ]{ }tBtxAtx +=& (3-3)
where A is an n x n matrix and B(t) is an n x 1 vector. Notice that constant
matrix A is independent of time, hence equation (3-3) is called linear time-
invariant system. If matrix A is a function of time, i.e. A(t), the solution of this
linear time-variant system becomes more complicated. If equation (3-3) were
treated as a scalar differential equation, i.e. n = 1, then it would be expressed
as the following:
( ) ( ) ( )tbtaxtx +=& (3-4)
where a and b are constant. The solution of equation (3-4) can be found as:
( ) ( ) ( ) τττ dbeexetxt
aatat∫
−+=0
0 (3-5)
where to = 0 is assumed and x(0) = xo = 0 at to = 0.
Similar to the solution for the scalar differential equation, if to ≠ 0, the general
solution in terms of n x n matrix would become
( ) [ ]( ) ( ) [ ]( ) [ ] ( ) τττ dBeetxetxt
t
AttAttA
∫−−−
+=0
00
0 (3-6)
Hence, the final solution of equation (3-3) is given by
( ) [ ] ( ) [ ]( ) ( ) τdτBexetxt
τtAtA∫
−+=0
0 (3-7)
The concept can be extended to the three dimensional analysis of plate. The
application of state equations is adopted if knowing the initial boundary and
loading conditions of the plate as the initial state of the system. The
displacements and stresses at an arbitrary position across thickness direction of
the plate can be solved subsequently by using transfer matrix approach.
The application of the above state space equations for plate will be discussed in
more details in the following chapter.
Chapter 3 State Space Method of 3D Elasticity
61
3.2 Governing Equations of Elasticity Problems
The general governing equations for elasticity problem are as follows:
Equilibrium equations [7]:
2
2
2
2
2
2
t
Wρf
z
σ
y
τ
x
τ
t
Vρf
z
τ
y
σ
x
τ
t
Uρf
z
τ
y
τ
x
σ
zzyzxz
y
yzyxy
x
xzxyx
∂∂=+
∂
∂+
∂
∂+
∂
∂∂∂=+
∂
∂+
∂
∂+
∂
∂∂∂=+
∂
∂+
∂
∂+
∂
∂
(3-8)
where fx, fy and fz are the body force per unit volume in the x, y and z direction,
respectively. ρ is the density coefficients of the material and U, V and W are the
in-plane and transverse displacement respectively.
Kinematic (strain-displacement relations) equations [7]:
y
U
x
Vγ
x
W
z
Uγ
z
V
y
Wγ
z
Wε
y
Vε
x
Uε
xy
zx
yz
z
y
x
∂∂+
∂∂=
∂∂+
∂∂=
∂∂+
∂∂=
∂∂=
∂∂=
∂∂=
(3-9)
where U, V and W represent the displacements along the coordinate axes x, y
and z, respectively.
Chapter 3 State Space Method of 3D Elasticity
62
Stress – strain relationship for a general anisotropic linearly elastic material [7]:
=
xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
γ
γ
γ
ε
ε
ε
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
τ
τ
τ
σ
σ
σ
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
(3-10)
where Cij (i,j = 1,2,3…..,6) are the material stiffness coefficients of anisotropic
body.
Compatibility equations:
In the strain-displacement relationships, there are six strain measures
xzxyzyxε,ε,ε,ε,ε and
yzε but only three independent displacements.
This means that there are six unknowns for only three independent variables.
As a result, there are constraints or compatibility equations exist, such as
xz
γ
z
ε
x
ε
zy
γ
y
ε
z
ε
yx
γ
x
ε
y
ε
zxxz
yzzy
xyyx
∂∂
∂=
∂
∂+
∂
∂
∂∂
∂=
∂
∂+
∂
∂
∂∂
∂=
∂
∂+
∂
∂
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
(3-11)
By using all the above equations, there are three methods to solve the elasticity
problem. Firstly, applying the displacements approach, secondly, using the
stresses approach and lastly using both approaches simultaneously which is
also known as hybrid method [7].
Chapter 3 State Space Method of 3D Elasticity
63
Displacement approach expresses stresses in terms of displacements following
by substitution of these stresses into the equilibrium equations. The solutions
expressed in terms of displacements are initially determined and followed by
strains and the stresses.
Stress approach determines the solution of six stresses firstly and satisfies the
compatibility conditions, equilibrium equations and stress boundary conditions.
Strains can be then determined afterwards. It is essential to note that in this
approach, the determination of displacements becomes more complicated as it
involves integration functions of the kinematic equation.
Hybrid approach seeks the solution for both displacement and some stress
components simultaneously.
3.3 State Equations for Simply Supported Orthotropi c Plate
A simply supported rectangular orthotropic thick plate is shown Figure 3.2
Figure 3.2: Coordinate system and plate dimension
a
b
1,U,x
3,W,z
2,V,y
h
Chapter 3 State Space Method of 3D Elasticity
64
Letting zyzxz
σZ,τY,τX === (X and Y are the transverse shear stresses. Z
is the normal stress).
And eliminating xyyx
τσσ ,, and the strains, by using the strain-displacement
and the equilibrium equations, the following expression can be obtained:
( )( )
−−+−
+−−−−−
−−
=
∂∂
W
Y
X
Z
V
U
CβCαC
βCβCαCξαβCC
αCαβCCβCαCξ
ξβα
βC
αC
W
Y
X
Z
V
U
z
000
000
000
000
0000
0000
751
5
2
4
2
6
2
63
163
2
6
2
2
2
29
8 (3-12)
where
2
22
449
558
337666
33
235
33
2
23224
33
2313123
33
2
13112
33
131
111
tρξ,
yβ
,x
α,C
C,C
C,C
C,CC,C
CC
,C
CCC,
C
CCCC,
C
CCC,
C
CC
∂∂=
∂∂=
∂∂=====−=
−=−=−=−=
where
Chapter 3 State Space Method of 3D Elasticity
65
( ) ( ) ( )
( ) ( ) ( )
yzzzyyxzzzxxxyyyxx
zxxzxyxzzxzyyzyxxy
xyzxyz
yxxyzzxxyzyyzxxzy
zyyxzxxyzzxyxxzyyzx
µEµE,µEµE,µEµE
,µµµµµµµµµQ
,GC,GC,GC
Q
µµEC,
Q
µµµEC,
Q
µµEC
Q
µµµEC,
Q
µµµEC,
Q
µµEC
===
−−−−=
===
−=
+=
−=
+=
+=
−=
21
11
1
665544
332322
131211
The derivation of the above equation (3-12) is shown as follows:
To eliminate three in-plane stresses, xyyx
τandσ,σ , and obtain the state
equation for elasticity, the following tasks are performed.
For orthotropic linear elasticity, from equation (3-9), stress-strain relationship
reduces to:
∂∂+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂∂∂∂∂
=
=
y
U
x
Vx
W
z
Uz
V
y
Wz
Wy
Vx
U
C
C
C
CCC
CCC
CCC
γ
γ
γ
ε
ε
ε
C
C
C
CCC
CCC
CCC
τ
τ
τ
σ
σ
σ
xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
66
55
44
332313
232212
131211
66
55
44
332313
232212
131211
00000
00000
00000
000
000
000
00000
00000
00000
000
000
000
(3-13)
Chapter 3 State Space Method of 3D Elasticity
66
From the third equation of (3-8) and neglecting the body force, fz,
y
τ
x
τ
t
Wρ
z
σ yzxzz
∂
∂−
∂
∂−
∂∂=
∂
∂2
2
(3-14)
From the third, fourth and fifth equations of (3-13), we can get,
x
Wτ
Cz
U
y
Wτ
Cz
V
σCy
V
C
C
x
U
C
C
z
W
xz
yz
z
∂∂−=
∂∂
∂∂−=
∂∂
+∂∂−
∂∂−=
∂∂
55
44
3333
23
33
13
1
1
1
(3-15)
Substitute xyx
τandσ from equation (3-13) and considering the first equation of
(3-15) into (3-8), we obtain,
x
σ
C
CV
yxC
yxC
CCC
Uy
CxC
CC
tρ
z
τ
z
xz
∂
∂−
∂∂∂+
∂∂∂
−
−
∂∂−
∂∂
−−
∂∂=
∂
∂
33
132
66
2
33
231312
2
2
662
2
33
2
13112
2
(3-16)
Substitute xyy
τandσ from equation (3-13) and considering the first equation
of (3-15) into (3-8), we obtain,
y
σ
C
CV
xC
yC
CC
tρ
Uyx
CyxC
CCC
z
τ
z
yz
∂
∂−
∂∂−
∂∂
−−
∂∂
+
∂∂∂+
∂∂∂
−−=
∂
∂
33
232
2
662
2
33
2
23222
2
2
66
2
33
231312
(3-17)
Chapter 3 State Space Method of 3D Elasticity
67
Let 2
22
tξ,
yβ,
xα,σZ,τY,τX
zyzxz ∂∂=
∂∂=
∂∂==== and use
449
558
337666
33
235
33
2
23224
33
2313123
33
2
13112
33
131
111
CC,
CC,
CC,CC,
C
CC
,C
CCC,
C
CCCC,
C
CCC,
C
CC
====−=
−=−=−=−=
Eqns. (3-14) to (3-17) can be simplified in the state equation as
( )( )
−−+−
+−−−−−
−−
=
∂∂
W
Y
X
Z
V
U
CβCαC
βCβCαCξαβCC
αCαβCCβCαCξ
ξβα
βC
αC
W
Y
X
Z
V
U
z
000
000
000
000
0000
0000
751
5
2
4
2
6
2
63
163
2
6
2
2
2
29
8 (3-18)
The eliminated stress components can be determined from (3-13) as
−−
=
Z
V
U
αCβC
CβCαC
CβCαC
τ
σ
σ
xy
y
x
066
543
132
(3-19)
For the derivation of eqns. (3-19), it can be shown below,
Substitute the first eqn.(3-15) into (3-13),
Chapter 3 State Space Method of 3D Elasticity
68
ZCVβCUαCσ
σC
C
y
V
C
CCC
x
U
C
CCσ
σCy
V
C
C
x
U
C
CC
y
VC
x
UCσ
z
WC
y
VC
x
UCσ
x
zx
zx
x
132
33
13
33
231312
33
2
1311
3333
23
33
13131211
131211
1
−+=∴
+∂∂
−+
∂∂
−=
+
∂∂−
∂∂−+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂=
Similarly for σy, substitute the first eqn.(3-15) into (3-13),
ZCVβCUαCσ
σC
C
y
V
C
CC
x
U
C
CCCσ
σCy
V
C
C
x
U
C
CC
y
VC
x
UCσ
z
WC
y
VC
x
UCσ
y
zy
zy
y
543
33
23
33
2
2322
33
231312
3333
23
33
13232212
232212
1
−+=∴
+∂∂
−+
∂∂
−=
+
∂∂−
∂∂−+
∂∂+
∂∂=
∂∂+
∂∂+
∂∂=
and taking the last expression of eqn. (3-13),
VαCUβCτ
y
U
x
VCτ
xy
xy
66
66
+=∴
∂∂+
∂∂=
The above elastic constants have a relation with engineering elastic stiffness
coefficients in the form of [27]:
Chapter 3 State Space Method of 3D Elasticity
69
( ) ( ) ( )
( ) ( ) ( )
yzzzyyxzzzxxxyyyxx
zxxzxyxzzxzyyzyxxy
xyzxyz
yxxyzzxxyzyyzxxzy
zyyxzxxyzzxyxxzyyzx
µEµE,µEµE,µEµE
,µµµµµµµµµQ
,GC,GC,GC
Q
µµEC,
Q
µµµEC,
Q
µµEC
Q
µµµEC,
Q
µµµEC,
Q
µµEC
===
−−−−=
===
−=
+=
−=
+=
+=
−=
21
11
1
665544
332322
131211
(3-20)
For Figure 3.2, the case of a simply supported plate, the boundary conditions
may be expressed as [27]:
x = 0 and a ; σx = 0, W = 0 and V = 0,
y = 0 and b; σy = 0, W = 0 and U = 0. (3-21)
Letting
( )
( )
( )
∑∑∞
=
∞
=
=
1 1m n
mn
mn
mn
b
yπnSin
a
xπmSinzW
b
yπnCos
a
xπmSinzV
b
yπnSin
a
xπmCoszU
W
V
U
(3-22)
and
( )
( )
( )
∑∑∞
=
∞
=
=
1 1m n
mn
mn
mn
b
yπnSin
a
xπmSinzZ
b
yπnCos
a
xπmSinzY
b
yπnSin
a
xπmCoszX
Z
Y
X
(3-23)
It can be noted that from eqn.(3-19), (3-21) and (3-22), the boundary conditions
of eqn. (3-21) and the state eqn. (3-18) are satisfied.
Chapter 3 State Space Method of 3D Elasticity
70
And finally, substitute equation (3-22) and (3-23) into (3-18), the following first-
order homogeneous ordinary differential equation can be found for each
combination of m and n [27]:
( )( )
−−++
++
−−
=
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
W
Y
X
Z
V
U
CηCξC
ηCηCξCξηCC
ξCξηCCηCξC
ηξ
ηC
ξC
W
Y
X
Z
V
U
dz
d
000
000
000
0000
0000
0000
751
5
2
4
2
663
163
2
6
2
2
9
8
(3-24)
or
( ) ( )zDRzRdz
dmnmn
= (3-25)
where b
n
a
m πηπξ == ,
Chapter 3 State Space Method of 3D Elasticity
71
( )
=
mn
mn
mn
mn
mn
mn
mn
W
Y
X
Z
V
U
zR and
( )( )
−−++
++
−−
=
000
000
000
0000
0000
0000
751
5
2
4
2
663
163
2
6
2
2
9
8
CηCξC
ηCηCξCξηCC
ξCξηCCηCξC
ηξ
ηC
ξC
D
From eqn. (3-24), a sixth order differential equation of any of the six
components can be obtained such as the equation of transverse displacement
Wmn [27],
02
2
4
4
6
6
=+++mno
mn
o
mn
o
mn WCdz
WdB
dz
WdA
dz
Wd (3-26)
where Ao, Bo and Co are the coefficient matrix.
The derivation of eqn. (3-26) is clearly shown below:
Initially, eliminate Xmn and Ymn from eqn. (3-24) by letting,
=
=
⇒
=
Z
V
U
B
W
Y
X
dz
dand
W
Y
X
A
Z
V
U
dz
d
W
Y
X
Z
V
U
B
A
W
Y
X
Z
V
U
dz
d
0
0
Chapter 3 State Space Method of 3D Elasticity
72
where
( )( )
−−++
++
=
−−
=
751
5
2
4
2
663
163
2
6
2
2
9
8
0
0
0
CηCξC
ηCηCξCξηCC
ξCξηCCηCξC
Band
ηξ
ηC
ξC
A
Differentiate the second part and substitute it into the first part, i.e.
=
=
=
=
=
W
Y
X
KKK
KKK
KKK
W
Y
X
dz
d
A.BKwhere
W
Y
X
.K
W
Y
X
.A.B
Z
V
U
dz
dB
W
Y
X
dz
d
333231
232221
131211
2
2
2
2
Therefore
WKYKXKdz
Wd
WKYKXKdz
Yd
WKYKXKdz
Xd
3332312
2
2322212
2
1312112
2
++=
++=
++=
(3-27)
Substitute Y from the third into the first eqn. (3-27), gives
WK
KKK
dz
Wd
K
KX
K
KKK
dz
Xd
WKWKXKdz
Wd
KKXK
dz
Xd
−++
−=∴
+
−−+=
32
3312132
2
32
12
32
3112112
2
1333312
2
3212112
2 1
(3-28)
Again, substitute Y from the third into the second eqn. (3-27), yields
Chapter 3 State Space Method of 3D Elasticity
73
( )( )WKKKK
dz
WdKXKKKK
dz
WdK
dz
XdK
dz
Wd
WKWKXKdz
Wd
KKXK
WKXKdz
Wd
Kdz
d
33223223
2
2
22312221322
2
332
2
314
4
2333312
2
322221
33312
2
322
2
1
1
−
++−=−−∴
+
−−+
=
−−
(3-29)
Multiply the term K31 to the both sides of eqn.(3-28) and then adding the
expression to eqn.(3-29). By doing this, we can eliminate the term 2
2
dz
Xd from
eqn.(3-27),
WK
KKKK
dz
Wd
K
KKX
K
KKKK
dz
XdK
−++
−=
32
331213312
2
32
1231
32
311211312
2
31
Add this expression to eqn.(3-29),
Chapter 3 State Space Method of 3D Elasticity
74
( ) ( )
−
+−
+
++
−
−+−=∴
−+−+
++
−=
−+−⇒
−++
−
+−++−=−
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKKX
WKKK
KKKKKKK
dz
Wd
K
KKKK
dz
Wd
K
KKKKKKKKX
WK
KKKK
dz
Wd
K
KKX
K
KKKK
WKKKKdz
WdKXKKKK
dz
WdK
dz
Wd
133132
313312
32233322
2
2
32
31122233
4
4
32
2
3112
113131222132
133132
313312322333222
2
32
31122233
4
4
32
2
3112113131222132
32
331213312
2
32
1231
32
31121131
332232232
2
22312221322
2
334
4
1
(3-30)
Express Y from the third eqn.(3-27), then substitute X from eqn.(3-30),
−−=
++=
WKXKdz
Wd
KY
WKYKXKdz
Wd
33312
2
32
3332312
2
1
Chapter 3 State Space Method of 3D Elasticity
75
WK
K
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKKK
K
dz
Wd
KY
32
33
133132
313312
32233322
2
2
32
3112
2233
4
4
32
2
3112
11313122213232
31
2
2
32
1
1
−
−
+−
+
++
−
−+−
−=
(3-31)
And finally substitute eqns.(3-30) and (3-31) into the first expression of eqn.(3-
27),
WKYKXKdz
Xd1312112
2
++=
Chapter 3 State Space Method of 3D Elasticity
76
WK
WK
K
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKKK
K
dz
Wd
K
K
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKKK
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKKdz
d
13
32
33
133132
313312
32233322
2
2
32
31122233
4
4
32
2
3112
11313122213232
31
2
2
32
12
133132
313312
32233322
2
2
32
31122233
4
4
32
2
3112
11313122213211
133132
313312
32233322
2
2
32
31122233
4
4
32
2
3112
1131312221322
2
1
1
1
1
+
−
−
+−
+
++
−
−+−
−
+
−
+−
+
++
−
−+−
=
−
+−
+
++
−
−+−
Chapter 3 State Space Method of 3D Elasticity
77
Rearrange the above expression, gives
WKWK
KK
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KKKK
KKKK
K
KK
dz
Wd
K
K
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKK
K
dz
Wd
KKK
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
KKKKKK
1332
1233
133132
313312
32233322
2
2
32
31122233
4
4
32
2
31121131
31222132
32
31122
2
32
12
133132
313312
32233322
2
2
32
311222334
4
32
2
3112
113131222132
11
2
2
133132
313312
32233322
4
4
32
311222336
6
32
2
3112
113131222132
1
+−
−
+−
+
++
−
−+
−−
+
−
+−
+
++−
−
+−
=
−
+−
+
++−
−
+−
Chapter 3 State Space Method of 3D Elasticity
78
W
K
KKKK
KKKK
WK
KK
K
KKKKKKKK
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
K
KK
dz
Wd
K
K
K
KKKKKKKK
WKK
K
KKK
KKKK
dz
Wd
K
KKKK
dz
WdK
dz
Wd
KKK
KKK
KKKK
dz
Wd
K
KKKK
dz
Wd
−
+−
+
−+−
−
−
+−
+
++−
−
−+−
+
−
+−
+
++−
=
−
+−
+
++−⇒
32
2
31121131
31222132
32
1233
32
2
3112113131222132
133132
313312
32233322
2
2
32
311222334
4
32
3112
2
2
32
12
32
2
3112113131222132
133132
313312
32233322
2
2
32
311222334
4
11
2
2
133132
313312
32233322
4
4
32
311222336
6
( )
WK
KKK
K
KKKKKKKK
WKK
K
KKK
KKKK
K
KK
dz
Wd
K
KKKK
K
KK
dz
Wd
K
KK
dz
Wd
K
K
K
KKKKKKKK
WKK
K
KKK
KKKK
Kdz
Wd
K
KKKKK
dz
WdK
dz
Wd
KKK
KKK
KKKK
dz
WdKKK
dz
Wd
−
−+−
+
−
+−
−
++
+−
−+−
+
−
+−
+
++−
=
−
+−
+−−−+⇒
32
331213
32
2
3112113131222132
133132
313312
32233322
32
31122
2
32
31122233
32
3112
4
4
32
31122
2
32
12
32
2
3112113131222132
133132
313312
32233322
112
2
32
31122233114
4
11
2
2
133132
313312
32233322
4
4
2233116
6
Chapter 3 State Space Method of 3D Elasticity
79
( )
0
1332
1233
32
2
3112113131222132
1132
31121331
32
31331232233322
2
2
32
12
32
2
3112113131222132
32
311211
32
31122233
133132
31331232233322
4
4
2233116
6
=
−
−+−
−
−
−+−
+
−+−
−
−
++
+
−+−
+−−−+
W
KK
KK
K
KKKKKKKK
KK
KKKK
K
KKKKKKK
dz
Wd
K
K
K
KKKKKKKK
K
KKK
K
KKKK
KKK
KKKKKKK
dz
WdKKK
dz
Wd
Therefore, for simplification,
02
2
4
4
6
6
=+++mno
mn
o
mn
o
mn WCdz
WdB
dz
WdA
dz
Wd
where
−
−+−
−
−
−+−=
−+−
−
−
++
+
−+−=
−−−=
1332
1233
32
2
3112113131222132
1132
31121331
32
31331232233322
32
12
32
2
3112113131222132
32
311211
32
31122233
133132
313312
32233322
223311
KK
KK
K
KKKKKKKK
KK
KKKK
K
KKKKKKKC
K
K
K
KKKKKKKK
K
KKK
K
KKKK
KKK
KKKKKKKB
KKKA
o
o
o
Chapter 3 State Space Method of 3D Elasticity
80
Since
( )( )
−−++
++
=
−−
=
751
5
2
4
2
663
163
2
6
2
2
9
8
0
0
0
CηCξC
ηCηCξCξηCC
ξCξηCCηCξC
Band
ηξ
ηC
ξC
A
( ) ( )[ ] ( )
( )[ ] ( )
( ) ( )
+−−
++−++++
++−++++
=
==
2
5
2
1957817
2
6
2
4
2
32
6
2
49
2
56385
63
2
2
2
6391
2
6
2
28
2
1
333231
232221
131211
2
2
ηCξCCCCηCCCξ
ξCηC
ξCηξCηCCηCCCCCξη
CCη
ξCξCCCCξηηCξCCξC
KKK
KKK
KKK
A.BK
The numerical results for the above equation (3-26) can be referred to Wu [27]
where the solutions of a sixth order differential equation governing the
transverse displacement Wmn and stresses can be obtained. It is also
interesting to note that various expressions of the solutions are also provided.
3.4 Conclusions
The first section of this chapter introduces the basic introduction of state space
method. It represents a physical system in the form of mathematical model
where a set of input, output and state variables are related by a first order
differential equation.
The idea of expression of a spring-mass damper system as one of the
application of ‘state space’ is that when the initial conditions of the system are
known, i.e. external loading and displacements when time = 0, the subsequent
displacements can be determined at any given time. The importance expression
Chapter 3 State Space Method of 3D Elasticity
81
of such spring mass damper system which express in-terms of a linear time-
invariant function, makes it suitable for the analysis of the three dimensional
analysis of laminated plate. Applying this concept to the laminated composite
plate, the plate itself is divided firstly, into several layers. For this particular case,
when the external loading is applied to the surface of the plate, and knowing the
initial boundary conditions of the overall plate, the subsequent displacements
and stresses can be determined throughout the thickness of the plate. The
larger the number of sublayers within the plate, the more accurate solutions will
be produced. The great advantage of this concept for the application of
laminated plate is due to fact that as each layer of the laminated plate consists
of different material properties, the displacements and stresses can be found
simultaneously. There will be continuity at the interfaces of each layer which
means that the displacements will be exactly the same between the bottom
surfaces of the layer to the top surface of the adjacent layer. However, the in-
plane stresses will not be the same at the interfaces because of the different
elastic modulus of that layer.
The second part of this chapter reviews and rederives the governing equations
of elasticity which are essential equations for solving the stresses and
displacements of any laminated structures.
A simply supported orthotropic plate is shown in the last section of this chapter.
It is importance to understand this section thoroughly as it links to the following
chapter where a clamped edges laminated plate is presented. The relationship
is such that the boundary conditions of the simply supported plate are treated or
modified by using a traction in the form of mathematical function so that all
edges of the simply supported are satisfied in x and y directions.
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
82
Chapter 4
STATE SPACE SOLUTION OF
CLAMPED EDGES LAMINATE D
PLATE
The investigation of 3D elasticity using state space method has been increased
in the past two decades and the application of such approach has been widely
applied to various fields. One of the significant works was performed by Fan
and Sheng on exact elasticity solution of partially clamped thick laminate in the
early 1990’s. The exact solutions obtained from their works have inspired the
author to explore further studies on various boundary and loading conditions of
laminate plate. In this and the next chapters, the state space method for
clamped plates with various loading and boundary conditions are presented.
The exact solution approach consists of determining the displacements of
rectangular plate by setting a general expression of displacement field
according to the boundary and loading conditions. The displacement field is
introduced to the equations of equilibrium which are then solved. The solving
technique can be applied to various loading and boundary conditions. In this
chapter, clamped edges plate with uniformly distributed loading is considered.
State space methods together with state transfer matrix, and with the aid of
programming code, are presented in this chapter to investigate the plate
behavior with various boundary, loading conditions and material properties. The
objective is to determine the exact elasticity solution of orthotropic plate with
various parameters by analytical analysis. The idea is to analyse the fixed
edges plate simply by applying traction to the edges of a simply supported plate
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
83
by superposition principle of elasticity. This principle is further explained in the
following section.
Knowing and understanding programming code is highly essential and utmost
importance in this work as massive iteration process is involved in order to get
the solutions. In this case, one of the available programming program, namely,
Mathematica (version 8), is used to create the codes and run the iteration
process.
4.1 State Space Solution of a Single Layer Plate
To understand briefly the process involved in the state transfer matrix, a single
layer of plate is considered as shown in Figure 4.1. Suppose a rectangular plate
of length b, width a and uniform thickness h with all edges being clamped.
Figure 4.1: A single layer plate [52]
The boundary conditions of the clamped plate are:
x = 0 and a ; U = V = W = 0,
y = 0 and b; U = V = W = 0.
a
b
x , U , 1
z , W , 3
y , V , 2
h
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
84
In order to satisfy the boundary conditions for clamped plates, the expressions
(3-22) which show the displacements expression for a simply supported plate
have to be treated in such a way it will satisfy the behaviour of clamped edges
plate. The treatment considers the combination of the bending of a simply
supported single lamina subjected to external transverse loading and in-plane
normal tractions along the simply supported edges. These tractions cause the
in-plane displacements of the simply supported to be nullified, resulting no
deformation to occur at all edges along the x and y directions. In another way to
explain this phenomenon, the in-plane normal tractions superpose the in-plane
displacements caused by the external transverse loading of the simply
supported plate. Consequently, U and V are zero at all edges of the plate.
Therefore, when all edges of the simply supported plate become fixed, the plate
is then behaves like a fully clamped edges structure. This means that only in-
plane displacements, U and V along the edges of a simply supported plate are
treated to behave as a fully clamped structure.
Such treatment can be expressed mathematically as,
V
U
fVV
fUU
+=
+=
where
VandU are the assumed in-plane displacements for a simply supported, i.e.
( )
( )
( )∑∑
∑∑
∑∑
∞ ∞
∞ ∞
∞ ∞
=
=
=
m nmn
m nmn
m nmn
b
yπnSin
a
xπmSinzWW
b
yπnCos
a
xπmSinzVV
b
yπnSin
a
xπmCoszUU
(4-1)
fU and fV are the specified functions that suppressed the in-plane
displacements of the plate along the edges. U and V are the resulted in-plane
displacements of a clamped edges plate.
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
85
The above plate is subjected to a uniformly distributed loading on the top
surface and all edges are simply supported and tractions are applied along the
edges so that all the in-plane displacements are zero along the edges. The
motivation for such a treatment for the boundaries is based on an application of
the superposition principle of elasticity: taking the corresponding simply support
plate as a template released structure, then applying the relevant tractions
(which are unknown now) along the edges of the released structure, and finally
ensuring the necessary deformation compatibility conditions to be satisfied.
For a single lamina plate, the specified functions fu and fv, can be assumed to
be linear to achieve excellent approximations of the unknowns. Obviously, for
greater accuracy, the plate must be divided into many thin layers as described
in more details in the following section. Providing all the interfaces preserve
continuity conditions of the state variables, external transverse loading are
applied on the surfaces and satisfies the clamped edges boundary conditions,
the solution for the laminated plate can be determined by solving a system of
linear algebra equation.
The following case of laminated plate comprises of many thin layers after
division process. The treatment is similar to a single thick homogenous plate as
described above.
4.2 State Space Solution of Laminated Plate
For the case of laminated plate having different material properties at each layer,
the following illustration is shown.
Considering a laminated plate composed of a number of different material
laminae. For an arbitrary ply in the laminated plate with clamped edges: jth ply
(Figure 4.2), we can assume that jth ply with clamped edges is equivalent to the
simply supported jth ply by adding assumed displacements function (traction) to
the simply supported plate. By applying this traction, all in-plane displacements
at the simply supported plate edges would become nullified, i.e. clamped edges.
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
86
A single ply is considered first and the lamination theory is taken into account
afterwards by using transfer method and recursive formulation.
The laminated composite structure comprises of layers which are bonded
together at the interface and still maintains the continuity. In this study, the
method used to solve the state space equations with respect to variable z-
coordinate is state transfer matrix approach which produces an analytical
solution satisfying all boundary conditions throughout all plies of the structure.
The idea is to create the relationship between the top and bottom surfaces of
the plate based on 3D elasticity and state space method, applying the existing
boundary conditions so that the initial displacements U, V and W can be
determined. Only knowing the values of these displacements, all the unknowns
constants can be solved which in turn gives the final exact solutions in term of
stresses and displacement at any z-location across the thickness of the plate.
Figure 4.2: Geometry and coordinate systems of the laminate [52]
(1)
(2)
( j )
( N )
b
x ,U,1
z , W,3
y ,V,2
h
(j+1)
a
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
87
Consider the jth ply of the laminate as shown in Figure 4.2, adding the in-plane
displacement (traction) functions ( ) ( ) ( ) ( ) ( ) ( )zxVzyUzyUj
ajj
,,,,, 00 and
( ) ( ),, zxV bj
along all edges, the displacement functions of the plate are
assumed as
( )
( )z,y,xfVV
z,y,xfUU
j
Vjj
j
Ujj
+=
+= (4-2)
where jj
VandU are the displacements assumed for simply supported plate
and W remains the same as the stated in eqn.(3-22), and [30]
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )zxVygzxVygzyxf
zyUxfzyUxfzyxf
bjj
jV
ajj
jU
,,,,
,,,,
20
1
20
1
+=
+= (4-3)
So the displacements of the jth ply, after applying traction, are assumed as [30]
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )z,xVygz,xVygVz,y,xfVV
z,yUxfz,yUxfUz,y,xfUU
b
jjj
j
Vjj
a
jjj
j
Ujj
2
0
1
2
0
1
++=+=
++=+= (4-4)
where f1(x), f2(x), g1(y) and g2(y) can be of any functions.
In this study, it is assumed that
( ) ( ) ( ) ( )b
yyg
b
yyg
a
xxf
a
xxf =−==−=
212111
These functions are substituted into eqns. (4-4) and yield the in-plane
displacement as [30]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )z,xVb
yz,xV
b
yVV
z,yUa
xz,yU
a
xUU
b
jjjj
a
jjjj
+
−+=
+
−+=
0
0
1
1
(4-5)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
88
Let
( )
( )
( )j
m n
mn
mn
mn
b
yπnSin
a
xπmSinzZ
b
yπnCos
a
xπmSinzY
b
yπnSin
a
xπmCoszX
Z
Y
X
∑∑∞
=
∞
=
=
1 1
and
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )jm
b
m
mm
n
a
n
nn
a
xπmSinzVz,xV
a
xπmSinzVz,xV
b
yπnSinzUz,yU
b
yπnSinzUz,yU
b
a
=
=
=
=
∑
∑
∑
∑
0
0
0
0
(4-6)
To verify the boundary conditions of the clamped edges plate are satisfied,
substitute x = 0 and a and y = 0 and b into eqns. (4-1), (4-5) and (4-6), they all
show that U ,V and W are equal to 0 along the edges of the plate, hence they
satisfy the boundary conditions of a clamped edges plate.
By adding equation (4-4) to (3-12) of simply supported plate as discussed in the
previous chapter, the state equation can be deduced to [30]
( ){ } ( ){ } ( ){ } jmnjmnjjmn
zBzRDzRz
+=∂∂
(4-7)
The derivation expression of eqn.(4-7) is shown below,
By differentiating
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
89
( ) ( )
( ) ( )
( ) ( )
( ) ( )ygz
Vyg
z
V
z
V
z
V
VygVygVV
and
xfz
Uxf
z
U
z
U
z
U
UxfUxfUU
bo
b
ao
a
21
2
0
1
21
2
0
1
∂∂+
∂∂+
∂∂=
∂∂
++=
∂∂+
∂∂+
∂∂=
∂∂
++=
(4-8)
Substitute the above eqn.(4-8) into eqn.(3-12), give
( ) ( )
( ) ( )xfz
Uxf
z
UWαXC
z
U
WαXCxfz
Uxf
z
U
z
U
ao
ao
218
821
∂∂−
∂∂−−=
∂∂∴
−=∂
∂+∂
∂+∂∂
(4-9)
Similarly,
( ) ( )
( ) ( )ygz
Vyg
z
VWβYC
z
V
WβYCygz
Vyg
z
V
z
V
bo
bo
219
921
∂∂−
∂∂−−=
∂∂∴
−=∂
∂+∂
∂+∂∂
(4-10)
0+−−=∂∂
YβXαz
Z (4-11)
Substitute eqn.(4-8) into the fourth expression of eqn.(3-12), yields
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
90
( ) ( )
( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )
( ) ( ) ( ) ( )( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )[ ]b''
aa""
b''
a
a""
b
a
a
b
a
b
a
VαygVαygCC
UβxfUβxfCUxfUxfC
ZαCVαβCCUβCαCz
X
ZαCVygαCCVygαCC
VαβCCUxfUxfβC
UxfUxfCUβCαCz
X
ZαCVygαβCCVygαβCC
VαβCCUxfUxfβC
UxfUxfαCUβCαCz
X
ZαCVygVygαβCCVαβCC
UxfUxfβCαCUβCαCz
X
ZαCVygVygVαβCC
UxfUxfUβCαCz
X
ZαCVαβCCUβCαCz
X
20
163
2
2
02
162
0
12
163
2
6
2
2
12
63
0
163
632
0
1
2
6
2
0
12
2
6
2
2
1263
0
163
632
0
1
2
6
2
0
1
2
2
2
6
2
2
12
0
16363
2
0
1
2
6
2
2
2
6
2
2
12
0
163
2
0
1
2
6
2
2
163
2
6
2
2
++
−+−+−
+++−−−=∂∂∴
++−+
−+−+−
++−+−−=∂∂
++−+
−+−+−
++−+−−=∂∂
+++−+
−+−−+−−=∂∂
++++
−++−−=∂∂
++−−−=∂∂
(4-12)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
91
Substitute eqn.(4-8) into the fifth expression of eqn.(3-12), yields
( ) ( )
( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
( ) ( ) ( )( ) ( ) ( ) ( )( )bb""
a''
b""
b
a''
b
a
b
a
VαygVαygCVygVygC
UβxfUβxfCC
ZβCVβCαCUαβCCz
Y
ZβCVygVygC
VαygVαygCVβCαC
UβxfUβxfCCUαβCCz
Y
ZβCVygVygβCαCVβCαC
UxfUxfαβCCUαβCCz
Y
ZβCVygVygVβCαC
UxfUxfUαβCCz
Y
ZβCVβCαCUαβCCz
Y
2
2
02
162
0
14
2
0
163
5
2
4
2
663
52
0
14
2
2
02
16
2
4
2
6
2
0
16363
52
0
1
2
4
2
6
2
4
2
6
2
0
16363
52
0
1
2
4
2
6
2
0
163
5
2
4
2
663
+−+
−++−
+−−++−=∂∂∴
++
−+−−−
+++−+−=∂∂
++−−+−−
+++−+−=∂∂
+++−−
++++−=∂∂
+−−++−=∂∂
(4-13)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
92
Substitute eqn.(4-8) into the sixth expression of eqn.(3-12), yields
( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( ) ( )( )
( ) ( )( )( ) ( )( )b''
a''
b''a''
ba
ba
VygVygC
UxfUxfCZCVβCUαCz
W
ZC
VygVygCVβCUxfUxfCUαCz
W
ZC
VygVygβCVβCUxfUxfαCUαCz
W
ZCVygVygVβCUxfUxfUαCz
W
ZCVβCUαCz
W
2
0
15
2
0
11751
7
2
0
1552
0
111
7
2
0
1552
0
111
72
0
152
0
11
751
+
+++++=∂
∂∴
+
+++++=∂
∂
+
+++++=∂
∂
++++++=∂
∂
++=∂
∂
(4-14)
Rearrange eqns.(4-9) to (4-14) in matrix form gives,
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
93
( )( )
( ) ( )
( ) ( )
( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )[ ]
( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ]
+++
+−
+−++−
++
−+−+−
∂∂−
∂∂−
∂∂−
∂∂−
+
−−+−
+−−−−−
−−
=
∂∂
b''a''
b
b""a''
b''
aa""
bo
ao
VygVygCUxfUxfC
VαygVαygC
VygVygCUβxfUβxfCC
VαygVαygCC
UβxfUβxfCUxfUxfC
ygz
Vyg
z
V
xfz
Uxf
z
U
W
Y
X
Z
V
U
CβCαC
βCβCαCξαβCC
αCαβCCβCαCξ
ξβα
βC
αC
W
Y
X
Z
V
U
z
2
0
152
0
11
2
2
02
16
2
0
142
0
163
20
163
2
2
02
162
0
12
21
21
751
5
2
4
2
6
2
63
163
2
6
2
2
2
29
8
0
000
000
000
000
0000
0000
(4-15)
Or the above expression can be simplified as
B
W
Y
X
Z
V
U
.D
W
Y
X
Z
V
U
z+
=
∂∂
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
94
Substituting equations (4-1), (3-23) and (4-6) into (4-7), the following first order
non-homogeneous ordinary differential equation of jth ply can be determined for
each combination of m and n [30]:
( ){ } ( ){ } ( ){ }jmnjmnjjmn
zBzRDzRdz
d += (4-16)
where
( ){ } ( ) ( ) ( ) ( ) ( ) ( )[ ]Tjmnmnmnmnmnmn
jmnzWzYzXzZzVzUzR =
( )( )
−−++
++−−
−−
=
000
000
000
0000
0000
0000
751
52
42
663
1632
62
2
9
8
CCC
CCCCC
CCCCC
C
C
Dj
ηξηηξξη
ξξηηξηξ
ηξ
( ){ }
( ) ( )
( ) ( )[ ]
j
a
nn
a
nn
jmn
UUηC
dz
dU
dz
dU
zB
+
+−
=
0
02
10
0
2
1
02
6
0
when m = 0 and n ≠ 0 (4-17)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
95
( ){ }
( ) ( )
( ) ( )[ ]j
b
mm
b
mm
jmn
VVξC
dz
dV
dz
dV
zB
+
+−
=
02
10
0
2
1
0
02
6
0
when m ≠ 0 and n = 0 (4-18)
( ){ }
( )( ) ( )
( )( ) ( )
( ) ( ) ( )[ ] ( )( ) ( ) ( )[ ]
( )( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
−−−−−−
−−+−−+
−−+
+−−
−−−
−−−
=
b
mm
a
nn
b
mm
a
nn
b
mm
a
nn
b
mm
a
nn
jmn
VVπncosbπn
CUUπmcos
aπm
C
VVπncosπn
ξCUUπmcos
aπm
ηCC
VVπncosbπn
ξCCUUπmcos
πm
ηC
dz
dV
dz
dVπncos
πn
dz
dU
dz
dUπmcos
πm
zB
0501
0
22
2
6063
0630
22
2
6
0
22
0
22
12
12
12
12
12
12
0
12
12
when m ≠ 0 and n ≠ 0 (4-19)
( ){ } { }0=jmn
zB when m = 0 and n = 0 (4-20)
The derivation of eqns.(4-17) to (4-20) are explained in a very detail manner at
the end of this section.
Each lamina is divided into a number of layers to ensure that each layer is thin.
For each thin layer, the approximations solutions can be obtained by assuming
a linear relationship with respect to z. After applying the external loads on the
top surface of the plate, considering the boundary conditions and continuity
conditions of the state variables at the interfaces of these thin layers, the
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
96
solution for thick plate fully clamped along four edges is deduced by solving a
linear algebra equation system.
As a result, the laminated can be composed of many laminae which have
different material properties and each of the lamina is divided into a number of
layers to ensure that each layer is thin. Then the approximate solution obtained
can arbitrarily approach the exact three dimensional solution if the number of
layers and the sinusoidal displacement mode numbers m and n are sufficiently
large.
Taking a thin layer of thickness hj, the solution of non-homogenous eqn. (4-16)
is given by
( ){ } ( )[ ] ( ){ } ( ){ }jmnjmnjjmn
zCRzGzR += 0 (4-21)
where z = 0 (top surface) and z = h (bottom surface)
( ){ } ( ) ( ) ( ) ( ) ( ) ( )[ ]Tjmnmnmnmnmnmn
jmnzWzYzXzZzVzUzR =
( ){ } ( ) ( ) ( ) ( ) ( ) ( )[ ]Tjmnmnmnmnmnmn
jmnWYXZVUR 0000000 =
( )[ ][ ]( )zD
jjezG = , [G(z)]j is the transfer matrix of the homogeneous plate
( ){ } [ ]( )( ){ } ττ
τdBezC
jmn
zzD
jmnj
∫−
=0
Dj is the same as eqn.(4-16)
Notice that the non-homogeneous state eqn. (4-21) is analogous to the eqn.(3-7)
in terms of the through thickness z-coordinate, for this reason, the application of
state space method of spring-mass damper system is applied to the laminated
composite plate.
It is clearly notice that the non-homogenous vector ( ){ }jmn
zB contains the
unknown constants which need to be solved initially. To determine these
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
97
unknown constants, boundary conditions of the plate are applied to the state
equation of the clamped edges plate which consists of many thin layers for each
ply. Only when the thickness of these layers are thin enough, it can be assumed
that all the unknown functions are linearly distributed across the thickness of the
thin layers and subsequently through the thickness of the plate.
By considering the continuity between the bottom surface of jth layer and the
subsequent top surface of j+1th layer , the relationship of displacement functions
as shown in equation (4-5) can be expressed as [30]:
( ) ( )[ ] ( ) ( )
( ) ( )[ ] ( ) ( )
( ) ( )[ ] ( ) ( )
( ) ( )[ ] ( ) ( )
+
−=
+
−=
+
−=
+
−=
+
+
+
+
j
jb
jj
jb
jjj
b
m
j
j
jj
j
jjjm
j
ja
jj
ja
jjj
a
n
j
j
jj
j
jjjn
d
zB
d
zBzV
d
zB
d
zBzV
d
zA
d
zAzU
d
zA
d
zAzU
1
0
1
00
1
0
1
00
1
1
1
1
(4-22)
where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )bjj
ajj
bjj
ajj
BandBAAandBBAA1
011
01
00 ,,,,,++++
are the end
unknown values of linear functions at the edges of the plate at jth and j+1th
layers, respectively and zj and dj are the local z-coordinate and thickness of the
jth layer, respectively.
Substitute eqn.(4-22) into (4-17), (4-18) and (4-19) [30]:
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
98
( ){ }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )
j
a
j
a
jjjj
ja
jj
j
a
j
a
jjj
jmn
AAAAd
zAAηC
d
AAAA
zB
−+−++
−+−
=
++
++
0
0
2
1
0
0
2
1
1
00
1
02
6
1
0
1
0
(4-23)
when m = 0 and n ≠ 0
( ){ }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )j
b
j
b
jjjj
jb
jj
j
b
j
b
jjj
jmn
BBBBd
zBBζC
d
BBBB
zB
−+−++
−+−
=
++
++
0
2
1
0
0
2
1
0
1
00
1
02
6
1
0
1
0
(4-24)
when m ≠ 0 and n = 0
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
99
( ){ }
( )( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( )
( )
( ) ( )
( ) ( ) ( ) ( )( )
( )( )
( ) ( )
( ) ( ) ( ) ( )( )
( )( )
( ) ( )
( ) ( ) ( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) ( )( )
( )
( ) ( )
( ) ( ) ( ) ( )( )j
j
b
jj
b
jj
j
b
jj
j
a
jj
a
jj
j
a
jj
j
b
jj
b
jj
j
b
jj
j
a
jj
a
jj
j
a
jj
j
b
jj
b
jj
j
b
jj
j
a
jj
a
jj
j
a
jj
j
b
j
b
jjj
j
a
j
a
jjj
jmn
BBBBd
z
BB
πCosnbπn
C
AAAAd
z
AA
πCosmaπm
C
BBBBd
z
BB
πCosnπn
ξC
AAAAd
z
AA
πCosmaπm
ηCC
BBBBd
z
BB
πCosnbπn
ξCC
AAAAd
z
AA
πCosmπm
ηC
d
BBBBπCosn
πn
d
AAAAπCosm
πm
zB
−−+
+−
−
−
−−+
+−
−−
−−+
+−
−
+
−−+
+−
−+
−−+
+−
−+
+
−−+
+−
−
−+−−−
−+−−−
=
++
++
++
++
++
++
++
++
0
1
0
1
0
5
0
1
0
1
0
1
0
1
0
1
0
22
2
6
0
1
0
1
0
63
0
1
0
1
0
63
0
1
0
1
0
22
2
6
1
00
1
22
1
00
1
22
12
12
12
12
12
12
0
12
12
(4-25) when m ≠ 0 and n ≠ 0
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
100
Following the recursive formulation and applying the boundary conditions, we
assume that all layers are interconnected to each other by bonding the
interfaces to preserve continuity i.e. the state vectors at the bottom of the jth
layer is the same as those to the top of subsequent layer, j+1th layer.
For instance, let zj = dj (the thickness of the layer) at layer jth,
( ){ } ( )[ ] ( ){ } ( ){ }jmnjmnjjjjmn
dCRdGdR += 0 (4-26)
For the subsequent j+1th layer, the state vector is expressed as
( ){ } ( )[ ] ( ){ } ( ){ }111111
0++++++
+=jmnjmnjjjjmn
dCRdGdR (4-27)
By virtue of continuity conditions at the interface between the layers, this gives
( ){ } ( ){ }1
0+
=jmnjjmn
RdR (4-28)
Therefore,
( ){ } ( )[ ] ( )[ ] ( ){ } ( ){ }( ){ }
1
11110
+
+++++
+=
jmn
jmnjmnjjjjjjmn
dC
dCRdGdGdR (4-29)
Eqn.(4-29) can summarized as
( ){ } [ ] ( ){ } [ ]Π+Π=+
01 mnjmn
RdR (4-30)
where
( )[ ] ( )[ ]( )[ ] ( ){ } ( ){ }
111
11
+++
++
+=Π
=Π
jmnjmnjj
jjjj
dCdC.dG
dG.dG
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
101
Repeating the above eqns.(4-26), (4-27) and (4-28) from the top surface to the
bottom surface of the laminate, the relationship of the top surface through the
bottom surface of the laminate can be obtained by using recursive formulation.
The non homogeneous vector Π contains the boundary unknown coefficients,
.BandB,A,A b
j
o
j
a
j
o
j
After doing the process of recursive formulation from the top surface, i.e. z = 0
through the bottom surface of the plate at z = h, we can get the expression
collectively as,
( )( )( )( )( )( )
( )( )( )( )( )( )
ΠΠΠΠΠΠ
+
ΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠΠ
=
6
5
4
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
0
0
0
0
0
0
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
mn
W
Y
X
Z
V
U
hW
hY
hX
hZ
hV
hU
(4-31)
Knowing the boundary and loading conditions of the laminated plate, the
relationship of top and bottom surfaces of the plate can now be established for
a uniformly distributed loading, as follows,
( ) ( ) ( ) ( )
( ) ( ) ( ) 0000
53116
000002
===
∞=−===
mnmnmn
mnmnmn
Z;hY;hX
............,,n,m,πmn
qZ;Y;X
(4-32)
Substitute eqns.(4-32) into the third, fourth and fifth eqns.(4-31), the top surface
displacements can be solved initially, i.e.
( )( )( )
ΠΠΠ
−
ΠΠΠ
ΠΠΠΠΠΠΠΠΠ
=
−
5
4
3
53
43
33
2
1
565251
464241
363231 16
0
0
0
πmn
q.
W
V
U
mn
mn
mn
(4-33)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
102
Therefore, the initial vector of ( ){ }0mn
R must be determined as this vector give
the initial displacement vector of ( ) ( ) ( )000mn
mnmn WandV,U . Considering
the first and second rows of eqn.(4-31), we can get the relationship of the state
vectors of Umn(z) and Vmn(z) in terms of ( ) ( ) ( )000mn
mnmn WandV,U at any
layers of the plate, i.e.
( )( )
( ) ( ) ( )( ) ( ) ( )
( )( )( )
( ) ( ) ( )( ) ( ) ( )
( )( )( )
( )( )
ΠΠ+
ΠΠΠΠΠΠ
+
ΠΠΠΠΠΠ
=
z
z
Y
X
Z
zzz
zzz
W
V
U
.zzz
zzz
zV
zU
mn
mn
mn
mn
mn
mn
mn
mn
1
1
252423
151413
262221
161211
0
0
0
0
0
0
(4-34)
Substitute eqn.(4-33) into (4-34) and considering eqn. (4-32), gives
( )( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( )( )( )
( )( )
ΠΠ+
ΠΠΠΠΠΠ
+
ΠΠΠ
−
ΠΠΠ
ΠΠΠΠΠΠΠΠΠ
ΠΠΠΠΠΠ
=
−
z
z
Y
X
Z
zzz
zzz
πmn
q
.
.zzz
zzz
zV
zU
mn
mn
mn
mn
mn
1
1
252423
151413
5
4
3
53
43
33
2
1
565251
464241
363231
262221
161211
0
0
0
16
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
103
( )( )
( ) ( ) ( )( ) ( ) ( )
( )( )
( )( )
ΠΠ+
ΠΠ
−
ΠΠΠ
−
ΠΠΠ
ΠΠΠΠΠΠΠΠΠ
ΠΠΠΠΠΠ
=
⇒
−
z
zz
z
πmn
q
πmn
q
.
.zzz
zzz
zV
zU
mn
mn
1
1
23
132
5
4
3
53
43
33
2
1
565251
464241
363231
262221
161211
16
16
(4-35)
To solve these ( ) ( )zVandzU mnmn which contains the unknowns, they must
be solved simultaneously within all the layers of the plate as further explained
below.
The following boundary conditions for clamped edges should be satisfied,
bandywhenWVU
aandxwhenWVU
00
00
========
(4-36)
Therefore, considering equations (4-1), (4-5), (4-6) and (4-36), we can easily
established the following expressions, [30]
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )∑∑ ∑
∑∑ ∑
∑∑ ∑
∑∑ ∑
=
+
−=
=
+
=
=
+
−=
=
+
=
m n m
b
mmn
nb
z
m n mm
mnz
m n n
a
nmn
ma
z
m n nn
mnz
a
xπmSinzV
a
xπmSinzVV
a
xπmSinzV
a
xπmSinzVV
b
yπnSinzU
b
yπnSinzUU
b
yπnSinzU
b
yπnSinzUU
01
0
01
0
00
00
(4-37)
By simplifying the above eqns.(4-37), for each m and n, the above equations
yields [30]
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
104
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∑
∑ ∑
∑
∑ ∑
∑
∑ ∑
∑
∑ ∑
=−+⇒
=
−+
=+⇒
=
+
=−+⇒
=
−+
=+⇒
=
+
nmn
nb
m
m nmn
nb
m
nmn
m
m nmn
m
mmn
ma
n
n mmn
ma
n
mmn
n
n mmn
n
zVzV
a
xπmSinzVzV
zVzV
a
xπmSinzVzV
zUzU
b
yπnSinzUzU
zUzU
b
yπnSinzUzU
01
01
0
0
01
01
0
0
0
0
0
0
(4-38)
Finally, we can solved Umn(z) and Vmn(z) with the relationship of eqns.(4-22) and
(4-38) at all layers across the thickness of the laminated plate. Consequently, all
the unknowns ( ) ( ) ( ) ( )bjj
ajj
BBAA ,,, 00 can be determined for each sub-layer by
substituting the displacements ( ) ( ) ( ) ( )bmm
ann
VandVUU 00 ,, from eqn.(4-22) into
(4-38). This can be achieved by forming the algebraic equations that can be
solved with the same number of unknown coefficients.
It is also interesting to note that when the applied loading is symmetric, we can
have further relationship such that ( ) ( ) ( ) ( )z,yUz,yU a
nn−=0
and ( ) ( ) ( ) ( )z,xVz,xV b
mm−=0 which in turn decreases the number of unknowns
and reduces the number of algebraic equations. This means that only the first
and third expressions on eqns.(4-38) are required to determine these unknowns.
After considering the applied external loading, boundary conditions and the
continuity conditions at the interfaces of the laminated plate, the analytical
solution of the plate with clamped edges is now formulated in this section by a
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
105
set of linear algebra equations which can be solved by any of mathematical
tools such as Mathematica, Matlab or Fortran etc. The final result would then
depend on the number of thin layers and the sinusoidal displacement mode
numbers of m and n. The larger the number of thin layers and of m and n, the
greater the accuracy of the exact three dimensional elasticity solution. In this
study the values of m = 1,3,5,7,9,…..99 and n = 1,3,5,7,9,…..29 are considered.
The results of this study will be verified with the numerical analysis which are
presented in the following chapter.
The derivation of non-homogeneous vector B mn(z)
In this section, the derivation of eqns.(4-17) to (4-20) are shown clearly. The
method used consists of an expansion of a function f(x) in terms of an infinite
sum of sines and cosines, which is known as Fourier series.
Generally, the Fourier series of a function f(x) is defined as [53],
( )
+
+= ∑∑∞
=
∞
= a
xπmSinb
a
xπmCosaaxf
mm
mmo
112
1 (4-39)
If f(x) is odd, then the series is reduced to a Fourier sine series, i.e.
( )
= ∑∞
= a
xπmSinbxf
mm
1
(4-40)
If f(x) is even, then the series is reduced to a Fourier cosine series, i.e.
( )
+= ∑∞
= a
xπmCosaaxf
mmo
12
1 (4-41)
Since
( )
−=a
xxf 1
1 , ( )
a
xxf =
2 (4-42)
and
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
106
( ) ( )
( ) ( )
( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )[ ]
( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ]
+++
+−
+−++−
++
−+−+−
−−
−−
=
b''a''
b
b""a''
b''
aa""
bo
ao
mn
VygVygCUxfUxfC
VαygVαygC
VygVygCUβxfUβxfCC
VαygVαygCC
UβxfUβxfCUxfUxfC
ygdz
dVyg
dz
dV
xfdz
dUxf
dz
dU
B
2
0
152
0
11
2
2
02
16
2
0
142
0
163
20
163
2
2
02
162
0
12
21
21
0
(4-43)
Integrate and expand f1(x) using Fourier series, i.e.,
( )
( )( )[ ] ( ) ( )
( )[ ] ( ) ( )
++=
−
++=
−
+
=
−
+=
−
+=
−=
∫ ∑∫∫
∫ ∑∫
∑
∞
=
∞
=
∞
=
πm
πmSinπma
πm
a.πmSin
πm
πmCos
πm
πmSinπma.a
πm
a.πmSin.a
πm
πmCosa
dxa
xπmCos
a
xπmCosadx
a
xπmCosadx
a
xπmCos
a
x
dxa
xπmCos
a
xπmCosaadx
a
xπmCos
a
x
a
xπmCosaa
a
xxf
m
m
a
mm
aa
a
mm
a
mm
4
221
4
221
1
1
1
022
022
0 100
0
0 10
0
101
( )( )22
0
120
2
10
πm
πmCosa:mwhen
a:mwhen
m
−=≠
=→ (4-44)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
107
Now repeat the above procedures for f2(x), i.e.
( )
( )( ) ( )[ ] ( ) ( )[ ]
πm
b.πmSinπma
πm
b.πmSin.a
πm
πmSinπmπmCosa
dxa
xπmCos
a
xπmCosbdx
a
xπmCosbdx
a
xπmCos
a
x
dxa
xπmCos
a
xπmCosbbdx
a
xπmCos
a
x
a
xπmCosbb
a
xxf
m
a
mm
aa
a
mm
a
mm
4
221 022
0 100
0
0 10
0
102
++=
++−
+
=
+=
+=
=
∫ ∑∫∫
∫ ∑∫
∑
∞
=
∞
=
∞
=
( )[ ]πmCosπm
b:mwhen
b:mwhen
m−−=≠
=→
12
0
2
10
22
0 (4-45)
Therefore, taking the first row of eqn.(4-43),
( )[ ] ( ) ( )xfdz
dUxf
dz
dUzB
a
n
o
n
mn 211−−=
when m = 0, n ≠ 0 :
( )[ ]
+−=
−
−=dz
dU
dz
dU
dz
dU
dz
dUzB
a
n
o
n
a
n
o
n
mn 2
1
2
1
2
11
(4-46)
when m ≠ 0, n = 0 : ( )[ ] 01
=zBmn
(4-47)
when m ≠ 0, n ≠ 0:
( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )[ ]
−−−=
−−−
−−=
dz
dU
dz
dUπmCos
πmzB
πmCosπmdz
dUπmCos
πmdz
dUzB
a
n
o
n
mn
a
n
o
n
mn
12
12
12
221
22221
(4-48)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
108
Similarly, repeat the above steps for the case of g1(y) and g2(y), as follows:
Integrate and expand g1(y) using Fourier series, i.e.,
( )
( )( )[ ] ( ) ( )
( )[ ] ( ) ( )
++=
−
++=
−
+
=
−
+=
−
+=
−=
∫ ∑∫∫
∫ ∑∫
∑
∞
=
∞
=
∞
=
πn
πnSinπna
πn
a.πnSin
πn
πnCos
πn
πnSinπnb.a
πn
a.πnSin.b
πn
πnCosb
dyb
yπnCos
b
yπnCosady
b
yπnCosady
b
yπnCos
b
y
dyb
yπnCos
b
yπnCosaady
b
yπnCos
b
y
b
yπnCosaa
b
yyg
n
n
b
nn
bb
b
nn
b
nn
4
221
4
221
1
1
1
022
022
0 100
0
0 10
0
101
( )( )22
0
120
2
10
πn
πnCosa:nwhen
a:nwhen
n
−=≠
=→ (4-49)
Now repeat the above procedures for g2(y), i.e.
( )
( )( ) ( )[ ] ( ) ( )[ ]
πn
b.πnSinπnb
πn
b.πnSin.b
πn
πnSinπnπnCosb
dyb
yπnCos
b
yπnCosbdy
b
yπnCosbdy
b
yπnCos
b
y
dyb
yπnCos
b
yπnCosbbdy
b
yπnCos
b
y
b
yπnCosbb
b
yyg
n
b
nn
bb
b
nn
b
nn
4
221 022
0 100
0
0 10
0
102
++=
++−
+
=
+=
+=
=
∫ ∑∫∫
∫ ∑∫
∑
∞
=
∞
=
∞
=
( )[ ]πnCosπn
b:nwhen
b:nwhen
n−−=≠
=→
12
0
2
10
22
0 (4-50)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
109
Therefore, taking the second row of eqn.(4-43),
( )[ ] ( ) ( )ygdz
dVyg
dz
dVzB
b
m
o
m
mn 212−−=
when m = 0, n ≠ 0 : ( )[ ] 02
=zBmn
(4-51)
when m ≠ 0, n = 0 :
( )[ ]
+−=
−
−=dz
dV
dz
dV
dz
dV
dz
dVzB
b
m
o
m
b
m
o
m
mn 2
1
2
1
2
12
(4-52)
when m ≠ 0, n ≠ 0:
( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )[ ]
−−−=
−−−
−−=
dz
dV
dz
dVπnCos
πnzB
πnCosπndz
dVπnCos
πndz
dVzB
b
m
o
m
mn
b
m
o
m
mn
12
12
12
222
22222
(4-53)
From the third row of eqn.(4-43),
( )[ ] 03
=zBmn
(4-54)
Now consider the fourth row of eqn.(4-43), i.e.
( )[ ] ( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( )[ ]b''
aa""
mn
VαygVαygCC
UβxfUβxfCUxfUxfCzB
2
0
163
2
2
02
162
0
124
++
−+−+−=
(4-55)
As
( )
( )
( ) 0
1
1
1
1
1
=
−=
−=
xf
axf
a
xxf
"
' and
( )
( )
( ) 0
1
2
2
2
=
=
=
xf
axf
a
xxf
"
' (4-56)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
110
( )
( )
( ) 0
1
1
1
1
1
=
−=
−=
yg
byg
b
yyg
"
' and
( )
( )
( ) 0
1
2
2
2
=
=
=
yg
byg
b
yyg
"
' (4-57)
Substitute eqns.(4-63) and (4-64) into (4-62), gives
( )[ ]( ) ( )
( ) ( )
( )
( )
( ) ( ) ( )
( )[ ] ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( )[ ] ( )
( )( ) ( ) ( )
+
−+
−
+
−=
+
−+
−
−
+
−
−−=
+
−+
−
+
−
−
+
−=
∑∑
∑∑
∑∑
∑∑
∑∑
∑
∑
∑
∑
m
b
mm
o
m
n
a
nn
o
nmn
m
b
mm
o
m
n
a
nn
o
nmn
m
b
mm
o
m
n
a
n
n
o
n
n
a
n
n
o
n
mn
xξCosVb
xξCosVb
ξCC
b
yπnSinU
a
x
b
yπnSinU
a
xηCzB
xξCosVξb
xξCosVξb
CC
yηSinUηa
xyηSinUη
a
xCzB
xξSinVdx
d
bxξSinV
dx
d
bCC
yηSinUdy
d
a
x
yηSinUdy
d
a
x
C
yηSinU
yηSinU
CzB
11
1
11
1
11
1
0
0
63
2
64
63
22
64
63
2
2
2
2
624
(4-58)
Consider the first part of eqn.(4-58), i.e.
( )[ ] ( )
+
−= ∑∑n
a
nn
o
nmn b
yπnSinU
a
x
b
yπnSinU
a
xηCzB 12
64
( )[ ] ( ) ( ) ( )
+
−= ao
mnU
a
xU
a
xηCzB 12
64 (4-59)
Now, take the second part of eqn.(4-58), i.e.
( )[ ] ( )( ) ( ) ( )
+
−+−= ∑∑m
b
mm
o
mmnxξCosV
bxξCosV
bξCCzB
11634
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
111
Applying the Fourier series of
( ) ( )( ) ( )[ ] ( )
n
b
nn
b
b
nn
b
nn
aπn
πnSinb
πn
aπnCosb
πn
πnCos
dyb
yπnSin
b
yπnSinady
b
yπnSina
πn
πnCos
dyb
yπnSin
b
yπnSinaady
b
yπnSin
b
b
yπnSinaa
b
−+−
=−
+
=−
+=
+=
∫ ∑∫
∫ ∑∫
∑
∞
=
∞
=
∞
=
2
21
2
11
1
1
1
0
0 100
0 10
0
10
(4-60)
( )[ ]( )[ ]
( )[ ]bπn
πnCos
πnSinbπn
πnCosa:nwhen
ba:nwhen
n
−=
−−
=≠
=→
12
21
120
10
0 (4-61)
By using eqn. (4-44), we can get
when m = 0, n ≠ 0 :
( )[ ] ( ) ( ) ( )( )[ ] ( ) ( ) ( )[ ]a
n
o
nmn
a
n
o
nmn
UUηCzB
UUηCzB
+
=
+
=
2
64
2
64
2
12
1
2
1
(4-62)
when m ≠ 0, n = 0 :
( )[ ] 04
=zBmn
(4-63)
By using eqns. (4-44), (4-45) and (4-61), we can get
when m ≠ 0, n ≠ 0 :
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
112
( )[ ] ( ) ( ) ( ) ( )( )( )( )
( )[ ] ( ) ( )[ ]( ) ( )[ ]( )
( )( ) ( )[ ] ( ) ( )[ ] ( )
( )[ ] ( )[ ] ( ) ( )[ ]( )
( )[ ] ( ) ( )[ ]b
m
o
m
a
n
o
nmn
b
m
o
m
a
n
o
nmn
b
m
o
ma
n
o
nmn
VVπnCosbπn
ξCC
UUπmCosπm
ηCzB
Vbπn
πnCosV
bπn
πnCosξCC
UπmCosπm
UπmCosπm
ηCzB
Vb
Vb
ξCCUa
xU
a
xηCzB
−−+
+−−=∴
−+
−−+
−
−−−=
+
−+−
+
−=
12
12
1212
12
12
1
1
1
63
22
2
6
4
63
2222
2
64
63
2
64
(4-64)
Now consider the fifth row of eqn.(4-43), i.e.
( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]b
b""a''
mn
VαygVαygC
VygVygCUβxfUβxfCCzB
2
2
02
16
2
0
142
0
1635
+−
+−++−=
(4-65)
Substitute eqns.(4-56) and (4-57) into (4-65), gives
( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ]
( )[ ] ( ) ( ) ( )
( ) ( )[ ]( )
( )
+
−−+−
+
−+−=
+−
+−++−=
∑
∑
∑∑
m
b
m
m
o
mb
n
a
nn
o
nmn
b
b""a''
mn
xξSinVdx
d
b
y
xξSinVdx
d
b
y
CVVC
yηSinUdy
d
ayηSinU
dy
d
aCCzB
VαygVαygC
VygVygCUβxfUβxfCCzB
2
2
2
2
6
0
4
635
2
2
02
16
2
0
142
0
1635
1
00
11
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
113
( )[ ] ( ) ( ) ( )
( ) ( )
( )[ ] ( ) ( ) ( )
( ) ( )
+
−+
+
−+−=
+
−−
+
−+−=
∑∑
∑∑
∑∑
∑∑
m
b
mm
o
m
n
a
nn
o
nmn
m
b
mm
o
m
n
a
nn
o
nmn
xξSinVξb
yxξSinVξ
b
yC
yηCosUηa
yηCosUηa
CCzB
xξSinVdx
d
b
yxξSinV
dx
d
b
yC
yηSinUdy
d
ayηSinU
dy
d
aCCzB
22
6
635
2
2
2
2
6
635
1
11
1
11
( )[ ] ( ) ( ) ( )
( ) ( )
+
−+
+
−+−=
∑∑
∑∑
m
b
mm
o
m
n
a
nn
o
nmn
xξSinVb
yxξSinV
b
yξC
yηCosUa
yηCosUa
ηCCzB
1
11
2
6
635
Therefore
when m = 0, n ≠ 0: ( )[ ] 05
=zBmn
(4-66)
when m ≠ 0, n = 0: from eqn.(4-49) and (4-50),
( )[ ] ( ) ( )( )[ ] [ ]b
m
o
mmn
b
m
o
mmn
VVξCzB
VVξCzB
+=
+
+=
2
65
2
65
2
12
1
2
10
(4-67)
when m ≠ 0, n ≠ 0: from eqns.(4-44), (4-45), (4-49) and (4-50),
( )[ ] ( ) ( )[ ] ( )( )
( )[ ] ( )[ ]
( )[ ] ( ) ( )[ ][ ] ( )[ ][ ]b
m
o
m
a
n
o
nmn
b
m
o
m
a
n
o
nmn
VVπnCosπn
ξCUU
aπm
πmCosηCCzB
VπnCosπn
VπnCosπn
ξC
Uaπm
πmCosU
aπm
πmCosηCCzB
−−+−−+
=
−−−+
−+
−−+−=
1212
12
12
1212
22
2
663
5
2222
2
6
635
(4-68)
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
114
Now, take the final sixth row of eqn.(4-43):
( )[ ] ( ) ( )[ ] ( ) ( )[ ]( )[ ]
+
−+
+
−=
+++=
ba
mn
b''a''
mn
Vb
Vb
CUa
Ua
CzB
VygVygCUxfUxfCzB
1111 0
5
0
16
2
0
152
0
116 (4-69)
Therefore
when m = 0, n ≠ 0: ( )[ ] 06
=zBmn
(4-70)
when m ≠ 0, n = 0: ( )[ ] 06
=zBmn
(4-71)
when m ≠ 0, n ≠ 0: using eqn.(4-61),
( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )[ ]
−+
−−
+
−+
−−=
b
m
o
m
a
n
o
nmn
Vbπn
πnCosV
bπn
πnCosC
Uaπm
πmCosU
aπm
πmCosCzB
1212
1212
5
16
( )[ ] ( )[ ][ ] ( )[ ][ ]b
m
o
m
a
n
o
nmnVVπnCos
bπn
CUUπmCos
aπm
CzB −−−−−−= 1
21
251
6
(4-72)
Re-arrange the above expressions of eqns.(4-46) to (4-48), (4-51) to (4-53), (4-
54), (4-62) to (4-64) and (4-70) to (4-72), the non-homogeneous vectors Bmn(z)
are given by,
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
115
( ){ }
( ) ( )
( ) ( )[ ]
j
a
nn
a
nn
jmn
UUηC
dz
dU
dz
dU
zB
+
+−
=
0
02
10
0
2
1
02
6
0
m = 0, n ≠ 0
( ){ }
( ) ( )
( ) ( )[ ]j
b
mm
b
mm
jmn
VVξC
dz
dV
dz
dV
zB
+
+−
=
02
10
0
2
1
0
02
6
0
m ≠ 0, n = 0
( ){ }
( )( ) ( )
( )( ) ( )
( ) ( ) ( )[ ] ( )( ) ( ) ( )[ ]
( )( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]j
b
mm
a
nn
b
mm
a
nn
b
mm
a
nn
b
mm
a
nn
jmn
VVπCosnbπn
CUUπCosm
aπm
C
VVπCosnπn
ξCUUπCosm
aπm
ηCC
VVπCosnbπn
ξCCUUπCosm
πm
ηC
dz
dV
dz
dVπCosn
πn
dz
dU
dz
dUπCosm
πm
zB
−−−−−−
−−+−−+
−−+
+−−
−−−
−−−
=
0501
0
22
2
6063
0630
22
2
6
0
22
0
22
12
12
12
12
12
12
0
12
12
m ≠ 0, n ≠ 0
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
116
4.3 Application of State Space Method to Laminated Plate
In order to clarify the formulations as stated in the previous section, the
following example of the application of state space method to laminated plate is
shown for clarity of the method being used. In this section, the formation of
algebraic equations are illustrated in the detailed manner for clarification
purpose, however, in order to solve the equations, they require an advance
computer programming software since they involve massive matrix problem
calculation and iteration processes. They are many different types of
programming softwares available such as Matlab, Fortran and so on. In this
study the latest version of Mathematica version 8 is used.
Consider a laminated plate subjected to uniformly distributed loading at the top
surface of the plate as shown in Figure 4.3. The plate consists of 3 plies in
which ply 1 is the same as ply 3, i.e. same material properties and thickness of
d1 with comprised of 3 no. of sublayers. For ply 2, there are 10 no. of sublayers
having a thickness of d2 each. The relationship of material properties between
ply1 and ply 2 is in the ratio of 2
11
1
11ply
ply
C
C. The overall thickness of the plate is h and
ply1 is equal to ply3 = 0.1h while ply 2 thickness is 0.8h.
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
117
Figure 4.3: View of a clamped edges laminated plate
For ply 1: the thickness of each sublayer is d1,
Sublayer 1: when z = 0, the unknown coefficient is A1
Sublayer 2: when z = 0, the unknown coefficient is A2
Sublayer 3: when z = 0, the unknown coefficient is A3
For ply 2: the thickness of each sublayer is d2,
Sublayer 4: when z = 0, the unknown coefficient is A4
Sublayer 5: when z = 0, the unknown coefficient is A5
Sublayer 6: when z = 0, the unknown coefficient is A6
Sublayer 7: when z = 0, the unknown coefficient is A7
Sublayer 8: when z = 0, the unknown coefficient is A8
Sublayer 9: when z = 0, the unknown coefficient is A9
Sublayer 10: when z = 0, the unknown coefficient is A10
Sublayer 11: when z = 0, the unknown coefficient is A11
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
118
Sublayer 12: when z = 0, the unknown coefficient is A12
Sublayer 13: when z = 0, the unknown coefficient is A13
For ply 3: the thickness of each sublayer is d1,
Sublayer 14: when z = 0, the unknown coefficient is A14
Sublayer 15: when z = 0, the unknown coefficient is A15
Sublayer 16: when z = 0, the unknown coefficient is A16
Sublayer 16: when z = d1, the unknown coefficient is A17
Because of continuity and they are perfectly bonded at the interfaces, the
relationship of displacement functions as stated in eqns.(4-22), the following
unknown coefficients can be established at x = 0,
Since ( ) ( )[ ] ( ) ( )
+
−=
+j
j
jj
j
jjjn d
zA
d
zAzU 0
1
00 1
( )[ ]
( )[ ]
( )[ ] 0
31
0
41
0
33
0
0
21
0
31
0
22
0
0
11
0
21
0
11
0
00103
00102
00101
01
Ad
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AU:sublayer
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AU:sublayer
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AU:sublayer
zwhen:PlyFor
n
n
n
=
+
−=
=
+
−=
=
+
−=
=
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
119
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ]
( )[ ] 0
132
0
142
0
1313
0
0
122
0
132
0
1212
0
0
112
0
122
0
1111
0
0
102
0
112
0
1010
0
0
92
0
102
0
99
0
0
82
0
92
0
88
0
0
72
0
82
0
77
0
0
62
0
72
0
66
0
0
52
0
62
0
55
0
0
42
0
52
0
44
0
001013
001012
001011
001010
00109
00108
00107
00106
00105
00104
02
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zwhen:PlyFor
n
n
n
n
n
n
n
n
n
n
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
120
( )[ ]
( )[ ]
( )[ ]
( )[ ] 0
171
10
171
10
16171
0
0
161
0
171
0
1616
0
0
151
0
161
0
1515
0
0
141
0
151
0
1414
0
1
116
001016
001015
001014
03
Ad
dA
d
dAdU:sublayer
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dorzwhen:Ply
n
n
n
n
=
+
−=
=
+
−=
=
+
−=
=
+
−=
=
Now, applying the first algebraic expression as stated in eqn.(4-38), i.e.
( ) ( ) ( )
( )
( )
( ) 003
002
001
1
0
3
0
3
2
0
2
1
0
1
0
=
+
=
+
=
+
=
+
∑
∑
∑
∑
mmn
mmn
mmn
jmmn
n
UA:sublayer
UA:sublayer
UA:sublayer
:PlyFor
zUzU
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
121
( )
( )
( )
( )
( )
( )
( )
( )
( )
( ) 0013
0012
0011
0010
009
008
007
006
005
004
2
13
0
13
12
0
12
11
0
11
10
0
10
9
0
9
8
0
8
7
0
7
6
0
6
5
0
5
4
0
4
=
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
=
+
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
mmn
mmn
mmn
mmn
mmn
mmn
mmn
mmn
mmn
mmn
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
:PlyFor
( )
( )
( )
( ) 016
0016
0015
0014
3
171
0
17
16
0
16
15
0
15
14
0
14
=
+
=
+
=
+
=
+
∑
∑
∑
∑
mmn
mmn
mmn
mmn
dUA:sublayer
UA:sublayer
UA:sublayer
UA:sublayer
:Ply
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
122
Now, applying the first expression as stated in eqn.(4-35) into the above
expression, i.e.
[ ]
[ ]
[ ]0
1616
3
0
1616
2
0
1616
1
1
3
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
3
2
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
2
1
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
1
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
:PlyFor
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
123
[ ]
[ ]
[ ]
[ ]0
1616
7
0
1616
6
0
1616
5
0
1616
4
2
7
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
7
6
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
6
5
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
5
4
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
4
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
:PlyFor
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
124
[ ]
[ ]
[ ]
[ ]0
1616
11
0
1616
10
0
1616
9
0
1616
8
11
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
11
10
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
10
9
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
9
8
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
8
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
125
[ ]
[ ]0
1616
13
0
1616
12
13
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
13
12
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
12
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
Π
Π
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
[ ]
[ ]0
1616
15
0
1616
14
3
15
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
15
14
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
14
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
:PlyFor
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
126
[ ]
[ ]0
1616
61
0
1616
61
17
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
17
16
1132
5
4
3
53
43
33
2
1
565251
464241
363231
161211
0
16
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
=
Π+Π−
ΠΠ
Π
−
ΠΠΠ
×
ΠΠΠΠΠΠΠΠΠ
ΠΠΠ
+
−
−
πmn
q
πmn
q
A:sublayer
πmn
q
πmn
q
A:sublayer
From the above expressions, it clearly shows that there are 17 unknown
coefficients with 17 available equations, therefore they can be solved
simultaneously to determine their solutions. Since the applied loading is
symmetric, i.e. ( ) ( ) ( ) ( )z,yUz,yU a
nn−=0 , this means that
0
22
0
11AA,AA aa −=−= and so on. Otherwise, the relationship is not applicable.
Notice also that solving the above equations is depending on the term m and n.
Once the unknowns are determined, the final solutions can be obtained in-terms
of displacements and stresses at one time. Depending on the computer
specifications, a standard desktop PC can takes up to about 5 hours to solve
the above problems.
4.4 Conclusions
This chapter deals with the application of state space method to the clamped
edges laminated plate. The mathematical concept of clamped edges laminated
plate is clearly derived and developed from the state equation of a simply
supported plate. The details of the process or procedures involved in the
Chapter 4 State Space Solution of Clamped Edges Laminated Plate
127
analysis of clamped edges laminated plate have been explained in a very detail
and systematic manner. For clarification and understanding purposes, thorough
steps have been presented with regard to the method of analyzing clamped
edges plate in determination of solving the unknown coefficients. Some
derivations of formulae or expressions have also been shown in this chapter.
It is clearly showed in this chapter that the nature of work of solving the problem
involves massive matrix calculations and iteration process. Therefore, the
knowledge and skills of using a programming software is highly prerequisite.
Since this task is based on the previous research performed by Fan [30], i.e. the
case of a partially clamped edges plate with 2 sides simply supported and 2
sides clamped, it brings the attention and interest to the author to solve a further
challenging task, i.e. to investigate a fully clamped edges laminated plate. This
is the novel solution presented in this study.
Further challenging works have been performed to analyse a fully clamped
edges laminated plate. This is because of the fact that there are new additional
terms need to be considered and that is solving the unknowns coefficient of B’s.
Applying the same principle as before but now the term
( ) ( ) ( )∑ =+n
mnm
zVzV 00 is added. As a result, there are 34 unknowns to be
solved for the problem, 17 unknowns each for A and B. The results of these
new analytical exact solutions are verified with numerical analysis and
presented in the next chapter.
Further cases related to the investigation on the exact solution of clamped
edges laminated plate are also presented in the following chapter including
different loading conditions and material properties.
Some of the derivations of expressions are also clearly presented in very detail
manner this chapter.
Chapter 5 Numerical Analysis of Laminated Plates
128
Chapter 5
NUMERICAL ANALYSIS OF
LAMINATED PLATES
In this chapter, numerical solutions by means of finite element method are
presented in order to validate the analytical works based on the 3D elasticity
and state space method. The objectives of this approach are to gain further
insight of composite plate behavior in such a way that the model (a continuum)
is discretized into simple geometric shapes called finite elements and to perform
some other parametric studies. Material properties and the governing
relationships are considered over these elements. FEM is an alternative
approach to solving the governing equations of a structural problem particularly
at the elastic phase.
The application together with their results of FEM to composites is discussed in
detail in the following chapter. Various particular structural problems are
illustrated to show the effectiveness of FEM, verifying the analytical solutions
etc.
5.1 Introduction
The finite element method has become a powerful tool to generate the
numerical solutions for a wide range of engineering problems since the
existence of the finite element method in the 1950s [54]. Solutions ranging from
deformation and stress analysis for various engineering structures such as
building and bridges, aircraft and others to temperature and flows problems, can
be determined. With the significant advancement of computer technology, one
can model and simulate the behavior using finite element method with relative
ease, efficient and economical manner. By assembling the model appropriately
Chapter 5 Numerical Analysis of Laminated Plates
129
with the precise material properties and considering the loading and constraints,
the response of the model can be accurately simulated and numerical solutions
would be obtained.
In this study, the shape of the model is only limited to rectangular plate,
however, if a more complicated geometry such as a rectangular plate with a cut-
out is analysed, the governing equations become increasingly more
complicated and thus it is difficult to solve with ordinary mathematical program.
In such cases, finite element method is very much useful.
One of the available commercial finite element packages is known as Abaqus. It
is a very versatile program with broad application. This program is used in this
study due to its great flexibility of choices in predicting the linear or non linear
behavior of structures and varieties options in material selection. Numerous
investigations have been performed towards the successful application of finite
element simulation to study the behaviour of composite structures for many
years. The scope of this study is to model and simulate the performance of
various composite rectangular plates. The details of work that have been
carried out towards producing the acquired results in this study are presented in
the subsequent sections.
5.2 Modelling of Composite Plate Using FEM
Creating the part of plate is the initial task in modeling composite plate using
FEM. Part module is used to build different parts of the model. In this study, the
models are created and divided into a number of different layers of composite
plate to represent the laminate having different material properties for each
layer. Later all the layers can be assembled to form the entire model.
Laminated composite plate has two significant distinctions over a metal. Firstly,
it comprises of layered material built up by stacking the plies of different
material properties and secondly, for each ply, it is not isotropic. It depends on
the direction of the fibre to give the desired stiffness which varies from each ply.
Careful considerations had been paid prior to the modeling of finite elements in
Chapter 5 Numerical Analysis of Laminated Plates
130
order to avoid mistakes and wrong interpretation such as sign conventions,
coordinate systems, selecting the right face of the member at the right
placement, applying the correct boundary conditions, etc. Otherwise, these
would lead to misleading results. Therefore, accurate modeling was a
prerequisite. So the tasks which involve modeling and analysing, require
considerable efforts and time consuming to arrive the correct element types,
analysis procedures, material properties and other factors. This procedure
requires meticulous thought in order to determine the precise results. The
following sections give the details explanation to determine the numerical
simulation behaviour.
Generally, finite element method is an approximate technique as such the
output results should be carefully evaluated before relied upon them for a
design application. Considering the number of elements, or the fineness of the
meshing, the type of elements to be used, choosing the appropriate analysis
type are some of the factors that can significantly affect the accuracy of the
model. As the models are increasingly created with more refined mesh or as
the number of elements is increased, the results should converge to the true
numerical solution.
5.2.1 Element types
There were many different type of elements used during the initial stages of the
analysis which in turn affect the quality of results. However, in Abaqus, there
are many different types of elements available, depending on the nature of the
problem to be simulate. These include solid, shell, beam, rigid, membrane
infinite, connector or truss elements. For each of these elements, there are
further types of elements depending on the number of nodes, linear or quadratic
elements, full or reduced integrations. The degrees of freedom are the
fundamental variables calculated during the analysis. The use of solid elements
is limited to three-dimensional brick elements that have only displacement
degrees of freedom. In contrast, conventional shell elements have displacement
and rotational degrees of freedom. Continuum shell elements look like three-
dimensional continuum solids, but their kinematic and constitutive behaviour is
Chapter 5 Numerical Analysis of Laminated Plates
131
similar to conventional shell elements. Continuum shell elements have only
displacement degrees of freedom. Various types of finite elements, implies
different assumptions made and various results are derived. For instance,
Kirchhoff plate elements are derived for the application of thin plates which only
account for flexural deformation. Mindlin plate elements can be used for the
application of moderately thick plates accounting for both flexural and shear
deformations.
There are numerous other applications or examples which can be simulated by
the numerical analysis using FEM including heat transfer or fluid mechanics
simulations. A very good result will also be achieved if using elements with
higher order shape function rather than those with lower order. Elements having
many nodes give accurate results than less number of nodes. Using Gaussian
quadrature numerical techniques, full integration type of element can give more
precise results than reduced integration type of element.
Reduced integration uses a lower-order integration to form the element stiffness,
hence reduces running time in comparison with full integration elements.
5.2.2 Assembling the model
All the parts of the composite plate created earlier with either composite solids
or shells can be put together (assembly) to get the required model. After doing
this, we can apply the necessary constraints and loads on the assembly. The
thickness, number of section points required for numerical integration through
the thickness of each layer, material properties, orientation associated with
each layer, correct boundary conditions are the basic specifications needed for
assembling and defining the composite model. This is very easily done in
Abaqus using a special tool from composite layup where the common inputs are
put in one table together as shown in Figure 5.1 . This tool is very useful and
convenient to use particularly for the case of composite consisting of many
layers and having various fibre orientations. The number of integration points
are also easily be specified from the same tool.
Chapter 5 Numerical Analysis of Laminated Plates
132
The unidirectional laminate plate is assembled by stacking up each lamina in
predetermined directions and thicknesses to obtain the desired stiffness. The
behaviour is still considered as transversely isotropic in a cross section
perpendicular to the fibres.
Figure 5.1: Table used to model composite solid and shell in Abaqus
The other beneficial use of assembling the model using composite lay-up tool is
that the model can be checked whether the layers are in the right order using
the ply stack plot option. If they are not positioned correctly, these can be
corrected from the composite lay-up tool table.
For all layers of elements, full bond is assumed at the interface which means
that delamination cannot be resulted during the loading. To do these constraints,
designated surface areas that need to be bonded must be assigned initially.
This can be done easily in Abaqus by creating a set of surface area in the
Assembly module. By assigning tie connection at every interface of the
Chapter 5 Numerical Analysis of Laminated Plates
133
elements after merging them together, the relationship between the bottom
surface of jth layer and the top surface of the j+1th layer can maintain continuity
right from the top surface through the bottom surface of the laminate.
5.2.3 Boundary Conditions
For the in-plane displacements, u and v, and transverse displacement, w
assigning the boundary conditions are straightforward in FEM. Depending on
the cases, we can set the following boundary conditions for the laminate plate:
• Clamped, restraining U, V and W and rotations
• Simply supported, restraining U or V, W and σy or σx
Accurate boundary conditions are one of the important parts of modeling
composite structures in finite element method. Establishing the right boundary
conditions is truly vital to determine the required results. If the boundary
conditions are not properly assigned, the results would be inaccurate even
though using the precise element type and fine meshing. Depending on the
loading and boundary conditions also, if the laminated plate is subjected to both
in-plane and transverse loading, the plate will experiencing both stretching and
bending. From Figure 4.2, partially clamped edges laminate means that both
edges or faces when x = 0 and a , are fixed in all directions. The remaining two
edges or faces are simply supported at y = 0 and b. For the case of fully
clamped, obviously, all four sides or faces are fixed in all directions. This implies
that both translation and rotations are constraint.
As far as the analytical calculation is concerned, if all the edges of the laminate
plate are fully clamped, and because of the loading distribution is symmetrical, it
can be simplified that ( ) ( ) ( ) ( ) 00 =−= z,xVz,xV b
and ( ) ( ) ( ) ( )z,yUz,yU a 0−= . For the case of partially clamped, however,
( ) ( ) ( ) ( ) 0,,0 == zxVzxV b (the edges are simply supported).
Chapter 5 Numerical Analysis of Laminated Plates
134
5.2.4 Analysis Type
In this study, one of the important assumptions considered in the analysis of
composite plate using FEM is the type of analysis. It is vital to be acknowledged
if accurate results are to be obtained. The type of analysis chosen in this study
is assumed to be a linear static elastic analysis. In practice, the behaviour of
composite structures may exhibit non-linear after they reach the elastic limit
whereby the matrix begins to cracking at a tensile stress or the fibre starts to
fracture. It is only considered that elastic macro-mechanics investigation is the
prime study in which failure criteria is not included. Six degrees of freedom are
chosen at each node for shell elements, three translations and three rotations.
This permits the transverse shear deformation to be occurred. However, only 3
rotations degree of freedom are available by default for continuum and solid
elements. Three Simpson-type integration points are used along the thickness
of each shell element.
5.3 Material Properties
Selecting the appropriate material properties of the composite structures into
Abaqus is an important task which is required a careful consideration.
Laminated plate, in particular, has different moduli and strength different
directions. It is essential to note that in this study, composite plate has been
assigned as orthogonal type of material as an input in FEM. This behaviour is
different from other materials such as isotropic metal which has the same
elastic properties in all directions.
The type of material considered in the finite element modeling is orthotropic.
This implies that the properties along the fibre direction (x) are significantly
different from the other two normal directions (y and z).
The basic type of material used in Abaqus is isotropic material which can be
applied to the majority of elements available in the library. For the case of
composite plate, anisotropic or orthotropic behaviour, different strength and
stiffness at different directions will be assigned to elements selected.
Chapter 5 Numerical Analysis of Laminated Plates
135
Here, the values of material properties that are inserted into Abaqus are in
terms of ratio between one layer to another layer with respect to their elastic
properties. This can be clearly shown in the following:
The reduced material properties used in the numerical and the analytical
analyses are [30],
262931.0,159914.0,266810.0,530172.0
,115017.0,543103.0,0831715.0,246269.0
11
66
11
55
11
44
11
33
11
23
11
22
11
13
11
12
====
====
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
There are 3 laminae considered in the laminate where material properties of top
lamina (ply 1) are the same as the bottom lamina (ply 3). However, the material
property of lamina 2 (middle ply) is such that stiffness property ratio of ply1/ply2
= 5 or
( )
( ) .52
11
111 =
C
C
5.4 Mesh
Meshing is one of the most important modules since accuracy of the results
may depend on the meshing of the assemblies. This module can generate
meshes and verify them. Before meshing can be done, sometimes, partitioning
of the part is required for accuracy and convenience purposes particularly for
the case of complex shape.
After partition tasks have been done, normally the entire model shall be turned
green to indicate that meshing can be proceed.
Another great advantage of using Abaqus is the flexibility of selecting the
procedure prior to meshing. One can choose meshing by the number, size,
shape and type of element.
Chapter 5 Numerical Analysis of Laminated Plates
136
In this study, since the shape of the entire model is dominantly rectangular, the
shape of the mesh chosen is hex-dominated type. Before the model is
submitted for analysis in the job module, mesh verification and sensitivity
checks can also be performed. More details explanation about FEM and the
results of FEM for various boundary, loading conditions and material properties
are further shown in the following chapter.
5.5 Partially Clamped Edges Composite Plate with Va riable
Thickness to Width Ratio
Here, a three plies laminate plate is modeled with a full 3D model, stacked on
each other, subjected to a uniformly distributed transverse loading and is
clamped on two sides (at x = 0 and a) and the other two edges are simply
supported (at y = 0 and b). Ply 1 is the top layer, ply 2 is the middle layer and
the bottom layer is ply3 as shown in Figure 5.2. Material properties are as
mentioned in the previous chapter, Section 5.3. As the loading distribution and
boundary conditions are symmetrical, it can be simplified that
( ) ( ) ( ) ( )z,yUz,yU a 0−= , i.e. clamped edges and ( ) ( ) ( ) ( ) 00 == z,xVz,xV b ,
i.e. simply supported edges. The thickness of ply 1 is the same as ply 3 and
equal to 0.1h while ply 2 thickness is equal to 0.8h.
Chapter 5 Numerical Analysis of Laminated Plates
137
Figure 5.2: Geometry of plate consists of three pli es
Given the wide variety of element types that are available in Abaqus, solid and
shell elements are selected and run for the analysis of composite plate. Based
on the results obtained for each type of element, they are then compared with
those results from the analytical calculations. From this comparison, only the
element type that behaves as close as the analytical is considered and selected
for the subsequent case studies. Element type for FE analysis is based on
specific characteristics including first (linear) or second (quadratic), full or
reduced integration, and hexahedra/quadrilaterals.
The three-dimensional brick elements (solid), C3D8 and C3D20, were used to
model the laminated composite solids. Three layers of different materials having
same orientations were specified in each solid element. All layers were stacked
in the z-coordinates.
The overall in-plane dimension of the plate, i.e. a = 1 and b = 1, while the
thickness, h was varied. The in-plane mesh size for each element was varied
from coarse (0.5 x 0.5) to fine (0.03 x 0.03) meshes in order to improve
convergence and accuracy of the results. However, the mesh size in the z-
a
b
x , u , 1
z , w , 3
y , v , 2
h
Ply 1
Ply 2
Ply 3
Chapter 5 Numerical Analysis of Laminated Plates
138
direction was kept constant, i.e. for ply 1 and ply 3, the mesh thickness was
assigned to 0.1h/3 (3 sublayers) and the thickness of ply 2 meshes was taken
as 0.8h/10 (10 sublayers). Mesh sensitivity tests were taken to confirm which
mesh size would be appropriate to use for particular cases. The results from a
particular case of laminate with h/a ratio of 0.2 are shown in Figure 5.3. The
results show that FEM results converged to the exact solutions when the in-
plane element’s mesh size reduced to 0.03 x 0.03 from 0.5 x 0.5.
-18
-16
-14
-12
-10
-8
-600.10.20.30.40.50.6mesh
size
stre
ss, s/
q
sx FEMsy FEMsx Exactsy Exact
Figure 5.3: Mesh sensitivity test results for stres ses at x = y = z = 0 for h/a = 0.2
The shell elements used in the FE analysis include conventional and continuum
shells. Shell elements that were used to model structures had thickness
significantly smaller than the other dimensions. Conventional shell elements use
this condition to discretize a body by defining the geometry at a reference
surface. In this case the thickness was defined through the section property
definition. Conventional shell elements generally have displacement and
rotational degrees of freedom.
In contrast, continuum shell elements discretize an entire three-dimensional
body. The thickness is determined from the element nodal geometry.
Continuum shell elements have only displacement degrees of freedom. From a
Chapter 5 Numerical Analysis of Laminated Plates
139
modeling point of view continuum shell elements look like three-dimensional
continuum solids, but their kinematic and constitutive behavior is similar to
conventional shell elements.
The interaction between one layer of continuum shell to the next layer had to be
defined like modelling the plate using solid element. Tie connection was chosen
to simulate such interaction to merge the nodes at the interfaces. In this study,
three section points was defined for Simpson’s integration points through the
shell thickness.
The non-dimensional results are presented in the Appendix A with varying the
thickness to width ratio (h/a) of 0.2, 0.3, 0.4, 0.5, 0.6, 0.8 and 1. Table A-1 to A-
31 and Figure A-1 to A-29, generally show that solid elements exhibit better
results than shell elements as a matter of fact that for stress/displacement
(bending) simulation, the degrees of freedom in solid consider only translations
whereas conventional shell elements calculate both translation and rotations at
each node. Solid elements agree well to the exact solutions in terms of the
displacement and stress distributions at the centre of the plate particularly solid
element of C3D20. Solid elements have two through-thickness translational
freedoms which allow through-thickness direct strains (and stresses) to occur
whilst shell element does not have. Thin shell elements also neglect transverse
shear stresses in Abaqus and these elements use Kirchhoff thin shell theory.
These hypotheses occur when the thickness to width ratio is less than 1/10 [55].
This ratio is used to separate the full three-dimensional plate bending problem
with a two-dimensional problem. Table A-1 also shows that shell elements do
not give precise deflection results as the thickness to width ratio increases. This
is due to the fact that shell elements are used to simulate structural behaviour in
which the thickness is significantly smaller than the in-plane dimensions.
Continuum shells also do not show good results for stresses as compared to
solid and the exact solution, as shown in Table A-3. As the thickness to width
ratio increases, continuum shells give poor stress results. This is because even
though continuum shells look like three-dimensional continuum solids from a
Chapter 5 Numerical Analysis of Laminated Plates
140
modeling point of view, their kinematic and constitutive behaviour is similar to
conventional shell elements. It is important to note that Abaqus assumes
transverse constant shear strains through the shell’s thickness. The transverse
shear stresses are zero at the shell surface and they are continuous between
layers.
However, FEM cannot give precise stress results by shell elements compared
to exact solutions at the boundary edges. FEM also gives poor result for
transverse shear stresses at the edges when solid elements are used. Instead,
shell elements exhibit good response compared to the exact solutions as shown
in Figure A-31. Conventional shell elements allow transverse shear deformation.
They use thick shell theory as the shell thickness increases and become
discrete Kirchhoff thin shell elements as the thickness decreases; the
transverse shear deformation becomes very small as the shell thickness
decreases. In exact solution, shear stresses are zero at the top and bottom
surfaces of the composite laminate at the edges and increases in magnitude
dramatically afterwards. However, FEM only takes average stresses near the
top and bottom surfaces of the plate.
In Abaqus, second order solid element (C3D20) provides higher accuracy than
first order solid (C3D8) since they capture stress concentrations more
effectively particularly in this case of bending-dominated problems. Fully
integrated linear isoparametric elements (C3D8) suffer from ‘shear locking’, they
cannot provide pure bending solution because they must shear at the numerical
integration points to respond with an appropriate kinematic behaviour
corresponding to the bending. This shear locks the element which resulting
them far too stiff response. Element type C3D20 can overcome this shear
locking deficiency.
From this analysis, it can also show that the maximum deflection and stresses
at the centre of the plate are reduced gradually as the thickness to width ratio
(h/a) reaches to about 0.6. Beyond this ratio, increasing the thickness of the
laminate is no longer effective as the plate deflection tends to become
horizontal as shown in Figure A-1.
Chapter 5 Numerical Analysis of Laminated Plates
141
Equally importance to be noted from this analysis is how to choose an
appropriate element type in the subsequent cases. It is clearly shown that only
solid element C3D20 exhibits a closer result to the exact solution and therefore,
this element would be selected hereafter.
5.6 Fully Clamped Edges Composite Plate
In this case, all geometries and loading conditions are the same as the previous
model (Figure 5.2) except that all the edges are fully clamped. This means that
along all edges, i.e. at x = 0, a and y = 0, b, then U = V = W = 0. Fully clamped
edges plate also means that the relationship of the unknowns are
( ) ( ) ( ) ( )z,yUz,yU a 0−= and ( ) ( ) ( ) ( )z,xVz,xV b−=0 .
There are no exact analytical solutions are available in the literature. As
indicated in the previous chapter, modelling the laminate plate using FEM with
the first order element such as C3D8 with fully integrated elements, suffer from
“locking” behaviour (both shear and volumetric locking). Shear locking occurs in
first-order, fully integrated elements which are subjected to bending. The
numerical formulation of such element gives rise to shear strains that do not
really exist. Therefore, this element is too stiff in bending. Volumetric locking
occurs in fully integrated elements for incompressible material. Although, there
is no incompressible materials to be used in this studies. Previous section has
shown that the element C3D8 is not as good as C3D20 in laminated analysis.
Due to these reasons, the results of FEM are based on solid element type
C3D20 which will be used from this point onwards.
The new breakthrough non-dimensional analytical and numerical results are
shown as follows:
(Note that the deflection curves show continuity across the thickness of the
plate. However, no continuity for in-plane stresses across the thickness
because of the different material properties. Therefore, three stress cases are
shown for FEM – x for ply 1; □ for ply 2 and ○ for ply 3)
Chapter 5 Numerical Analysis of Laminated Plates
142
Table 5-1: Displacement and stresses distribution o f fully clamped laminate when h/a = 0.2
Exact FEM z = 0 14.41962 14.57520
2211 b
yanda
xatqh
WC== z = h 13.62818 13.78035
Ply1 Top -5.70218 -7.27574 Ply1 Bottom -5.75164 -4.44513
Ply2 Top -1.26698 -1.00880 Ply2 Bottom 1.11187 0.85842
Ply3 Top 5.60194 4.32076 22
byand
axat
qx ==
σ
Ply3 Bottom 5.54484 7.15232 Ply1 Top -5.29157 -6.67196
Ply1 Bottom -4.95703 -4.68203 Ply2 Top -1.15272 -1.10204
Ply2 Bottom 0.93845 0.89455 Ply3 Top 4.75117 4.51241
22
byand
axat
q
y ==σ
Ply3 Bottom 5.10697 6.50327
Ply1 Top 44.80379 32.29940 Ply1 Bottom -12.27516 -8.49153
Ply2 Top -2.45503 -1.69830 Ply2 Bottom 1.36364 1.03895
Ply3 Top 6.81822 5.19473 2
0b
yandxatqx ==
σ
Ply3 Bottom -34.59811 -26.48580 Ply1 Top 10.36055 7.95430
Ply1 Bottom -2.83854 -2.09119 Ply2 Top -0.56771 -0.41824
Ply2 Bottom 0.31534 0.25586 Ply3 Top 1.57666 1.27930
20
byandxat
q
y ==σ
Ply3 Bottom -8.00056 -6.52261 Ply1 Top 0 6.07206
Ply1 Bottom 1.93712 5.72504 Ply2 Top 1.93712 1.14501
Ply2 Bottom 1.34124 0.79127 Ply3 Top 1.34124 3.95636
20
byandxat
qxz ==
τ
Ply3 Bottom 0 4.02236
Ply1 Top 13.58259 10.79870 Ply1 Bottom -1.02771 0.75091
Ply2 Top -0.20554 0.15018 Ply2 Bottom -0.10236 -0.32949
Ply3 Top -0.51180 -1.64745
02
== yanda
xatqx
σ
Ply3 Bottom -10.53502 -9.01476 Ply1 Top 30.83717 23.81460
Ply1 Bottom -2.33325 1.65600 Ply2 Top -0.46665 0.33120
Ply2 Bottom -0.23239 -0.72664 Ply3 Top -1.16196 -3.63318
02
== yanda
xatq
yσ
Ply3 Bottom -23.91813 -19.88050 Ply1 Top 0 7.74460
Ply1 Bottom 2.24200 6.14487 Ply2 Top 2.24200 1.22897
Ply2 Bottom 1.58581 0.79948 Ply3 Top 1.58581 3.99738
02
== yanda
xatq
yzτ
Ply3 Bottom 0 5.11093
Chapter 5 Numerical Analysis of Laminated Plates
143
0.0
0.2
0.4
0.6
0.8
1.0
13.6 13.8 14.0 14.2 14.4 14.6
W C11/qhz/
h
Exact FEM
Figure 5.4: Displacement (W C 11/qh) distribution through the thickness of
the plate at x = a/2 and y = b/2 when h/a = 0.2
0
0.2
0.4
0.6
0.8
1
-8 -6 -4 -2 0 2 4 6 8
sx/q
z/h
ExactFEMFEMFEM
Figure 5.5: Stress ( σx/q) x = a/2 and y = b/2 when h/a = 0.2
Chapter 5 Numerical Analysis of Laminated Plates
144
0
0.2
0.4
0.6
0.8
1
-8 -6 -4 -2 0 2 4 6 8sy/q
z/h
ExactFEMFEMFEM
Figure 5.6: Stress ( σy/q) x = a/2 and y = b/2 when h/a = 0.2
0
0.2
0.4
0.6
0.8
1
-40 -20 0 20 40sx/q
z/h
ExactFEMFEMFEM
Figure 5.7: Stress ( σx/q) x = 0 and y = b/2 when h/a = 0.2
Chapter 5 Numerical Analysis of Laminated Plates
145
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15sy/q
z/h
ExactFEMFEMFEM
Figure 5.8: Stress ( σy/q) x = 0 and y = b/2 when h/a = 0.2
0.0
0.2
0.4
0.6
0.8
1.0
-15 -10 -5 0 5 10 15
sx/q
z/h
ExactFEMFEMFEM
Figure 5.9: Stress ( σx/q) x = a/2 and y = 0 when h/a = 0.2
Chapter 5 Numerical Analysis of Laminated Plates
146
0
0.2
0.4
0.6
0.8
1
-30 -20 -10 0 10 20 30 40sy/q
z/h
ExactFEMFEMFEM
Figure 5.10: Stress ( σy/q) x = a/2 and y = 0 when h/a = 0.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7txz/q
z/h
ExactFEMFEMFEM
Figure 5.11: Stress ( τxz/q) at x = 0 and y = b/2 when h/a = 0.2
Chapter 5 Numerical Analysis of Laminated Plates
147
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10tyz/q
z/h
ExactFEMFEMFEM
Figure 5.12: Stress ( τyz/q) at x = a/2 and y = 0 when h/a = 0.2
Chapter 5 Numerical Analysis of Laminated Plates
148
Table 5-2: Displacement and stresses distribution o f fully clamped laminate when h/a = 0.4
Exact FEM z = 0 2.94212 2.96443
2211 b
yanda
xatqh
WC== z = h 2.11877 2.13772
Ply1 Top -2.19954 -2.41687 Ply1 Bottom -1.01566 -0.73370
Ply2 Top -0.32295 -0.26708 Ply2 Bottom 0.16532 0.10059
Ply3 Top 0.85980 0.53519 22
byand
axat
qx ==
σ
Ply3 Bottom 1.83882 2.19132 Ply1 Top -1.96879 -2.35161
Ply1 Bottom -1.34919 -1.20984 Ply2 Top -0.43553 -0.408388
Ply2 Bottom 0.20992 0.192513 Ply3 Top 1.09549 1.00714
22
byand
axat
q
y ==σ
Ply3 Bottom 1.78259 2.14072
Ply1 Top 21.79929 16.3769 Ply1 Bottom -10.23391 -8.43135
Ply2 Top -2.04678 -1.68627 Ply2 Bottom 0.84678 0.77621
Ply3 Top 4.23392 3.88103 2
0b
yandxatqx ==
σ
Ply3 Bottom -10.42915 -8.39733 Ply1 Top 5.04093 4.03311
Ply1 Bottom -2.36652 -2.07637 Ply2 Top -0.47331 -0.41528
Ply2 Bottom 0.19581 0.19116 Ply3 Top 0.97906 0.955769
20
byandxat
q
y ==σ
Ply3 Bottom -2.41167 -2.06799 Ply1 Top 0 3.24192
Ply1 Bottom 1.08675 2.8687 Ply2 Top 1.08675 0.573739
Ply2 Bottom 0.48941 0.2449 Ply3 Top 0.48941 1.2245
20
byandxat
qxz ==
τ
Ply3 Bottom 0 1.16783
Ply1 Top 6.77574 5.85498 Ply1 Bottom -1.88233 -1.72143
Ply2 Top -0.37647 -0.34429 Ply2 Bottom 0.07239 0.07289
Ply3 Top 0.36194 0.364467
02
== yanda
xatqx
σ
Ply3 Bottom -3.58084 -3.2661 Ply1 Top 15.38327 12.9122
Ply1 Bottom -4.27354 -3.79633 Ply2 Top -0.85471 -0.75927
Ply2 Bottom 0.16434 0.160754 Ply3 Top 0.82172 0.803769
02
== yanda
xatq
yσ
Ply3 Bottom -8.12976 -7.20283 Ply1 Top 0 4.31943
Ply1 Bottom 1.27907 3.22337 Ply2 Top 1.27907 0.644674
Ply2 Bottom 0.63278 0.279716 Ply3 Top 0.63278 1.39858
02
== yanda
xatq
yzτ
Ply3 Bottom 0 1.7419
Chapter 5 Numerical Analysis of Laminated Plates
149
0
0.2
0.4
0.6
0.8
1
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
W C11/qhz/
h
Analytical FEM
Figure 5.13: Displacement (W C 11/qh) distribution through the thickness of
the plate at x = a/2 and y = b/2 when h/a = 0.4
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3sx/q
z/h
ExactFEMFEMFEM
Figure 5.14: Stress ( σx/q) x = a/2 and y = b/2 when h/a = 0.4
Chapter 5 Numerical Analysis of Laminated Plates
150
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3sy/q
z/h
ExactFEMFEMFEM
Figure 5.15: Stress ( σy/q) x = a/2 and y = b/2 when h/a = 0.4
0
0.2
0.4
0.6
0.8
1
-20 -10 0 10 20 30sx/q
z/h
ExactFEMFEMFEM
Figure 5.16: Stress ( σx/q) at x = 0 and y = b/2 when h/a = 0.4
Chapter 5 Numerical Analysis of Laminated Plates
151
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4 6sy/q
z/h
ExactFEMFEMFEM
Figure 5.17: Stress ( σy/q) at x = 0 and y = b/2 when h/a = 0.4
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5txz/q
z/h
ExactFEMFEMFEM
Figure 5.18: Stress ( τxz/q) at x = 0 and y = b/2 when h/a = 0.4
Chapter 5 Numerical Analysis of Laminated Plates
152
0
0.2
0.4
0.6
0.8
1
-6 -4 -2 0 2 4 6 8sx/q
z/h
ExactFEMFEMFEM
Figure 5.19: Stress ( σx/q) at x = a/2 and y = 0 when h/a = 0.4
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15 20sy/q
z/h
ExactFEMFEMFEM
Figure 5.20: Stress ( σy/q) at x = a/2 and y = 0 when h/a = 0.4
Chapter 5 Numerical Analysis of Laminated Plates
153
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5tyz/q
z/h
ExactFEMFEMFEM
Figure 5.21: Stress ( τyz/q) at x = a/2 and y = 0 when h/a = 0.4
Chapter 5 Numerical Analysis of Laminated Plates
154
Table 5-3: Displacement and stresses distribution o f fully clamped laminate when h/a = 0.6
Exact FEM z = 0 1.48584 1.48930
2211 b
yanda
xatqh
WC== z = h 0.66226 0.66219
Ply1 Top -1.90919 -1.82563 Ply1 Bottom -0.09684 -0.05983
Ply2 Top -0.14097 -0.13321 Ply2 Bottom -0.00536 -0.02171
Ply3 Top 0.00582 -0.07443 22
byand
axat
qx ==
σ
Ply3 Bottom 1.11317 1.16011 Ply1 Top -1.43183 -1.63643
Ply1 Bottom -0.58389 -0.48249 Ply2 Top -0.28494 -0.26417
Ply2 Bottom 0.04925 0.04044 Ply3 Top 0.29132 0.24938
22
byand
axat
q
y ==σ
Ply3 Bottom 0.90977 1.03978
Ply1 Top 16.37247 12.70100 Ply1 Bottom -8.60849 -7.31159
Ply2 Top -1.72170 -1.46232 Ply2 Bottom 0.46260 0.42136
Ply3 Top 2.31299 2.10680 2
0b
yandxatqx ==
σ
Ply3 Bottom -4.65173 -3.79350 Ply1 Top 3.78601 3.12785
Ply1 Bottom -1.99065 -1.80061 Ply2 Top -0.39813 -0.36012
Ply2 Bottom 0.10697 0.10377 Ply3 Top 0.53486 0.51884
20
byandxat
q
y ==σ
Ply3 Bottom -1.07568 -0.93422 Ply1 Top 0 2.64572
Ply1 Bottom 0.85546 2.18789 Ply2 Top 0.85546 0.43758
Ply2 Bottom 0.24314 0.11425 Ply3 Top 0.24314 0.57123
20
byandxat
qxz ==
τ
Ply3 Bottom 0 0.51622
Ply1 Top 4.93674 4.53661 Ply1 Bottom -1.85880 -1.93444
Ply2 Top -0.37176 -0.38689 Ply2 Bottom 0.06516 0.07725
Ply3 Top 0.32580 0.38627
02
== yanda
xatqx
σ
Ply3 Bottom -1.64874 -1.55840 Ply1 Top 11.20810 10.00470
Ply1 Bottom -4.22011 -4.26608 Ply2 Top -0.84403 -0.85322
Ply2 Bottom 0.14794 0.17037 Ply3 Top 0.73967 0.85185
02
== yanda
xatq
yσ
Ply3 Bottom -3.74320 -3.43678 Ply1 Top 0 3.45644
Ply1 Bottom 0.98750 2.46116 Ply2 Top 0.98750 0.49223
Ply2 Bottom 0.33504 0.14465 Ply3 Top 0.33504 0.72327
02
== yanda
xatq
yzτ
Ply3 Bottom 0 0.82099
Chapter 5 Numerical Analysis of Laminated Plates
155
0.0
0.2
0.4
0.6
0.8
1.0
0.6 0.8 1.0 1.2 1.4 1.6
W C11/qh
z/h
Exact sol FEM
Figure 5.22: Displacement (W C 11/qh) distribution through the thickness of
the plate at x = a/2 and y = b/2 when h/a = 0.6
0
0.2
0.4
0.6
0.8
1
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5sx/q
z/h
ExactFEMFEMFEM
Figure 5.23: Stress ( σx/q) at x = a/2 and y = b/2 when h/a = 0.6
Chapter 5 Numerical Analysis of Laminated Plates
156
0
0.2
0.4
0.6
0.8
1
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5sy/q
z/h
ExactFEMFEMFEM
Figure 5.24: Stress ( σy/q) at x = a/2 and y = b/2 when h/a = 0.6
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15 20sx/q
z/h
ExactFEMFEMFEM
Figure 5.25: Stress ( σx/q) at x = 0 and y = b/2 when h/a = 0.6
Chapter 5 Numerical Analysis of Laminated Plates
157
0
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3 4 5
sy/q
z/h
ExactFEMFEMFEM
Figure 5.26: Stress ( σy/q) at x = 0 and y = b/2 when h/a = 0.6
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4 6sx/q
z/h
ExactFEMFEMFEM
Figure 5.27: Stress ( σx/q) at x = a/2 and y = 0 when h/a = 0.6
Chapter 5 Numerical Analysis of Laminated Plates
158
0
0.2
0.4
0.6
0.8
1
-10 -5 0 5 10 15sy/q
z/h
ExactFEMFEMFEM
Figure 5.28: Stress ( σy/q) at x = a/2 and y = 0 when h/a = 0.6
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
txz/q
z/h
ExactFEMFEMFEM
Figure 5.29: Stress ( τxz/q) at x = 0 and y = b/2 when h/a = 0.6
Chapter 5 Numerical Analysis of Laminated Plates
159
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4tyz
z/h
ExactFEMFEMFEM
Figure 5.30: Stress ( τyz/q) at x = a/2 and y = 0 when h/a = 0.6
The tasks of seeking analytical solutions of fully clamped laminate plate were
very challenging indeed, due to the limitation of computer capacity used. Unless
a supercomputer having a massive memory and fast processor is used, this
task cannot be completed with cost-effective manner. However, using a
standard desktop computer, the author had divided the tasks into smaller
components in order to get the intermediate results (to solve the unknowns).
Once all these smaller components were solved, they were then combined
altogether to get the final results. Careful considerations were taken during this
laborious, lengthy and time consuming process of breaking the smaller tasks as
human errors are very high. A standard PC can only generate up to about
twelve loops (cycle per run) which takes to about one hour. The analysis
precision was set up to the accuracy of 1000 decimal places. Once the
unknown results are determined, the prime results of displacements and
stresses can be found and it took about five to six hours to get them by using
the machine with Linux operation system and ten to eleven hours if Windows
operating machine is used.
Chapter 5 Numerical Analysis of Laminated Plates
160
In general, the results obtained from the exact solutions and FEM agree well
except the transverse shear stresses τxz and τyz at x = 0, y = b/2 and x = a/2,
y = b/2, respectively as shown in Figure 5.29 and Figure 5.30. FEM exhibits
poor results of transverse shear stresses at the boundary. FEM does not
preserve the continuity of the interfaces from the top through the bottom of the
laminated plate.
Stress results at ply1 and ply 3 when x = a/2 and y = b/2 at h/a = 0.2, do not
show smooth curve compared to FEM results as shown in Figure 5.5. More
results need to be obtained to verify this behaviour such as stress results for
thickness to width ratio of 0.1 and also by increasing the number of sub-division
(sub-layers) within the ply1 and ply3.
As the thickness to width ratio increases from 0.2 to 0.6, the deflection and
stresses curves between exact and FEM solutions agree well to each other
apart from the ratio of 0.2 as shown in Figure 5.4, Figure 5.5, Figure 5.13 and
Figure 5.22. Further investigation needs to be carried out to check the exact
solutions including for the case of thickness to width ratio of 0.1 or less and
most importantly increasing the number of sub-layers to capture the deflection
and stress results. For each of these curve, since loading is applied at the top
surface of the laminated plate, generally, the maximum and minimum deflection
are located at the top and bottom surfaces, respectively, except h/a = 0.2.
Transverse shear stresses txz and tyz are maximum at top and bottom
surfaces in FEM while they are zero values obtained by the exact solutions.
This is because of FEM treatment over the nodes of the element near the top
and bottom surfaces of the plate. FEM may calculates the average stresses of
the top and bottom nodes within the same individual element to give the
resultant stress at the outer surface’s nodes. Conversely, the exact solutions
give zero stress at the top fibre and increases dramatically immediately below
the top fibre. This also applies the same response for the bottom fibre of the
plate. These stresses distribution can be seen from Figure 5.11, Figure 5.18
and Figure 5.30
Chapter 5 Numerical Analysis of Laminated Plates
161
5.7 Laminated Plate Subjected To Hydrostatic Loadin g
In this model, laminated plate is subjected to hydrostatic loading having
boundary conditions of 2 sides fixed (at x = 0 and a) and 2 sides simply
supported (at y = 0 and b) with h/a = 0.4.
Figure 5.31: Partially clamped plate subjected to h ydrostatic loading for
h/a = 0.4
The loading is applied linearly increase from zero at x = 0 and qo at x = a as
shown in Figure 5.31, or this can be expressed as a
xqq o= . From this
unsymmetrical loading, it is important to be noted and applied in the analytical
method of exact solution that ( ) ( ) ( ) ( )z,yUz,yU a 0−≠ . This means that all the
unknown coefficients of oa AA11
−≠ and so on. However the relationship of
( ) ( ) ( ) ( ) 00 == z,xVz,xV b are still used in this model since the edges are
simply supported, hence they are no unknown coefficients exists.
The results obtained from the analysis are as follows:
x
z
y qo
Chapter 5 Numerical Analysis of Laminated Plates
162
Table 5-4: Exact solution versus FEM for plate subj ected to hydrostatic
loading
4.0=a
h
Exact Solution FEM z = 0 1.77429 1.77587
2211 b
yanda
xathq
WC
o
== z = h 1.35222 1.35399
Ply1 Top -1.67846 -1.52918
Ply1 Bottom -0.55238 -0.46341 Ply2 Top -0.17746 -0.15283
Ply2 Bottom 0.07050 0.07255 Ply3 Top 0.37709 0.37935
22
byand
axat
q
σ
o
x ==
Ply3 Bottom 1.43574 1.43890
Ply1 Top -1.85418 -1.63832 Ply1 Bottom -1.10110 -0.98244
Ply2 Top -0.30988 -0.27967 Ply2 Bottom 0.20065 0.20403
Ply3 Top 1.03720 1.04311 22
byand
axat
q
σ
o
y ==
Ply3 Bottom 1.71577 1.72043
Ply1 Top 10.86507 13.52490 Ply1 Bottom -5.66625 -7.55287
Ply2 Top -1.13325 -1.51057 Ply2 Bottom 0.60321 0.704527
Ply3 Top 3.01603 3.52264 2
0b
yandxatq
σ
o
x ==
Ply3 Bottom -5.52563 -6.18245
Ply1 Top 0 0.60958 Ply1 Bottom 0.62205 0.62353
Ply2 Top 0.62205 0.12471 Ply2 Bottom 0.31860 0.10641
Ply3 Top 0.31860 0.53204 2
0b
yandxatq
τ
o
xz ==
Ply3 Bottom 0 0.50125
Table 5-4 shows that FEM produces inaccurate results for stresses at the edges
of the plate especially transverse shear stresses. However, in-plane stresses
away from the edges agree well.
Chapter 5 Numerical Analysis of Laminated Plates
163
5.8 Laminated Plate with increasing number of sub-l ayers
In this section, a laminated plate which is similar to the model as shown in
Figure 5.2, i.e. the plate is partially clamped, subjected to uniformly distributed
loading on the top surface of the plate and having material properties as stated
in section 5.3, is analysed. The only variable considered in this investigation is
the number of sub-layers within the plies.
There are three cases taken into investigation:
Ply 1 Ply 2 Ply 3
Case No. of
sub-
layers
Thickness
(mm)
No. of
sub-
layers
Thickness
(mm)
No. of
sub-
layers
Thickness
(mm)
Total
no. of
sub-
layers
1 3 0.013 10 0.032 3 0.013 16
2 4 0.01 11 0.0291 4 0.01 19
3 5 0.008 12 0.0267 5 0.008 22
All the three cases, the plate’s thickness to width ratio are kept the same value
of 0.4. This means that the thickness of ply 1 (= ply 3) and ply 2 are 0.1h and
0.8h, respectively.
From the investigation, the results are obtained at the centre of the plate, i.e. x =
a/2 and y = b/2, as follows:
Table 5-5: The exact solutions and FEM of Case 1
Deflection, qh
WC11 Stress
q
σx Stress
q
σy
Ply h
z
Exact FEM Exact FEM Exact FEM 0 3.54859 3.55175 -3.05251 -3.05836 -3.28738 -3.27664
0.0333 3.54438 3.54768 -2.32003 -2.31467 -2.82134 -2.82669 0.0667 3.53883 3.54228 -1.59978 -1.60413 -2.37990 -2.38977
1
0.1 3.53216 3.53568 -0.92631 -0.92682 -1.95543 -1.96488
Chapter 5 Numerical Analysis of Laminated Plates
164
0.1 3.53216 3.53568 -0.30544 -0.30566 -0.55727 -0.55933 0.18 3.40284 3.40640 -0.20759 -0.20854 -0.43974 -0.44132 0.26 3.28177 3.28535 -0.14124 -0.14177 -0.33269 -0.33398 0.34 3.17055 3.17413 -0.09571 -0.09619 -0.23384 -0.23496 0.42 3.07040 3.07398 -0.06342 -0.06389 -0.14113 -0.14204 0.5 2.98221 2.98580 -0.03859 -0.03911 -0.05259 -0.05324
0.58 2.90658 2.91018 -0.01692 -0.01746 0.03367 0.03332 0.66 2.84378 2.84738 0.00569 0.00523 0.11973 0.11977 0.74 2.79372 2.79730 0.03447 0.03422 0.20818 0.20868 0.82 2.75584 2.75943 0.07690 0.07720 0.30229 0.30339
2
0.9 2.72896 2.73253 0.14479 0.14510 0.40652 0.40806 0.9 2.72896 2.73253 0.75795 0.75870 2.07964 2.08622
0.9333 2.72202 2.72558 1.42482 1.42813 2.51492 2.52268 0.9667 2.71392 2.71745 2.13358 2.13352 2.96670 2.97367
3
1 2.70444 2.70798 2.87148 2.87780 3.43154 3.44086
Table 5-6: The exact solutions and FEM of Case 2
Deflection, qh
WC11 Stress
q
σx Stress
q
σy
Ply h
z
Exact FEM Exact FEM Exact FEM 0 3.55034 3.55200 -3.05226 -3.05830 -3.28891 -3.27682
0.025 3.54733 3.54908 -2.50739 -2.49741 -2.93711 -2.93806 0.05 3.54350 3.54538 -1.95572 -1.95522 -2.59958 -2.60681
0.075 3.53903 3.54098 -1.42821 -1.43161 -2.27391 -2.28255 1
0.1 3.53393 3.53593 -0.92889 -0.92678 -1.95703 -1.96495 0.1 3.53393 3.53593 -0.30597 -0.30564 -0.55761 -0.55933
0.1727 3.41605 3.41810 -0.21532 -0.21598 -0.45018 -0.45157 0.2455 3.30488 3.30693 -0.15181 -0.15208 -0.35165 -0.35280 0.3182 3.20161 3.20368 -0.10676 -0.10703 -0.26021 -0.26127 0.3909 3.10725 3.10933 -0.07423 -0.07453 -0.17432 -0.17524 0.4636 3.02255 3.02463 -0.04939 -0.04977 -0.09243 -0.09320 0.5364 2.94804 2.95010 -0.02869 -0.02914 -0.01316 -0.01373 0.6091 2.88404 2.88610 -0.00910 -0.00957 0.06497 0.06466 0.6818 2.83065 2.83273 0.01264 0.01224 0.14365 0.14366 0.7545 2.78773 2.78980 0.04094 0.04071 0.22497 0.22537 0.8273 2.75473 2.75680 0.08188 0.08214 0.31153 0.31244
2
0.9 2.73072 2.73280 0.14502 0.14509 0.40688 0.40808 0.9 2.73072 2.73280 0.75906 0.75866 2.08140 2.08632
0.925 2.72562 2.72768 1.25452 1.25747 2.40628 2.41239 0.95 2.71988 2.72193 1.77472 1.77609 2.74070 2.74641
0.975 2.71344 2.71548 2.31708 2.31573 3.08409 3.08903 3
1 2.70619 2.70825 2.87128 2.87768 3.43359 3.44102
Chapter 5 Numerical Analysis of Laminated Plates
165
Table 5-7: The exact solutions and FEM of Case 3
Deflection, qh
WC11 Stress
q
σx Stress
q
σy
Ply h
z
Exact FEM Exact FEM Exact FEM 0 3.55130 3.55213 -3.05045 -3.05829 -3.28948 -3.27694
0.02 3.54897 3.54988 -2.62132 -2.60804 -3.00783 -3.00528 0.04 3.54607 3.54710 -2.17144 -2.16988 -2.73310 -2.73856 0.06 3.54275 3.54385 -1.74416 -1.74349 -2.46962 -2.47635 0.08 3.53902 3.54015 -1.32509 -1.32914 -2.21035 -2.21858
1
0.1 3.53489 3.53605 -0.93035 -0.92675 -1.95804 -1.96500 0.1 3.53489 3.53605 -0.30627 -0.30562 -0.55783 -0.55932
0.1667 3.42660 3.42783 -0.22178 -0.22237 -0.45888 -0.46017 0.2333 3.32385 3.32508 -0.16105 -0.16124 -0.36765 -0.36870
0.3 3.22760 3.22883 -0.11672 -0.11691 -0.28256 -0.28357 0.3667 3.13864 3.13988 -0.08409 -0.08431 -0.20247 -0.20340 0.4333 3.05760 3.05885 -0.05912 -0.05940 -0.12617 -0.12699
0.5 2.98497 2.98620 -0.03878 -0.03914 -0.05255 -0.05323 0.5667 2.92105 2.92230 -0.02062 -0.02104 0.01947 0.01898 0.6333 2.86603 2.86728 -0.00232 -0.00274 0.09110 0.09084
0.7 2.81992 2.82118 0.01887 0.01852 0.16372 0.16374 0.7667 2.78252 2.78378 0.04669 0.04650 0.23908 0.23943 0.8333 2.75337 2.75463 0.08613 0.08642 0.31923 0.32003
2
0.9 2.73169 2.73293 0.14515 0.14507 0.40709 0.40807 0.9 2.73169 2.73293 0.75969 0.75865 2.08240 2.08639
0.92 2.72765 2.72888 1.15291 1.15615 2.34130 2.34665 0.94 2.72321 2.72445 1.56490 1.56616 2.60709 2.61191 0.96 2.71835 2.71958 1.98812 1.98933 2.87778 2.88248 0.98 2.71302 2.71425 2.42837 2.42623 3.15499 3.15875
3
1 2.70714 2.70838 2.87048 2.87763 3.43459 3.44111
Chapter 5 Numerical Analysis of Laminated Plates
166
0
0.2
0.4
0.6
0.8
1
2.7 2.9 3.1 3.3 3.5 3.7
Deflection WC 11/qh
z/h
Case 1Case 2Case 3FEM
Figure 5.32: Central deflection across the thicknes s of plate h/a=0.4
0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3 4sx/q
z/h
Case 1
Case 2
Case 3
FEM
Figure 5.33: Stress,sx of plate h/a = 0.4 at x = a/2 y = b/2
Chapter 5 Numerical Analysis of Laminated Plates
167
0
0.2
0.4
0.6
0.8
1
-4 -3 -2 -1 0 1 2 3 4sy/q
z/h
Case 1Case 2Case 3FEM
Figure 5.34: Stress,sy of plate h/a = 0.4 at x = a/2 y = b/2
The above Tables 5-5 to 5-7 and Figures 5.32 to 5.34 show the results of exact
solutions and FEM of laminated plate of h/a = 0.4 with the variable of number of
sub-layers within the plies. From this investigation, it clearly shows that by
increasing the number of sub-layers, the values of deflection and stresses are
slightly enhanced. If more sub-layers are provided in the analysis, further
precise results can be obtained.
5.9 Conclusions
The comparisons of results between the exact solution and FEM analysis for
the laminated composite plate subjected to various parameters have been
presented in this chapter. The investigation of laminated plate having different
boundary, loading conditions and dimension of the laminated plate, have been
carried out to emphasize the effectiveness of the exact solution method. To
verify the analytical solutions, finite element program, Abaqus/CAE is used
interms of displacement and stresses results.
Chapter 5 Numerical Analysis of Laminated Plates
168
In the beginning of the modelling and analysis of the laminated plate with FEM,
various type of elements which are available in Abaqus library are used
including solids, conventional and continuum shells. The purpose of this initial
task is to obtain which element type best describe of the laminated plate
deformation and stresses as compared to the exact solution. From the findings,
solid elements C3D20 have proven close relationship behaviour of the
laminated plate corresponding to the exact solutions. From the outcome of this
investigation also verified that the remaining analysis would be based on
modeling of the laminated plate with solid element C3D20.
The novelty of solutions presented in this study is based from the work of
analytical solution of a fully clamped edges laminated plate. It is interesting to
note all the exact solutions agree well with the numerical solutions except when
the plate thickness to width ratio of 0.2 and the transverse shear stresses at the
outer surfaces of the plate edges. Both normal stresses and deflection at the
centre of the plate shows significant difference between the exact and the
numerical solutions. From this observation, further investigations are required to
best describe the response particularly for the case of thickness to width ratio is
less than 0.2 and also by increasing the number of sublayers of the laminate.
The other case presented in this chapter is the analysis of laminated plate
subjected to hydrostatic loading. The importance point to note from this case is
regarding the unsymmetrical loading type which shows that the relationship of
the unknown coefficients are no longer related.
It has been shown clearly that better and more precise results of displacements
and stresses would be obtained when the numbers of sub-layers have
increased within the plies. The results of exact solutions closely agree with the
FEM.
Chapter 5 Numerical Analysis of Laminated Plates
169
The other most important point that needs to be highlighted from these works is
that the analytical method involves massive matrix calculations whereby it is
common to encounter a problem at one point. There is a point of an exceptional
stage where the equations fail to be solved. This problem is known as
singularities.
For instance, the function ( ) ,x
xf1= has a singularity at x = 0, where the
solution is not defined or it falls to ∞± .
For the case of laminated plate, singularities are extremely important since
matrix calculations involve inverse matrix problem solving, where complex
analysis or solutions can be encountered. The issue of singularity can be
reduced or overcome by introducing high precision analysis. All the analytical
methods carried out in this investigation have been set to a very high precision,
between 400 (for a partially clamped edges plate) to 1000 decimal places (for a
fully clamped edges plate).
Chapter 6 Flexural Deformation of RC Slab with FRP
170
Chapter 6
FLEXURAL DEFORMATION OF RC
SLAB WITH FRP
In this chapter, the deflection of RC slab strengthened with FRP using FEM is
reviewed. This task was investigated during my first and second year of PhD
study. The objective of this study is to model and analyse FRP strengthened RC
slab using FEM program, Abaqus, so that its numerical flexural deformation
behaviour would correspond to the performance of composition materials of the
structure tested experimentally in the laboratory. To verify the accuracy of the
modelling deflection of the structures, the results from FEM simulation were
then compared with those of full scale FRP strengthened RC slabs testing
which was carried out in the University of Manchester [51] by others.
6.1 Numerical Modelling
Modelling of FRP strengthened RC slab using FEM is not as easy and straight
forward as compared with modeling composite laminates as a matter of fact of
their inherent material properties particularly for concrete and steel. This is
because concrete is a brittle material and steel is ductile when subjected to
tensile loading. By combining both materials allow them to behave beyond their
elastic region before they reach failure stage. Unlike FRP which dominantly
behaves in linear elastic only, the input data of plastic behaviour for concrete
and steel are mandatory to capture the structural response beyond elastic limit.
The full input data that was inserted into Abaqus and actual behaviour of these
materials which are tested and recorded in the laboratory are shown in Figure
6.1. The concrete in compression data is shown in Figure 6.1(a), steel bars in
tension in Figure 6.1(b) and FRP in tension in Figure 6.1(c).
Chapter 6 Flexural Deformation of RC Slab with FRP
171
Damaged plasticity model was selected in the modeling of FRP strengthened
RC slab because it provides different characteristics of concrete in tension and
compression.
The basic specimens parameters recorded in the laboratory are required for the
input data of modeling the RC slab with FEM. The concrete 28 days cube
compressive strength of 35 MPa and the average Poisson’s ratio of 0.2 were
noted [51]. The concrete was designed having a slump of 50mm with 10 mm
aggregate size, free water-cement ratio of 0.48 and 410kg/m3 of cement content
[51].
The ultimate tensile strength, modulus of elasticity and Poisson’s ratio of FRP
were 2970 N/mm2 ,172000 N/mm2 and 0.29 (average), respectively [51].
Tensile test of steel bar reinforcement noted in the laboratory showed that the
yield strength and ultimate strength reached up to 570 N/mm2 and 655 N/mm2,
respectively [51].
0
5
10
15
20
25
30
35
40
45
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
Strain
Str
ess
(MP
a)
(a)
Chapter 6 Flexural Deformation of RC Slab with FRP
172
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2
Strain
Str
ess
(MP
a)
(b)
0
500
1000
1500
2000
2500
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016
Strain
Str
ess
(MP
a)
(c)
Figure 6.1: Stress and strain relationship of (a) c oncrete (b) steel and (c) FRP [51]
Chapter 6 Flexural Deformation of RC Slab with FRP
173
Damage states of concrete in tension and compression are characterized
independently by two hardening variables, the plastic strains in tension and
compression, respectively. Cracking (tension) and crushing (compression) in
concrete are represented by increasing the values of the hardening (softening)
variables. These variables control the evolution of the yield surface and the
degradation of the elastic stiffness. Damage is calculated by the reduction of
strength against cracking and crushing strain as proposed by Jankowiak [56] as
shown in Table 6-1.
Table 6-1: The evolution of the damage variable for compression and tension
Compression Tension
Stress
(MPa) Plastic strain
Damage in
Compression
Stress
(MPa)
Cracking
Strain
Damage
in
Tension
16.39928 0 0 2.12947 0 0 16.61235 0.0000290544 0 1.50804 0.0008027 0.2918 18.15366 0.0000388282 0 1.08781 0.0016054 0.4892 21.66902 0.0000677221 0 0.81292 0.0024081 0.6183 24.92055 0.00010442 0 0.63589 0.0032108 0.7014 27.90273 0.000149086 0 0.52087 0.0040136 0.7554 30.60989 0.000201887 0 0.44294 0.0048163 0.792 33.03621 0.000262997 0 0.38591 0.005619 0.8188 35.17569 0.000332593 0 0.33996 0.0064217 0.8404 37.02219 0.000410858 0 0.29963 0.0072244 0.8593 38.56936 0.000497979 0 0.26220 0.0080271 0.8769 39.81070 0.000594149 0 0.22657 0.0088298 0.8936 40.73951 0.000699567 0 0.19253 0.0096325 0.9096 41.34890 0.000814438 0 0.16027 0.0104353 0.9247 41.63178 0.000938971 0 0.13006 0.011238 0.9389 41.65226 0.000982667 0 0.10219 0.0120407 0.952 41.50121 0.001114853 0.0036 0.07680 0.0128434 0.9639 41.03288 0.00125643 0.0149 0.05397 0.0136461 0.9747 40.24036 0.001407601 0.0339 0.03366 0.0144488 0.9842 39.11654 0.001568579 0.0609 0.01572 0.0152515 0.9926 37.65408 0.00173958 0.096 0 0.0160542 1 35.84544 0.001920828 0.1394 33.68287 0.002112553 0.1913 31.15837 0.002314991 0.2519 28.26372 0.002528387 0.3214
Chapter 6 Flexural Deformation of RC Slab with FRP
174
The above Table 6-1 values are obtained from the proposed expression by
Jankowiak [56] as shown below,
Damage in compression = stressecompressivimummax
stressecompressivstressecompressivimummax −
Damage in compression = stresstensileimummax
stresstensilestresstensileimummax −
The process of modeling of FRP strengthened RC slab is equivalently similar to
modeling composite laminate as described in the previous chapter. The
significant differences between them are the contribution of materials plastic
deformation and interaction of FRP and concrete interfaces. This will be
discussed in more detailed manner in the following section.
Having said that material properties provide major influence on the deformation
behaviour of the structures, these materials are modeled individually in Abaqus
to represent numerical simulation of the overall non linear structural behaviour.
Concrete material is conveniently modeled using Concrete Damaged Plasticity
in which Abaqus allows the flexibility of choosing the post-cracking behaviour
particularly the tensile property. It is also assumed that the main two failure
criteria are tensile cracking or compressive crushing of the concrete material.
Note that there is also another type of concrete modeling which is available in
Abaqus, namely, concrete smeared cracking. This model is intended for
concrete behaviour for relatively monotonic loadings under fairly low pressures.
Furthermore, damaged plasticity option is most suitable to predict flexural
effects but doest not consider for shear modeling in the post-cracking response.
The other important material properties inserted into Abaqus include:
Chapter 6 Flexural Deformation of RC Slab with FRP
175
Table 6-2: Basic material properties used in FEM
Concrete Steel CFRP
Density (kg/m3) 2400 7850 1700
Elastic Modulus
(MPa) 27385 200,000
150,000 (E1)
10,000 (E2)
Poisson ratio 0.2 0.3 0.3 ( υ12)
6.2 Geometric Properties of the Model
Schematic diagrams (not to scale) of the simply supported FRP strengthened
RC slab are shown in Figure 6.2
As we can see that the RC slab is symmetrical in geometric and loading
conditions, only a quarter of the original dimension of the simply supported RC
slab is modelled and analysed in FEM as shown in Figure 6.3. This means that
the model is simplified into a 800 x 800 x 150 mm dimension. A concentrated
loading of area 250 x 250 mm is applied at the centre of the top surface of the
slab. RC slab is also reinforced with eight numbers of 12mm diameter steel bars
uniformly distributed in each direction. Such RC slab is then strengthened with
four CFRP sheets, each of them has 100 mm width and 1.2 mm thick which are
bonded to the bottom surface of the slab. Note that the position of FRP shown
in Figure 6.3 is being lowered down for illustration purpose only.
Chapter 6 Flexural Deformation of RC Slab with FRP
176
(a) RC slab without CFRP
1600
1600
150
Steel bars
250
250
Chapter 6 Flexural Deformation of RC Slab with FRP
177
(b) RC slab with CFRP
Figure 6.2: View of RC slab model (a) without FRP ( b) with FRP
1600
1600
150
FRP
Chapter 6 Flexural Deformation of RC Slab with FRP
178
Figure 6.3: FEM of CFRP strengthened RC slab
6.3 Element Type
The tasks of meshing and choosing element types for FRP strengthened RC
slab are exactly similar to modeling composite laminates as described in the
previous chapter. As mentioned earlier, Abaqus has a vast element library to
choose from, depending on the model type, one can select any element to suit
the true model behaviour. Choosing an element can be based on the specific
element characteristics such as first or second order; full or reduced integration;
hexahedra/quadrilaterals etc.
In this case, a discrete modeling is adopted for RC slab in which all materials
are modelled separately and individually. There are five elements chosen to
simulate the behaviour of each material or component. These are described as
below,
CFRP Rigid support
Chapter 6 Flexural Deformation of RC Slab with FRP
179
6.3.1 Concrete Slab
Three dimensional solid elements are used to simulate non linear analysis of
concrete slab material. Abaqus Standard solid elements have two categories,
first category is first order (linear) interpolation and the other is second order
(quadratic) elements which use quadratic interpolation to calculate
displacements degree of freedom. Typically, an eight node first order brick
element is known as C3D8 while a quadratic element has 20 node, i.e. C3D20.
Solid (continuum) elements are more accurate and suitable for concrete slab, if
not distorted, particularly for quadrilaterals and hexahedra shape. In this study,
only solid element of C3D8 is used since this type of element is suitable to
simulate a structure with shear loading problem which resulting no difficulty with
‘shear locking’ phenomenon.
6.3.2 Steel reinforcement bars
Truss elements are assigned to model steel reinforcement bars as they are
suitable for line like structural components that carry axial loading along the axis
or at the centre line of the element. A two node three dimensional straight truss
element (T3D2) is selected which uses linear interpolation. An alternative
element to model steel bar is by using rebar element. However, Abaqus/CAE
does not provide results or outputs in the visualization module when using rebar
element.
All steel bar reinforcements which are assigned as truss elements, embedded
into the host element (solid) which is concrete slab. This means that the
translational degrees of freedom at each embedded node of truss elements are
eliminated and constrained to and shared with the three dimensional solid
element nodes. It is important to note that Abaqus only considers host elements
(concrete slab) which can have only translational degrees of freedom and the
number of translational degrees of freedom at a node of embedded element
(steel bars) must be identical to the number of translational degrees of freedom
at a node of the host elements (solid).
Chapter 6 Flexural Deformation of RC Slab with FRP
180
6.3.3 FRP sheets
As the thickness of the FRP sheets is quite smaller than the other dimensions, it
is more economic in computation cost to model them using a general
conventional shell element (S4). Due to its thin dimension, Abaqus conveniently
defined its thickness through the section property definition, rather than using
element nodal geometry with continuum shell elements. Conventional shell
elements S4 have three displacements and two rotational degrees of freedom
whereas continuum shell elements have only displacement degrees of freedom.
S4 is chosen in this study as it gives accurate solutions to out of plane bending
of FRP strengthened RC slab, it is suitable for large strain analysis and it
neglects transverse shear deformation as the shell thickness decreases. This
type of element is based on thin shell theory, i.e. the Kirchhoff theory.
6.3.4 Interaction of FRP and concrete slab
An interaction was allowed within this region to simulate any response during
loading particularly delamination of FRP. Like adhesive material, cohesive
element was used to model the delamination phenomenon at the interface of
concrete slab and FRP. This interaction is called the traction – separation laws
which describe the constitutive behaviour of cohesive elements. Cohesive
element COH3D8 was assigned to model such interaction in such a way that
the top nodes of cohesive elements were connected to the bottom nodes of
solid elements (concrete slab) whilst the bottom nodes of the cohesive elements
were joined together with the shell nodes (FRP). A cohesive element is an
independent element that is not embedded in another element. To assign this
element, a new thin layer was created between concrete and FRP surfaces.
The minimum thickness Abaqus allows in CAE is a layer of 0.0001 mm. After
partition of either concrete or FRP surface to allow for cohesive designated area,
then cohesive property can be assigned to this new section.
Damage of the interface was assumed to initiate when the maximum nominal
stress ratio reached a value of one in any direction. Damage initiation was
defined using the maximum nominal stress values and the progression of
damage at the interfaces was modeled using the mixed mode energy
Chapter 6 Flexural Deformation of RC Slab with FRP
181
independent behaviour. After damage initiation, the damage evolution variable
until failure was modeled using linear softening law. The strength values of
damaged property using Maximum damage option for nominal stresses can be
determined from either the material manufacturer or through experimental test
(peel test for normal and lap shear test for tangential). If only one value of
strength was provided, this value can be used for all three directions.
Figure 6.4: Damage traction-separation response use d in FEM [55]
Damage initiation is assumed when maximum nominal stress ratio reaches to
one as expressed below,
1=
o
t
to
s
so
n
n
t
t,
t
t,
t
tmax
where o
t
o
s
o
ntandt,t represent the peak values of the nominal stress when the
deformation is purely normal to the interface or purely in the first and second
shear direction, respectively. In this study, due to the lack of information, the
values of MPa.tandMPa.t,MPa.t o
t
o
s
o
n7537538641 === were assumed
[51].
Traction (stress)
Separation or slip (displacement)
damage initiation
damage evolution
Gf
Chapter 6 Flexural Deformation of RC Slab with FRP
182
Damage evolution was defined based on the energy that is required to
propagate a tensile crack of unit area, known as the fracture energy. The
fracture energy is equal to the area under the traction-separation curve.
Damage evolution with mixed mode ratio of energy is opted in the modeling and
fracture energy, Gf, of concrete can be estimated using the equation [57].
70.
fcmo
cm
Fof f
fGG
= (FEM input of Gf = 0.757 Nmm/mm2)
where fcmo = 10 MPa
GFo is the base value of fracture energy, depending on the maximum aggregate
size. GFo equal to 0.0275 Nmm/mm2 was used in the modeling.
fcm is the average compressive strength of concrete.
6.3.5 Boundary condition
In order to simulate the behaviour of RC slab using FEM close enough to the
test result, the slab’s symmetric boundary conditions are modeled using a rigid
support. The contact area between the surface of concrete and the support has
to be defined into tangential and normal surfaces. Frictionless is opted in the in
plane tangential interaction and the normal component only applies when the
slab comes into contact with the rigid support. This normal interaction will cease
when slab surface is no longer in contact with the rigid support and it can only
happen at the four corners of the slab being uplifted during the high magnitude
loading stage.
RC slab is positioned horizontally on a rigid support which is modeled using a
three dimensional four node bilinear quadrilateral surface element (R3D4). This
element is defined as master surfaces which in contact with the concrete slab.
Chapter 6 Flexural Deformation of RC Slab with FRP
183
6.4 Tension Stiffening
Throughout the overall process of modeling and analyzing FRP strengthened
RC slab using Abaqus, the task of getting the ideal plastic behaviour of the
structure to match closely with the experimental test results was in fact the most
difficult, challenging and time consuming job. There were many trials and error
jobs undertaken during this stage.
The model requires input data of the uniaxial tensile and compressive response
of plain concrete for damaged plasticity as shown in Figure 6.5 and Figure
6.1(a), respectively.
Figure 6.5: Concrete tensile response characterized by damaged plasticity
From Figure 6.5, it shows that at initial stage, the stress-strain response gives a
linear elastic behaviour until it reaches the failure stress, ft (about 2 MPa). This
failure stress corresponds to the beginning of micro-cracking in the concrete
slab. Since concrete is weak in tension, the failure strain of concrete is so low
and from this point, the formation of micro cracks is represented
macroscopically with a softening stress strain response which induces strain
localization in the concrete material.
cr
nnσ
ft
εt cr
nnε
Chapter 6 Flexural Deformation of RC Slab with FRP
184
In Abaqus, steel bars reinforcement are embedded into the concrete, however,
concrete behaviour is considered independently of the rebar. The effects related
to steel bar reinforcement and concrete interface such as bond slip and dowel
action, are modelled approximately by “tension stiffening” into the concrete
modeling. This stiffening simulates load transfer across the cracks through the
steel bars.
Such concrete strain softening response is the most crucial part to determine
the non elastic behaviour of RC slab by FEM in this study. This post failure
behaviour also allows for the effects of the steel bars reinforcement interaction
with the concrete to be simulated in a simpler manner.
As modeling approach, tension stiffening is required in the concrete damaged
plasticity and there are three different methods of tension stiffening available in
Abaqus. Firstly tension stiffening can define the post failure by stress-strain
relation, or stress-displacement or lastly by applying a fracture energy cracking
criterion.
There are three important parameters need to be defined for the descending
branch of the tensile stress-strain relation or tension stiffening:
1) The tensile strength of concrete at which a fracture zone started
2) The area under the stress-strain curve (fracture energy, Gf)
3) The shape of the descending branch
Out of these parameters, the first two are considered as material constants,
however, the shape of the descending branch varies depending on which
formulation to be used. Choosing the shape of this descending branch is the
utmost critical decision in modeling the non linear behaviour of RC slab, as it
influences the true behaviour of the structures as per experimental test results.
Chapter 6 Flexural Deformation of RC Slab with FRP
185
6.4.1 Non-linear tension stiffening stress-strain r elation
This relation gives the stiffening behaviour of concrete in terms of cracking
strain proposed by Hordijk [58] as shown in Figure 6.6, which shows an
exponential stress – strain descending path curve.
Figure 6.6: Exponential tension stiffening (Hordijk ) [58]
The above descending exponential curve can be plotted using the Hordijk
equation:
where the parameters c1 = 3 and c2 = 6.93
t
fcr
ult,nn f.h
G.ε 1365=
h is the crack bandwidth. For solid elements, the default is Vh = where V is
the volume of the element.
cr
nnσ
ft
εt cr
nnε
Gf
cr
ult,nnε
Chapter 6 Flexural Deformation of RC Slab with FRP
186
6.4.2 Linear tension stiffening stress-strain relat ion
In this case, the descending branch is linear as shown in Figure 6.7,
Figure 6.7: Linear tension stiffening [58]
The relation of the above linear descending stress-strain is given by
( )
−=
cr
ult,nn
cr
nn
t
cr
nn
cr
nnε
ε
fεσ 1
cr
nnσ
ft
εt cr
nnε
Gf
cr
ult,nnε
Chapter 6 Flexural Deformation of RC Slab with FRP
187
6.4.3 Multi-linear tension stiffening stress-strain relation
Multi-linear descending tension stiffening can also be used to define the plastic
behaviour of the model. This type of tension stiffening is used in this study as
shown in Figure 6.8,
Figure 6.8: Multi-linear tension stiffening [58]
The selection of the above shape of tension stiffening curve can be painstaking
tasks to do for modelling the inelastic behaviour of RC slab. Therefore choosing
the appropriate curve is vital and time consuming process in the modeling of
non linear analysis of RC slab.
In this study, multi-linear descending tension stiffening is used which exhibits
good relationship to the available test results.
The results of the numerical solutions are explained in the following section.
cr
nnσ
ft
εt cr
nnε
Gf
cr
ult,nnε
Chapter 6 Flexural Deformation of RC Slab with FRP
188
6.5 FEM Against Experimental Test Results
Getting the results of numerical non-linear analysis of FRP strengthened RC
slab using Abaqus took less time than obtaining analytical results using a
standard PC desktop. On average, Abaqus required about one quarter the time
of running the programming code using Mathematica. Depending on the mesh
size, element and model type, method of integration etc, Abaqus normally
needs about 3 to 4 hours for a complete run.
After spending tremendous time and effort in modeling the structures using
Abaqus, the most vital experience towards modeling FRP strengthened RC slab
using concrete damaged plasticity is gained using an ideal tension stiffening
behaviour of concrete. As there are many options available with regard to
tension stiffening including, linear, bi-linear, multi-linear and exponential,
choosing a suitable type of tension stiffening behaviour is crucial to ensure the
true response of the structure as close as to experimental test results.
Tension stiffening behaviour defines the post failure for cracked concrete which
allows for the effects of interaction between steel bars reinforcement and
concrete. Estimation of the tension stiffening effect depends on some factors
such as the ratio of reinforcement, the quality of bond between steel bars and
the concrete, the relative size of the concrete aggregate compared to the
diameter of steel bars and the mesh size. One can start to assume that
stiffening behaviour to be linear stress – strain relationship from maximum to
zero stress at a total strain of about 10 times the strain at failure. The typical
tensile strain value of concrete at failure is 0.00016. This means that tension
stiffening reduces the stress linearly from maximum to zero at a total strain of
about 0.0016 (or may be less than) can be sufficiently applied for initial
estimation. Numerical solutions are shown based on various tension stiffening
behaviour and element types as shown in Figure 6.9 and Figure 6.10,
respectively.
Chapter 6 Flexural Deformation of RC Slab with FRP
189
From Figure 6.9 shows the relationship of the applied loading to central
deflection of RC slab by varying the tension stiffening effect as described in
Section 6.4 against the results obtained from the experimental test [51].
0
50
100
150
200
250
300
0 10 20 30 40
Central Deflection (mm)
Load
(kN
) Exp TestBi-LinearLinearExponential
Figure 6.9: Relationship of load against central de flection of RC slab by variable tension stiffening response
0
50
100
150
200
250
300
0 10 20 30 40
Central Deflection (mm)
Load
(kN
)
Exp TestFEM C3D8FEM C3D8RFEM C3D20
Figure 6.10: Relationship of load against central d eflection of RC slab by variable element types
Chapter 6 Flexural Deformation of RC Slab with FRP
190
0
50
100
150
200
250
300
350
400
450
0 10 20 30
Central Deflection (mm)
Load
(kN
)RC slab FEM
RC slab experimental
RC slab with CFRPFEMRC slab with CFRPexperimental
Figure 6.11: Load against central deflection of un- strengthened and strengthened of RC slab with CFRP
From Figure 6.9 and Figure 6.10, the numerical solutions explicitly show that
multi-linear tension stiffening behaviour and three dimensional solid element
type C3D8 produce better results in modeling of RC slab with Abaqus as
compared to experimental test results. Rigid elements are used to allow the
contact interaction between the slab surface and the supports or the base. This
would simulate the corner of the slab being uplifted during the loading.
Same conclusion can be made with the analysis of RC slab strengthened with
CFRP as shown in Figure 6.11. The response generally agrees well for both
unstrengthened and strengthened RC slab with FRP.
This investigation shows the effectiveness of CFRP in strengthening of concrete
structures, both in serviceability and ultimate limit states. The use of CFRP in
this study has increased the ultimate RC slab flexural strength to about 43%
and also has reduced the central deflection to approximately 58%. In FEM, after
strengthening with CFRP, the serviceability load has increased to about 37%. It
was also noted that FRP strengthened RC slab was failed due to concrete
crushing whilst FRPs materials are still intact with concrete at failure.
Chapter 6 Flexural Deformation of RC Slab with FRP
191
6.6 Concrete slab reinforced with FRP
In this section, only linear analysis of layered plate or slab is considered and
analysed using the exact state space method. The purpose of this study is to
evaluate the effectiveness of state space method to concrete slab with CFRP
over the classical laminated plate theory. Their differences between these
approaches are being compared in-terms of determination of the central
deflection of the plate.
The effectiveness of a single sheet of typical CFRP reinforced concrete slab
subjected to uniformly distributed loading is investigated. For the given
mechanical properties of a single CFRP sheet, flexural deformation of concrete
slabs with various thickness is analysed.
The slab to be considered, as shown in Figure 6.12 (a), is originally designed
for the serviceability within the elastic limits. Due to slab degradation or some
other unspecified reasons, the structure is not able to bear more loading in
elastic range. In order to restore its elastic design and service, CFRP sheet is
bonded to the bottom of the slab. As a result of this retrofitting work, the whole
composite slab structure consists of two layers with different material properties.
These two material layers can be further divided many thin sub-layers.
The slab considered now consists of subdivided layers as shown in Figure 6.12
(b).
Chapter 6 Flexural Deformation of RC Slab with FRP
192
( a )
( b )
Figure 6.12: Geometry and coordinate systems of the layered slab
CFRP
a
b
x
z
y
h
concrete
hc
hCFRP
(1) d1
(2) d2
( j ) dj
( N ) dN
a
b
x ,u
z , w
y , v
h
Chapter 6 Flexural Deformation of RC Slab with FRP
193
Material properties used for both analytical and numerical analysis are shown
below:
Concrete: Compressive strength 30MPa
Elastic’s modulus 33GPa Poisson ratio 0.2 and Shear modulus 13.75GPa.
CFRP sheets: Longitudinal elastic modulus 135 GPa (E1)
Transverse elastic modulus 10 GPa (E2) Poisson’s ratio 0.3 (ν12) Shear moduli 5 GPa (G12) and 3.85 GPa (G23).
The calculated or reduced material parameters which will be inserted in the
analytical and numerical analyses, based on the above properties, are
For concrete:
GPaCC
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
67363750
3750375012502501250
11
11
44
11
55
11
66
11
33
11
23
11
13
11
22
11
12
.,.
,.,.,,.,.,,.
==
=======
For CFRP ply:
1048950136364013636403034830
0936930119153030348301191530753313
11
44
11
55
11
66
11
33
11
23
11
13
11
22
11
12
11
11
.,.,.,.
,.,.,.,.,.
====
=====
c
f
c
f
c
f
c
f
c
f
c
f
c
f
c
f
c
f
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
where cij
fij
CandC are the stiffness coefficients of CFRP and concrete,
respectively.
Boundary Conditions
In order to simplify the calculation, only a partly clamped slab is considered, two
opposite edges x = 0 and a are clamped and other two edges y = 0 and b, are
simply supported.
Chapter 6 Flexural Deformation of RC Slab with FRP
194
Geometric Conditions
In this case study, the in-plane dimensions are taken from the previous model,
however, the overall thickness are varies. The thickness of concrete slab are
varies from hc of 0.0988, 0.1988, 0.2988, 0.3988 to 0.5988 however FRP
thickness remains unchanged (hFRP = 0.0012). Therefore, the overall thickness
(h) of the composite slabs considered are 0.1, 0.2, 0.3. 0.4 and 0.6.
Considering the boundary conditions and the loading case, these imply that in
the current numerical analysis
( ) ( ) ( ) ( ) ( ) ( ) ( )zyUzyUandVzxV ab ,,0, 00 −=== .
Exact solution against FEM
In order to verify the analytical method of the slab strengthened with CFRP,
comparisons are provided with the Finite Element Method (FEM).
Since it is an elastic analysis, a proportional amount of the loading, qo, assigned
for the finite element analysis, is 1Pa. Quadratic elements with 20 nodes
(C3D20) were used to model the laminate structures and perfectly bonding (tie)
was also chosen at the interface between the concrete and CFRP material. All
other material, geometric parameters and boundary conditions were the same
as the analytical analysis. However, various cases with different ratio of
thickness to width (h/a) are considered.
The results are presented as follows:
Chapter 6 Flexural Deformation of RC Slab with FRP
195
Table 6-3: Deflection distribution (WC 11/qh) for h/a = 0.1 W C11/qh
No CFRP With FRP z/h
Exact FEM Exact FEM
0 27.52862 27.49960 26.34378 26.32090 0.076 27.57216 27.54290 26.38548 26.36240 0.152 27.60689 27.57790 26.41876 26.39580 0.228 27.63361 27.60470 26.44441 26.42160 0.304 27.65259 27.62370 26.46268 26.43990 0.38 27.66400 27.63520 26.47374 26.45100
0.456 27.66797 27.63920 26.47773 26.45500 0.532 27.66457 27.63580 26.47471 26.45200 0.608 27.65379 27.62500 26.46468 26.44200 0.684 27.63557 27.60680 26.44760 26.42490 0.76 27.60979 27.58100 26.42335 26.40070
0.836 27.57627 27.54750 26.39176 26.36910 0.912 27.53476 27.50590 26.35259 26.32990 0.988 27.48461 27.45600 26.30529 26.28280 0.988 27.48461 27.45600 26.30529 26.28280 0.992 27.48172 27.45310 26.30136 26.27890 0.996 27.47880 27.45020 26.29740 26.27500
1 27.47586 27.44730 26.29340 26.27110
Table 6-4: Deflection distribution (WC 11/qh) for h/a = 0.2 W C11/qh
No CFRP With FRP z/h
Exact FEM Exact FEM
0 4.53008 4.53352 4.45659 4.45992 0.0765 4.54146 4.54503 4.46752 4.47098 0.1529 4.54829 4.55206 4.47400 4.47765 0.2294 4.55146 4.55530 4.47690 4.48062 0.3058 4.55142 4.55529 4.47665 4.48041 0.3823 4.54853 4.55242 4.47364 4.47741 0.4588 4.54307 4.54696 4.46811 4.47189 0.5352 4.53516 4.53905 4.46022 4.46400 0.6117 4.52480 4.52870 4.44997 4.45375 0.6882 4.51188 4.51577 4.43723 4.44101 0.7646 4.49613 4.50001 4.42178 4.42554 0.8411 4.47717 4.48103 4.40322 4.40697 0.9175 4.45448 4.45831 4.38107 4.38479 0.994 4.42730 4.43117 4.35458 4.35834 0.994 4.42730 4.43117 4.35458 4.35834 0.996 4.42651 4.43040 4.35350 4.35727
0.99800 4.42573 4.42962 4.35241 4.35619 1 4.42494 4.42883 4.35131 4.35511
Chapter 6 Flexural Deformation of RC Slab with FRP
196
Table 6-5: Deflection distribution (WC 11/qh) for h/a = 0.3 W C11/qh
No CFRP With FRP z/h
Exact FEM Exact FEM
0 1.88921 1.89294 1.87608 1.87958 0.07662 1.88468 1.88870 1.87139 1.87517 0.15323 1.87713 1.88132 1.86370 1.86764 0.22985 1.86756 1.87180 1.85401 1.85800 0.30646 1.85667 1.86094 1.84301 1.84704 0.38308 1.84504 1.84933 1.83130 1.83535 0.45969 1.83308 1.83739 1.81927 1.82334 0.53631 1.82100 1.82531 1.80714 1.81121 0.61292 1.80881 1.81312 1.79493 1.79899 0.68954 1.79631 1.80061 1.78244 1.78650 0.76615 1.78312 1.78741 1.76931 1.77335 0.84277 1.76865 1.77291 1.75494 1.75897 0.91938 1.75211 1.75633 1.73859 1.74257 0.99600 1.73245 1.73666 1.71920 1.72318 0.99600 1.73245 1.73666 1.71920 1.72318 0.99733 1.73207 1.73628 1.71868 1.72266 0.9987 1.73169 1.73590 1.71815 1.72214
1 1.73130 1.73552 1.71763 1.72161
Table 6-6: Deflection distribution (WC 11/qh) for h/a = 0.4 W C11/qh
No CFRP With FRP z/h
Exact FEM Exact FEM
0 1.15025 1.15331 1.14675 1.14964 0.0767 1.13383 1.13727 1.13025 1.13351 0.1534 1.11513 1.11870 1.11148 1.11488 0.2301 1.09526 1.09889 1.09155 1.09500 0.3068 1.07521 1.07888 1.07143 1.07492 0.3835 1.05578 1.05948 1.05193 1.05545 0.4602 1.03755 1.04127 1.03364 1.03718 0.5368 1.02087 1.02459 1.01688 1.02043 0.6135 1.00579 1.00951 1.00175 1.00529 0.6902 0.99212 0.99583 0.98803 0.99157 0.7669 0.97938 0.98307 0.97527 0.97878 0.8436 0.96678 0.97043 0.96268 0.96617 0.9203 0.95324 0.95684 0.94921 0.95265 0.997 0.93734 0.94091 0.93345 0.93685 0.997 0.93734 0.94091 0.93345 0.93685 0.998 0.93711 0.94068 0.93314 0.93654 0.999 0.93688 0.94045 0.93282 0.93622
1 0.93665 0.94022 0.93250 0.93590
Chapter 6 Flexural Deformation of RC Slab with FRP
197
Table 6-7: Deflection distribution (WC 11/qh) for h/a = 0.6
W C11/qh
No CFRP With FRP z/h
Exact FEM Exact FEM
0 0.72483 0.72724 0.72445 0.72679 0.0768 0.68927 0.69217 0.68886 0.69169 0.1535 0.65159 0.65461 0.65117 0.65411 0.2303 0.61325 0.61633 0.61281 0.61581 0.3071 0.57587 0.57897 0.57540 0.57842 0.3838 0.54088 0.54396 0.54037 0.54337 0.4606 0.50939 0.51242 0.50883 0.51178 0.5374 0.48212 0.48510 0.48151 0.48441 0.6142 0.45941 0.46233 0.45874 0.46157 0.6909 0.44115 0.44401 0.44042 0.44319 0.7677 0.42677 0.42957 0.42599 0.42870 0.8445 0.41522 0.41795 0.41440 0.41705 0.9212 0.40487 0.40753 0.40405 0.40664 0.998 0.39350 0.39608 0.39273 0.39524 0.998 0.39350 0.39608 0.39273 0.39524
0.9987 0.39338 0.39597 0.39258 0.39509 0.9993 0.39327 0.39586 0.39243 0.39493
1 0.39316 0.39575 0.39227 0.39478
0
0.2
0.4
0.6
0.8
1
26.0 26.5 27.0 27.5 28.0
WC11/qh
z/h
Concrete (Exact)
Concrete with FRP(Exact)Concrete (FEM)
Concrete with FRP(FEM)
Figure 6.13: Deflection distribution (W C 11/qh) at x=a/2, y = b/2 h/a = 0.1
Chapter 6 Flexural Deformation of RC Slab with FRP
198
0
0.2
0.4
0.6
0.8
1
4.3 4.4 4.4 4.5 4.5 4.6 4.6
WC11/qh
z/h
Concrete (Exact)
Concrete with FRP(Exact)Concrete (FEM)
Concrete with FRP(FEM)
Figure 6.14: Deflection distribution (W C 11/qh) at x=a/2, y = b/2 h/a = 0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.70 1.75 1.80 1.85 1.90
W C11/qh
z/h
Concrete (Exact)
Concrete with FRP(Exact)Concrete (FEM)
Concrete with FRP(FEM)
Figure 6.15: Deflection distribution (W C 11/qh) at x=a/2, y = b/2 h/a = 0.3
Chapter 6 Flexural Deformation of RC Slab with FRP
199
0
0.2
0.4
0.6
0.8
1
0.90 0.95 1.00 1.05 1.10 1.15 1.20
WC11/qh
z/h
Concrete (Exact)
Concrete with FRP(Exact)Concrete (FEM)
Concrete with FRP(FEM)
Figure 6.16: Deflection distribution (W C 11/qh) at x=a/2, y = b/2 h/a = 0.4
0
0.2
0.4
0.6
0.8
1
0.35 0.45 0.55 0.65 0.75
WC11/qh
z/h
Concrete (Exact)
Concrete with FRP(Exact)Concrete (FEM)
Concrete with FRP(FEM)
Figure 6.17: Deflection distribution (W C 11/qh) at x=a/2, y = b/2 h/a = 0.6
Chapter 6 Flexural Deformation of RC Slab with FRP
200
From Table 6-3 to Table 6-7 or Figure 6.13 to Figure 6.17, it can be seen that
CFRP can effectively be used to reduce the deflection of the concrete slab
under service load within the elastic stage. For the typical material property of
CFRP as mentioned before, a single sheet of CFRP can reduce the deflection
of the slab up to the thickness to width ratio reaches to 0.4. When the thickness
ratio of the slab increases beyond 0.4, a single sheet of CFRP is insufficient and
may require more sheets of CFRP in order to reduce the deflection. This can
be shown in Figure 6.16 and Figure 6.17 where the contribution from a single
CFRP can be neglected.
From Table 6-3 and Figure 6.13, the exact solutions show that the magnitudes
are slightly higher than the FEM results. However, the FEM outputs are slightly
overestimated than the exact solutions when h/a equals to 0.2 and higher.
It can also be shown that the maximum deflection for plate with thickness to
width ratio (h/a) of 0.1 is located at about half the thickness of the plate and
gradually, the position of maximum deflection moves towards the top surface of
the plate as the ratio increases.
6.7 Conclusions
Modelling of RC slab strengthening with FRP using finite element analysis
program, Abaqus has been reviewed in this chapter. The main objective of this
task is to review and simulate the flexural deflection of the structure
corresponding to the available experimental test result.
From the outcome of the simulation work, the numerical solution of FEM
matches well to the test result which shows the significant enhancement of
loading capacity of RC slab and reducing it’s central deflection after FRP is
applied.
The central deflection of concrete slab with and without FRP agrees well
between the exact solutions and FEM results.
Chapter 7 Conclusions and Recommendations
201
Chapter 7
CONCLUSIONS AND
RECOMMENDATIONS
7.1 Conclusions
The analysis of laminated composite plate based on the 3D elasticity and state
space method under general boundary conditions have been presented.
All analytical solutions are compared and verified with numerical solutions
obtained from the finite element program, Abaqus. Modeling of plate using solid
element is more sufficient than shell element as FEM solutions agree well in
general. However, FEM results obviously fail to satisfy the traction free
conditions on the top and bottom surfaces of the plate. This means that FEM
predicts a magnitude of transverse shear stresses on the top and bottom
surfaces of the plate’s boundary edges, whereas these stresses are still zero
values obtained from the exact solution. The accuracy of FEM solutions are
determined using solid elements with higher order shape function (quadratic)
rather than low order shape function (linear).
Solid elements provide a more accurate solution than shell elements as
compared to the exact solutions because of the following:
• The transverse shear effects of solid elements are predominant than
shell elements.
• Solid elements do not ignore the normal stress.
• Solid elements provide accurate interlaminate stresses including near
localized regions of complex loading or geometry.
Chapter 7 Conclusions and Recommendations
202
By using transfer matrix method and preserving the continuity conditions at the
interfaces and imposing the in-plane traction along the edges which satisfies the
boundary conditions, exact elasticity solution is very accurate since it satisfies
the elasticity equations at every point of the laminate plate. Modeling the
composite plates using solids elements of C3D20 are more precise than using
shell elements. These are due to the fact that solids elements have more effect
on transverse stresses and interlaminar stresses which are obtained and
calculated by Abaqus based on the theory of linear elasticity.
From this study, it concludes that exact 3D elasticity solution can produces
exact solution of composite plate of any reduced material properties, boundary
conditions and loading conditions. Solid element C3D20 provides very good
result compared with the 3D exact elasticity solution with respect to deflection
and in-plane stresses. Meanwhile, shell element shows a better result than solid
elements for transverse shear stresses. FEM does not get an exact solution for
the transverse shear stresses.
The breakthrough and novel exact solutions of fully clamped composite plate
have been achieved in this study. Such new results can be used as a
benchmark solution for further analysis.
The use of computer machine with Linux operating system is an advantage in-
terms of computing efficiency rather than using machine with Windows
operating system.
FEM model was also enabled to predict the performance of FRP strengthened
RC slab that was tested in the laboratory. Both the numerical solutions and
experimental test results verified that ultimate flexural capacity of RC slab
strengthened with CFRP has increased by 43%. Solid element C3D8 and multi-
linear response are the ideal element type and best possible tension stiffening
behaviour to describe the performance of flexural deformation of FRP
strengthened RC slab.
Chapter 7 Conclusions and Recommendations
203
The accuracy of the numerical results depends on the thickness to width ratio,
the point of interest and the material properties of the laminated plate.
7.2 Future Works Recommendations
The following tasks can be performed for future investigation with regard to 3D
elasticity of laminated composite plates:
• Increasing number of m and n as these would improve the accuracy of
the results of analytical solutions.
• Increasing the number of sublayers within each ply of laminate. The
thinner the sublayer, the better the results.
• In the FEM analysis, the solution can give only an approximate solution
such that equilibrium is satisfied on average over an element. FEM does
not preserve the continuity between the interfaces. More accuracy of the
results can be obtained when using fine meshes but keeping in mind that
fine meshes generate more elements, hence costing computing process
to run.
• The novel solutions of fully fixed clamped edges laminated plate can be
used as benchmark for further analysis with different loading conditions
and material properties.
• Numerical simulations of FRP strengthened RC slab using Abaqus were
only modeled and analysed using concrete damaged plasticity which
mainly accounts for load – deflection relationship. From this adopted
method, the choices of various tension stiffening greatly influence the
global flexural response of such model. Further analysis using smeared
crack modeling can be used to incorporate the shear modeling and crack
propagation.
• Having developed code of analytical method, the exact solutions of
structural performance of any reduced material properties and boundary
conditions can be further explored such as the graphene structures.
• Further investigation of the simulation can be performed to get better
accuracy such as laminated plate having different material properties.
Chapter 7 Conclusions and Recommendations
204
• It is interesting to analyse further on the laminated plate having thickness
to width ratio of less than 0.2, particularly to compare the central
deflections and in-plane stresses at the centre of the plate.
References
205
REFERENCES
1. Bank, L.C., Composites for Construction: Structural Design with FRP materials. 2006: John Wiley & Sons, USA.
2. Tong, L., Mouritz, A.P., and Bannister, M.K., 3D Fibre Reinforced Polymer Composites. 2002: Elsevier Science Ltd, UK.
3. Jones, R.M., Mechanics of Composite Materials. 1999: Philadelphia, Pa. ; London : Taylor & Francis
4. Love, A.E.H., The Small Free Vibrations and Deformation of a Thin Elastic Shell. Philosophical Transactions of the Royal Society of London. A, 1888. 179: p. 491-546.
5. Murthy, M.V.V., An Improved Transverse Shear Deformation Theory for Laminated Anisotropic Plates. 1981, NASA Technical Paper: Virginia, USA.
6. Timoshenko, S. and Woinowsky-Krieger, S., Theory of Plate and Shells. Second Edition ed. 1959: McGraw-Hill.
7. Wu, Z.J., Wu, H.J., and Han, F., Elasticity. 2010: Beijing Institute of Technology Press.
8. Reissner, E., The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics, 1945. 12: p. 69-77.
9. Mindlin, R.D., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics, 1951. 18: p. 31-38.
10. Wang, C.M., et al., Relationships between bending solutions of Reissner and Mindlin plate theories. Engineering Structures, 2001. 23(7): p. 838-849.
11. Ambartsumyan, S.A., Theory of anisotropic plates. 1970: Technomic Publishing Co. Stamford Conn.
12. Pagano, N.J. and Halpin, J.C., Influence of end constraints in the testing of anisotropic bodies. Journal of Composite Materials, 1968. 2(4): p. 18-31.
13. Whitney, J.M. and Leissa, A.W., Analysis of heterogeneous anisotropic plates. Journal of Applied Mechanics, 1969. 28: p. 261-266.
14. Whitney, J.M., The effects of transverse shear deformation on the bending of laminate plates. Journal of Composite Materials, 1969. 3(4): p. 534-547.
15. Puppo, A.H. and Evensen, H.A., Interlaminar shear in laminated composites under generalised plane stress. Journal of Composite Materials, 1970. 4(2): p. 204-220.
16. Pipes, R.B. and Daniel, I.M., Moire analysis of the interlaminar shear edge effect in laminated composites. Journal of Composite Materials, 1971. 5(2): p. 255-259.
17. Pipes, R.B. and Pagano, N.J., Interlaminar stresses in composite laminates under uniform axial extension Journal of Composite Materials, 1970. 4(4): p. 538-548.
18. Pipes, R.B. and Pagano, N.J., The influence of stacking sequence of laminate strength. Journal of Composite Materials, 1971. 5(1): p. 50-57.
References
206
19. Rybicki, E.F., Approximate three dimensional solutions for symmetrical laminates under inplane loading. Journal of Composite Materials, 1971. 5(3): p. 354-360.
20. Lo, K.H., Christensen, R.M., and Wu, E.M., Stress solution determination for high order plate theory. International Journal of Solids and Structures, 1978. 14(8): p. 655-662.
21. Pagano, N.J., Exact solutions for composites laminates in cylindrical bending. Journal of Composite Materials, 1969. 3: p. 398-411.
22. Pagano, N.J., Exact solutions for rectangular bidirectional composites and sandwich plates Journal of Composites Materials, 1970. 4: p. 20-34.
23. Srinivas, S. and Rao, A.K., Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. International Journal of Solids and Structures, 1970. 6(11): p. 1463-1481.
24. Bahar, L.Y., A state space approach to elasticity. Journal of the Franklin Institute, 1975. 299(1): p. 33-41.
25. Iyengar, K.T.S.R. and Pandya, S.K., Analysis of Orthotropic Rectangular Thick Plates. Fibre Science & Technology, 1983. 18(1): p. 19-36.
26. Vlasov, V.Z., Method of Initial Functions in Problems of the Theory of Elasticity and Structural Mechanics. Proceedings: Ninth International Congress of Applied Mechanics, 1957. 6: p. 321 - 330.
27. Wu, Z.J. and Wardenier, J., Further investigation on the exact elasticity solution for anisotropic thick rectangular plates. International Journal of Solids and Structures, 1998. 35(7-8): p. 747-758.
28. Fan, J. and Ye, J.Q., An exact solution for the statics and dynamics of laminated thick plates with orthotropic layers. International Journal of Solids and Structures, 1990. 26(5-6): p. 655-662.
29. Fan, J. and Ye, J.Q., A series solution of the exact equation for thick orthotropic plates. International Journal of Solids and Structures, 1990. 26(7): p. 773-778.
30. Fan, J. and Sheng, H.Y., Exact Solution for thick laminate with clamped edges. Acta Mechanica Sinica, 1992. 24(5): p. 574 - 583.
31. Rogers, T.G., Watson, P., and Spencer, A.J.M., An Exact Three-Dimensional Solution for Normal Loading of Inhomogeneous and Laminated Anisotropic Elastic Plates of Moderate Thickness. Proceedings: Mathematical and Physical Sciences, 1992. 437(1899): p. 199-213.
32. Agbossou, A. and Mougin, J.P., A layered approach to the non-linear static and dynamic analysis of rectangular reinforced concrete slabs. International Journal of Mechanical Sciences, 2006. 48(3): p. 294-306.
33. Sheng, H.Y., Wang, H., and Ye, J.Q., State space solution for thick laminated piezoelectric plates with clamped and electric open-circuited boundary conditions. International Journal of Mechanical Sciences, 2007. 49(7): p. 806-818.
34. Li, R., et al., On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates. Applied Mathematics Letters, 2009. 22(12): p. 1821-1827.
35. Hollaway, L.C. and Leeming, M.B., Strengthening of Reinforced Concrete Structures Using Externally Bonded FRP Composites in Structural and Civil Engineering. 1999: Woodhead Publishing Limited.
References
207
36. Kollar, L.P. and Springer, G.S., Mechanics of Composite Structures. 2003: Cambridge University Press.
37. Cambridge, U.o. DoITPoMS (Dissemination of IT for the promotion of Materials Science) Mechanics of Fibre-Reinforced Composites 2000.
38. Andrade, C., Durability and repair of concrete structures. Targeted research action on environementally friendly construction technologies. 2001: Brussels: TRA-EFCT.
39. Santos Neto, A.B.d.S. and La Rovere, H.L., Composite concrete/GFRP slabs for footbridge deck systems. Composite Structures, 2010. 92(10): p. 2554-2564.
40. El Maaddawy, T. and Soudki, K., Strengthening of reinforced concrete slabs with mechanically-anchored unbonded FRP system. Construction and Building Materials, 2008. 22(4): p. 444-455.
41. Foret, G. and Limam, O., Experimental and numerical analysis of RC two-way slabs strengthened with NSM CFRP rods. Construction and Building Materials, 2008. 22(10): p. 2025-2030.
42. Michel, L., et al., Criteria for punching failure mode in RC slabs reinforced by externally bonded CFRP. Composite Structures, 2007. 81(3): p. 438-449.
43. El-Sayed, A., El-Salakawy, E., and Benmokrane, B., Shear strength of one-way concrete slabs reinforced with fiber-reinforced polymer composite bars. Journal of Composites for Construction, 2005. 9(2): p. 147-157.
44. Rochdi, E.H., et al., Ultimate behavior of CFRP strengthened RC flat slabs under a centrally applied load. Composite Structures, 2006. 72(1): p. 69-78.
45. Zhang, B., Masmoudi, R. and Benmokrane, B., Behaviour of one-way concrete slabs reinforced with CFRP grid reinforcements. Construction and Building Materials, 2004. 18(8): p. 625-635.
46. Ebead, U.A. and Marzouk, H., Tension-stiffening model for FRP-strengthened RC concrete two-way slabs. Materials and Structures/Materiaux et Constructions, 2005. 38(276): p. 193-200.
47. Pesic, N. and Pilakoutas, K., Flexural analysis and design of reinforced concrete beams with externally bonded FRP reinforcement. Materials and Structures/Materiaux et Constructions, 2005. 38(276): p. 183-192.
48. Ashour, A.F., El-Refaie, S.A., and Garrity, S.W., Flexural strengthening of RC continuous beams using CFRP laminates. Cement and Concrete Composites, 2004. 26(7): p. 765-775.
49. Mosallam, A.S. and Mosalam, K.M., Strengthening of two-way concrete slabs with FRP composite laminates. Construction and Building Materials, 2003. 17(1): p. 43-54.
50. Sheikh, S.A., Performance of concrete structures retrofitted with fibre reinforced polymers. Engineering Structures, 2002. 24(7): p. 869-879.
51. Abdullah, A.M., Analysis of Repaired/Strengthened RC Structures Using Composite Materials: Punching Shear. 2010, PhD Thesis, University of Manchester, UK.
52. Ye, J.Q., Laminated Composite Plates and Shells: 3D Modelling. 2002: Springer London.
53. Stroud, K.A., Engineering Mathematics. Fifth Edition ed. 2001: Industrial Press Inc. New York.
References
208
54. Chandrupatla, T.R. and Belegundu, A.D., Introduction to finite elements in engineering. Third ed. 2002, Upper Saddle River, N.J.: Prentice Hall.
55. Systemes, D., Abaqus CAE User's Manual, Theory Manual version 6.11. version 6.11 ed.
56. Jankowiak, T. and Lodygowski, T., Identification of Parameters of Concrete Damage Plasticity Constitutive Model. Foundations of Civil and Environmental Engineering, 2005. 6: p. 53-59.
57. CEB-FIP, CEB-FIP MODEL CODE 1990 - DESIGN CODE. 1990. 58. Beton, C.E.I.D., RC elements Under Cyclic Loading. 1996: Thomas
Telford.
List of Publications
209
LIST OF PUBLICATIONS
1. Wu, Z.J. and Kamis, E., Influence of Element Type on the Simulation of
Laminated Composite Plates with Clamped Edges, Conference of the
Association for Computational Mechanics in Engineering (ACME), 27-
28th March 2012, the University of Manchester, UK.
2. Kamis, E. and Wu, Z.J., Exact Elasticity Solution for Slabs Reinforced
with Fibre Reinforced Plastic Sheet. (to be submitted for journal
publication)
3. Kamis, E. and Wu, Z.J., Exact Elasticity Solution for Thick Laminate with
Fully Clamped Edges. (to be submitted for journal publication)
Appendix A
210
APPENDIX A
Table A-1: Maximum deflection at x = a/2, y = b/2 a nd z = 0 (W C11/ qh)
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 19.85492 19.70885 19.85000 20.70275 20.69185 20.17570 0.3 6.90785 6.90893 6.90500 6.96923 6.96373 6.42797 0.4 3.54859 3.54093 3.54863 3.33113 3.32780 3.58780 0.5 2.26852 2.28758 2.26940 1.91869 1.91804 2.50204 0.6 1.65476 1.67465 1.65685 1.24354 1.24166 1.94725 0.8 1.08207 1.08860 1.08541 0.64374 0.64250 1.35578 1 0.79664 0.80336 0.80043 0.39407 0.39321 1.00998
Table A-2: Stress ( σx/q) at x = a/2, y = b/2 at Ply1 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -9.95082 -8.59663 -10.02990 -8.82711 -8.89627 -8.52921 0.3 -4.78541 -3.82362 -4.80254 -3.58893 -3.61438 -3.19393 0.4 -3.05251 -2.78390 -3.05836 -1.86807 -1.87838 -1.68732 0.5 -2.36148 -2.12438 -2.36150 -1.12867 -1.13209 -0.93771 0.6 -2.11374 -1.86817 -2.10372 -0.750821 -0.752578 -0.62842 0.8 -2.07925 -1.80791 -2.07229 -0.40063 -0.400703 -0.46830 1 -2.20737 -1.89060 -2.20740 -0.24901 -0.248722 -0.61819
Table A-3: Stress ( σx/q) at x = a/2, y = b/2 at Ply1 bottom surface Shell
Solid Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -6.05797 -7.37290 -6.10997 -7.06169 -7.11702 -6.41338 0.3 -2.15883 -3.04784 -2.16349 -2.87114 -2.8915 -2.31356 0.4 -0.92630 -1.14816 -0.92682 -1.49445 -1.50270 -0.53669 0.5 -0.39640 -0.60238 -0.39780 -0.902934 -0.905671 -0.37342 0.6 -0.09078 -0.28668 -0.08217 -0.600657 -0.602062 -0.07150 0.8 0.39008 0.14918 0.385447 -0.320504 -0.320562 0.21217 1 0.75763 0.47419 0.751947 -0.199208 -0.198978 0.48004
Appendix A
211
Table A-4: Stress ( σx/q) at x = a/2, y = b/2 at Ply2 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -1.33206 -0.89997 -1.34265 -1.41234 -1.42341 -1.15223 0.3 -0.55165 -0.39958 -0.55260 -0.57423 -0.57830 -0.51587 0.4 -0.30544 -0.24894 -0.30566 -0.29889 -0.30054 -0.29689 0.5 -0.20013 -0.15254 -0.20063 -0.18059 -0.18114 -0.18462 0.6 -0.13951 -0.09136 -0.13811 -0.12013 -0.12041 -0.12297 0.8 -0.04267 -0.00883 -0.04391 -0.06410 -0.06411 -0.06317 1 0.03440 0.06658 0.03281 -0.03984 -0.03980 0.01011
Table A-5: Stress ( σx/q) at x = a/2, y = b/2 at Ply2 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 1.20446 0.75934 1.20057 1.41234 1.42341 1.26250 0.3 0.40391 0.25438 0.40489 0.57423 0.57830 0.63055 0.4 0.14479 0.09623 0.14510 0.29889 0.30054 0.35164 0.5 0.03422 0.00865 0.03436 0.18059 0.18114 0.24374 0.6 -0.01846 -0.02499 -0.01627 0.12013 0.12041 0.16136 0.8 -0.04070 -0.03512 -0.04044 0.06410 0.06411 0.07619 1 -0.03030 -0.02275 -0.03070 0.03984 0.03980 0.03883
Table A-6: Stress ( σx/q) at x = a/2, y = b/2 at Ply3 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 6.05059 7.29380 6.02718 7.06169 7.11702 7.50375 0.3 2.05010 2.95178 2.05395 2.87114 2.89150 3.76482 0.4 0.75795 0.97869 0.75870 1.49445 1.50270 1.77609 0.5 0.20891 0.39527 0.20889 0.90293 0.90567 1.46746 0.6 -0.05258 0.11370 -0.04073 0.60066 0.60206 0.97488 0.8 -0.16710 -0.07120 -0.16577 0.32050 0.32056 0.44430 1 -0.12414 -0.07534 -0.12603 0.19921 0.19898 0.19152
Appendix A
212
Table A-7: Stress ( σx/q) at x = a/2, y = b/2 at Ply3 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 9.93898 8.51696 9.94639 8.82711 8.89627 9.86828 0.3 4.68826 3.71646 4.70426 3.58893 3.61438 4.67811 0.4 2.87148 2.59650 2.87780 1.86807 1.87838 2.88275 0.5 1.98177 1.75051 1.98580 1.12867 1.13209 2.35680 0.6 1.41810 1.23765 1.43358 0.75082 0.75258 1.45682 0.8 0.74686 0.62972 0.75264 0.40063 0.40070 0.74341 1 0.37058 0.30528 0.37364 0.24901 0.24872 0.35918
Table A-8: Stress ( σy/q) at x = a/2, y = b/2 at Ply1 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -8.39846 -7.42995 -8.36945 -8.43043 -8.44158 -7.61809 0.3 -4.88857 -4.24907 -4.87596 -4.90654 -4.91795 -2.92017 0.4 -3.28739 -3.08263 -3.27664 -3.21709 -3.22801 -2.06006 0.5 -2.46848 -2.29218 -2.45607 -2.25525 -2.26214 -1.15395 0.6 -2.04166 -1.87835 -2.03460 -1.65395 -1.66190 -0.97315 0.8 -1.74207 -1.55455 -1.73062 -0.98911 -0.99458 -0.87882 1 -1.69364 -1.47021 -1.68334 -0.65272 -0.65658 -0.74259
Table A-9: Stress ( σy/q) at x = a/2, y = b/2 at Ply1 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -6.03856 -6.87907 -6.04680 -6.74434 -6.75327 -6.03275 0.3 -3.23138 -3.83999 -3.24440 -3.92523 -3.93436 -1.93642 0.4 -1.95543 -2.10412 -1.96488 -2.57367 -2.58241 -1.21315 0.5 -1.27051 -1.41120 -1.27773 -1.80420 -1.80971 -0.28650 0.6 -0.85195 -1.00163 -0.86480 -1.32316 -1.32952 -0.20818 0.8 -0.36474 -0.53530 -0.37395 -0.79129 -0.79566 -0.19897 1 -0.04711 -0.24367 -0.05303 -0.52217 -0.52526 -0.10487
Appendix A
213
Table A-10: Stress ( σy/q) at x = a/2, y = b/2 at Ply2 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -1.37431 -1.07036 -1.37621 -1.34887 -1.35066 -1.05234 0.3 -0.81207 -0.70828 -0.81470 -0.78505 -0.78687 -0.27633 0.4 -0.55728 -0.51682 -0.55933 -0.51474 -0.51648 -0.20335 0.5 -0.42122 -0.38748 -0.42298 -0.36084 -0.36194 -0.17546 0.6 -0.33821 -0.30733 -0.34091 -0.26463 -0.26590 -0.13176 0.8 -0.23984 -0.21748 -0.24216 -0.15826 -0.15913 -0.10985 1 -0.17140 -0.15009 -0.17323 -0.10444 -0.10505 -0.09614
Table A-11: Stress ( σy/q) at x = a/2, y = b/2 at Ply2 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 1.24154 0.93874 1.24407 1.34887 1.35066 1.39135 0.3 0.67748 0.57370 0.67873 0.78505 0.78687 0.89135 0.4 0.40652 0.36932 0.40806 0.51474 0.51648 0.47131 0.5 0.24954 0.22763 0.25095 0.36084 0.36194 0.23290 0.6 0.15422 0.14212 0.15411 0.26463 0.26590 0.11719 0.8 0.05594 0.05622 0.05607 0.15826 0.15913 0.05490 1 0.01900 0.02190 0.01925 0.10444 0.10505 0.03846
Table A-12: Stress ( σy/q) at x = a/2, y = b/2 at Ply3 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 6.24685 7.08642 6.25398 6.74434 6.75327 5.64233 0.3 3.42967 4.03891 3.43441 3.92523 3.93436 3.64233 0.4 2.07965 2.22285 2.08622 2.57367 2.58241 2.35653 0.5 1.30001 1.42405 1.30602 1.80420 1.80971 1.16448 0.6 0.82600 0.92383 0.82505 1.32316 1.32952 0.85562 0.8 0.33002 0.39179 0.33072 0.79129 0.79566 0.47691 1 0.13285 0.16716 0.13422 0.52217 0.52526 0.31045
Appendix A
214
Table A-13: Stress ( σy/q) at x = a/2, y = b/2 at Ply3 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 8.56515 7.63727 8.57717 8.43043 8.44158 7.20677 0.3 5.08548 4.46198 5.09223 4.90654 4.91795 5.20677 0.4 3.43155 3.24181 3.44086 3.21709 3.22801 3.38912 0.5 2.43217 2.27495 2.44150 2.25525 2.26214 1.92122 0.6 1.75310 1.61652 1.75459 1.65395 1.66190 1.44386 0.8 0.88926 0.80637 0.89382 0.98911 0.99458 0.92394 1 0.43075 0.38684 0.43557 0.65272 0.65658 0.14320
Table A-14: Stress ( σx/q) at x = 0, y = b/2 at Ply1 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 53.55596 25.92480 51.06000 10.84380 12.84330 11.67580 0.3 30.38884 16.16450 28.61140 3.21713 3.85948 3.37407 0.4 21.73011 14.26170 20.16670 1.18875 1.45989 1.31707 0.5 17.59548 11.92760 16.13850 0.56742 0.63724 0.94346 0.6 15.26024 10.46910 14.44660 0.22809 0.30617 0.72718 0.8 12.67571 5.78926 11.60800 0.05030 0.08334 0.57393 1 11.11321 5.31766 10.48180 0.00617 0.02370 0.41177
Table A-15: Stress ( σx/q) at x = 0, y = b/2 at Ply1 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -16.62360 -8.72296 -15.55646 -8.67506 -10.27460 -5.46955 0.3 -13.80722 -5.30263 -13.34220 -2.57370 -3.08758 -1.40093 0.4 -11.33251 -4.24720 -10.71385 -0.95100 -1.16791 -1.18536 0.5 -9.71725 -4.22684 -9.72375 -0.45393 -0.50979 -1.17945 0.6 -8.65996 -4.18473 -8.52141 -0.18247 -0.24493 -1.15555 0.8 -7.37854 -4.06174 -7.30472 -0.04024 -0.06667 -0.91967 1 -6.54162 -3.87882 -6.68188 -0.00494 -0.01896 -0.15760
Appendix A
215
Table A-16: Stress ( σx/q) at x = 0, y = b/2 at Ply2 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -3.32472 -1.96878 -3.11129 -1.73501 -2.05492 -2.24324 0.3 -2.76144 -1.02041 -2.46845 -0.51474 -0.61752 -0.53707 0.4 -2.26650 -0.92410 -2.19277 -0.19020 -0.23358 -0.23589 0.5 -1.94345 -0.91762 -1.94475 -0.09079 -0.10196 -0.14060 0.6 -1.73199 -0.89218 -1.70428 -0.03649 -0.04899 -0.13660 0.8 -1.47571 -0.87093 -1.46094 -0.00805 -0.01333 -0.11322 1 -1.30832 -0.82207 -1.33638 -0.00099 -0.00379 -0.09172
Table A-17: Stress ( σx/q) at x = 0, y = b/2 at Ply2 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 2.29242 1.21956 2.13553 1.73501 2.05492 0.80398 0.3 1.72281 0.47893 1.59960 0.51474 0.61752 0.14515 0.4 1.20642 0.40724 1.18510 0.19020 0.23358 0.13769 0.5 0.84555 0.34860 0.78371 0.09079 0.10196 0.11031 0.6 0.59353 0.27241 0.56496 0.03649 0.04899 0.09602 0.8 0.28733 0.15399 0.28961 0.00805 0.01333 0.00283 1 0.13305 0.07573 0.13778 0.00099 0.00379 0.00103
Table A-18: Stress ( σx/q) at x = 0, y = b/2 at Ply3 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 11.46209 3.96288 10.67764 8.67506 10.27460 5.57556 0.3 8.61407 2.44794 7.62841 2.57370 3.08758 0.72574 0.4 6.03208 2.05442 5.92550 0.95100 1.16791 0.18843 0.5 4.22776 1.59429 3.91855 0.45393 0.50979 0.08405 0.6 2.96764 0.90020 2.82479 0.18247 0.24493 0.04370 0.8 1.43665 0.87276 1.44806 0.04024 0.06667 0.03115 1 0.66523 0.43092 0.68892 0.00494 0.01896 0.02500
Appendix A
216
Table A-19: Stress ( σx/q) at x = 0, y = b/2 at Ply3 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -43.27342 -24.67300 -41.38450 -10.84380 -12.84330 -10.33740 0.3 -19.89811 -14.10180 -18.93650 -3.21713 -3.85948 -2.83846 0.4 -11.05129 -5.32642 -10.87167 -1.18875 -1.45989 -1.54643 0.5 -6.74559 -3.39907 -6.20385 -0.56742 -0.63724 -1.38473 0.6 -4.31346 -2.25281 -4.05856 -0.22809 -0.30617 -0.90930 0.8 -1.84821 -1.01757 -1.80302 -0.05030 -0.08334 -0.38314 1 -0.78724 -0.44697 -0.79056 -0.00617 -0.02370 -0.05320
Table A-20: Stress ( σy/q) at x = 0, y = b/2 at Ply1 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 12.38443 4.43123 10.58658 2.50754 2.95316 9.91810 0.3 7.02720 2.62853 5.59638 0.74393 0.88647 4.98424 0.4 5.02493 2.27565 4.47389 0.27489 0.33507 3.91270 0.5 4.06883 1.78167 3.72813 0.13121 0.14615 3.21063 0.6 3.52882 1.47681 3.31147 0.05274 0.07016 2.39296 0.8 2.93117 0.56568 2.85869 0.01163 0.01905 1.35899 1 2.56985 0.46908 2.58132 0.00143 0.00538 1.00101
Table A-21: Stress ( σy/q) at x = 0, y = b/2 at Ply1 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -3.84409 -2.86587 -3.50718 -2.00603 -2.36253 -3.42708 0.3 -3.19282 -1.26099 -3.04696 -0.59515 -0.70918 -2.93172 0.4 -2.62056 -0.70182 -2.39221 -0.21991 -0.26805 -2.25293 0.5 -2.24704 -0.63168 -2.14838 -0.10497 -0.11692 -1.91373 0.6 -2.00255 -0.62473 -1.97542 -0.04219 -0.05613 -1.63903 0.8 -1.70623 -0.60119 -1.79892 -0.00931 -0.01524 -1.05918 1 -1.51270 -0.56507 -1.51612 -0.00114 -0.00431 -0.55429
Appendix A
217
Table A-22: Stress ( σy/q) at x = 0, y = b/2 at Ply2 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -0.76882 -0.27429 -0.72144 -0.40121 -0.47251 -0.68447 0.3 -0.63857 -0.18152 -0.60940 -0.11903 -0.14184 -0.55059 0.4 -0.52412 -0.17116 -0.47844 -0.04398 -0.05361 -0.45600 0.5 -0.44941 -0.17005 -0.42968 -0.02099 -0.02338 -0.36924 0.6 -0.40051 -0.16253 -0.39509 -0.00844 -0.01123 -0.30263 0.8 -0.34125 -0.16009 -0.35979 -0.00186 -0.00305 -0.15917 1 -0.30254 -0.14452 -0.30323 -0.00023 -0.00086 -0.11266
Table A-23: Stress ( σy/q) at x = 0, y = b/2 at Ply2 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 0.53011 0.24512 0.51965 0.40121 0.47251 0.49402 0.3 0.39839 0.19052 0.34468 0.11903 0.14184 0.33559 0.4 0.27898 0.10037 0.26389 0.04398 0.05361 0.23407 0.5 0.19553 0.05112 0.19300 0.02099 0.02338 0.17203 0.6 0.13725 0.01223 0.13913 0.00844 0.01123 0.10993 0.8 0.06644 0.01108 0.07132 0.00186 0.00305 0.04471 1 0.03077 0.00549 0.03393 0.00023 0.00086 0.04070
Table A-24: Stress ( σy/q) at x = 0, y = b/2 at Ply3 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 2.65052 1.88650 2.69822 2.00603 2.36253 1.48415 0.3 1.99194 1.15356 1.93237 0.59515 0.70918 0.86714 0.4 1.39487 0.47822 1.31945 0.21991 0.26805 0.46858 0.5 0.97764 0.26768 0.96501 0.10497 0.11692 0.28540 0.6 0.68625 0.21913 0.69566 0.04219 0.05613 0.28068 0.8 0.33221 0.12534 0.33624 0.00931 0.01524 0.17606 1 0.15383 0.06299 0.16049 0.00114 0.00431 0.08370
Appendix A
218
Table A-25: Stress ( σy/q) at x = 0, y = b/2 at Ply3 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 -10.00666 -3.23849 -9.21567 -2.50754 -2.95316 -9.43712 0.3 -4.60129 -1.30403 -3.95940 -0.74393 -0.88647 -4.02563 0.4 -2.55553 -0.80716 -2.43107 -0.27489 -0.33507 -2.09981 0.5 -1.55987 -0.48183 -1.52781 -0.13121 -0.14615 -1.43784 0.6 -0.99746 -0.30183 -0.99949 -0.05274 -0.07016 -0.96739 0.8 -0.42739 -0.12360 -0.41562 -0.01163 -0.01905 -0.94773 1 -0.18204 -0.04931 -0.18063 -0.00143 -0.00538 -0.14923
Table A-26: Stress ( τxz/q) at x = 0, y = b/2 at Ply1 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 0 9.12096 7.15914 0 0 0 0.3 0 5.45091 4.41834 0 0 0 0.4 0 3.95007 3.51389 0 0 0 0.5 0 3.26464 3.02200 0 0 0 0.6 0 2.88679 2.75652 0 0 0 0.8 0 2.49997 2.49024 0 0 0 1 0 2.29488 2.34987 0 0 0
Table A-27: Stress ( τxz/q) at x = 0, y = b/2 at Ply1 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 2.44791 8.95765 6.79592 1.89790 2.02112 0 0.3 1.61330 5.29573 4.12603 1.10793 1.18713 0 0.4 1.24783 3.77662 3.17580 0.75340 0.81141 0 0.5 1.05043 3.05966 2.63791 0.58592 0.60881 0 0.6 0.93774 2.65064 2.32812 0.44773 0.48542 0 0.8 0.81018 2.20789 1.98315 0.31728 0.34520 0 1 0.73559 1.95501 1.78877 0.24616 0.26835 0
Appendix A
219
Table A-28: Stress ( τxz/q) at x = 0, y = b/2 at Ply2 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 2.44791 1.79153 1.35918 1.89790 2.02112 0 0.3 1.61330 1.05915 0.82521 1.10793 1.18713 0 0.4 1.24783 0.75532 0.63516 0.75340 0.81141 0 0.5 1.05043 0.61193 0.52758 0.58592 0.60881 0 0.6 0.93774 0.53013 0.46562 0.44773 0.48542 0 0.8 0.81018 0.44158 0.39663 0.31728 0.34520 0 1 0.73559 0.39100 0.35775 0.24616 0.26835 0
Table A-29: Stress ( τxz/q) at x = 0, y = b/2 at Ply2 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 1.85606 1.49682 1.01324 1.89790 2.02112 0 0.3 1.01449 0.73564 0.49866 1.10793 1.18713 0 0.4 0.63348 0.41783 0.30542 0.75340 0.81141 0 0.5 0.42755 0.26235 0.19934 0.58592 0.60881 0 0.6 0.29410 0.17191 0.13566 0.44773 0.48542 0 0.8 0.14387 0.07752 0.06488 0.31728 0.34520 0 1 0.06902 0.03504 0.03078 0.24616 0.26835 0
Table A-30: Stress ( τxz/q) at x = 0, y = b/2 at Ply3 top surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 1.85606 7.48408 5.06619 1.89790 2.02112 0 0.3 1.01449 3.67822 2.49331 1.10793 1.18713 0 0.4 0.63348 2.08914 1.52711 0.75340 0.81141 0 0.5 0.42755 1.31175 0.99669 0.58592 0.60881 0 0.6 0.29410 0.85955 0.67830 0.44773 0.48542 0 0.8 0.14387 0.38762 0.32441 0.31728 0.34520 0 1 0.06902 0.17521 0.15388 0.24616 0.26835 0
Appendix A
220
Table A-31: Stress ( τxz/q) at x = 0, y = b/2 at Ply3 bottom surface
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 0 7.51593 5.14023 0 0 0 0.3 0 3.65195 2.41301 0 0 0 0.4 0 2.04101 1.41982 0 0 0 0.5 0 1.26171 0.88786 0 0 0 0.6 0 0.81381 0.58169 0 0 0 0.8 0 0.35591 0.26254 0 0 0 1 0 0.15608 0.11844 0 0 0
Table A-32: Stress ( τxy/q) at x = 0, y = b/2 across the thickness
Shell Solid
Conventional Continuum
Linear Quadratic Linear Quadratic Linear h/a Exact
C3D8 C3D20 S4 S8R SC8R
0.2 0 0 0 0 0 0 0.3 0 0 0 0 0 0 0.4 0 0 0 0 0 0 0.5 0 0 0 0 0 0 0.6 0 0 0 0 0 0 0.8 0 0 0 0 0 0 1 0 0 0 0 0 0
Appendix A
221
0
5
10
15
20
25
0.2 0.4 0.6 0.8 1
h/a
W C
11/q
h
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-1: Deflection (WC 11 / qh) at x = a/2 , y = b/2 at z = 0
-12
-10
-8
-6
-4
-2
00.2 0.4 0.6 0.8 1h/a
sx/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-2: Stress ( σx/q) at x = a/2 and y = b/2 Ply1 top
Appendix A
222
-8
-6
-4
-2
0
2
0.2 0.4 0.6 0.8 1h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-3: Stress ( σx/q) at x = a/2 and y = b/2 Ply1 bottom
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.2 0.4 0.6 0.8 1h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-4: Stress ( σx/q) at x = a/2 and y = b/2 Ply2 top
Appendix A
223
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-5: Stress ( σx/q) at x = a/2 and y = b/2 Ply2 bottom
-1
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-6: Stress ( σx/q) at x = a/2 and y = b/2 Ply3 top
Appendix A
224
0
2
4
6
8
10
12
0.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-7: Stress ( σx/q) at x = a/2 and y = b/2 Ply3 bottom
-9
-8
-7
-6
-5
-4
-3
-2
-1
00.2 0.4 0.6 0.8 1
h/a
sy/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-8: Stress ( σy/q) at x = a/2 and y = b/2 Ply1 top
Appendix A
225
-8
-7
-6
-5
-4
-3
-2
-1
00.2 0.4 0.6 0.8 1
h/a
sy/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-9: Stress ( σy/q) at x = a/2 and y = b/2 Ply1 bottom
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.00.2 0.4 0.6 0.8 1
h/a
sy/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-10: Stress ( σy/q) at x = a/2 and y = b/2 Ply2 top
Appendix A
226
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-11: Stress ( σy/q) at x = a/2 and y = b/2 Ply2 bottom
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-12: Stress ( σy/q) at x = a/2 and y = b/2 Ply3 top
Appendix A
227
0
2
4
6
8
10
0.2 0.4 0.6 0.8 1.0
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-13: Stress ( σy/q) at x = a/2 and y = b/2 Ply3 bottom
0
10
20
30
40
50
60
0.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-14: Stress ( σx/q) at x = 0 and y = b/2 Ply1 top
Appendix A
228
-18
-16
-14
-12
-10
-8
-6
-4
-2
00.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-15: Stress ( σx/q) at x = 0 and y = b/2 Ply1 bottom
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.00.2 0.4 0.6 0.8 1
h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
`
Figure A-16: Stress ( σx/q) at x = 0 and y = b/2 Ply2 top
Appendix A
229
0.0
0.5
1.0
1.5
2.0
2.5
0.2 0.4 0.6 0.8 1h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-17: Stress ( σx/q) at x = 0 and y = b/2 Ply2 bottom
0
2
4
6
8
10
12
14
0.2 0.4 0.6 0.8 1h/a
sx/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-18: Stress ( σx/q) at x = 0 and y = b/2 Ply3 top
Appendix A
230
-50
-40
-30
-20
-10
00.2 0.4 0.6 0.8 1
h/a
sx/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-19: Stress ( σx/q) at x = 0 and y = b/2 Ply3 bottom
0
2
4
6
8
10
12
14
0.2 0.4 0.6 0.8 1h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-20: Stress ( σy/q) at x = 0 and y = b/2 Ply1 top
Appendix A
231
-5.0
-4.0
-3.0
-2.0
-1.0
0.00.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-21: Stress ( σy/q) at x = 0 and y = b/2 Ply1 bottom
-0.8
-0.6
-0.4
-0.2
0.00.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-22: Stress ( σy/q) at x = 0 and y = b/2 Ply2 top
Appendix A
232
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-23: Stress ( σy/q) at x = 0 and y = b/2 Ply2 bottom
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.2 0.4 0.6 0.8 1
h/a
sy/
q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-24: Stress ( σy/q) at x = 0 and y = b/2 Ply3 top
Appendix A
233
-12
-10
-8
-6
-4
-2
00.2 0.4 0.6 0.8 1
h/a
sy/
q Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-25: Stress ( σy/q) at x = 0 and y = b/2 Ply3 bottom
0
2
4
6
8
10
0.2 0.4 0.6 0.8 1
h/a
txz
/q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-26: Stress ( τxz/q) at x = 0 and y = b/2 Ply1 bottom
Appendix A
234
0.0
0.5
1.0
1.5
2.0
2.5
0.2 0.4 0.6 0.8 1.0
h/a
txz
/q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-27: Stress ( τxz/q) at x = 0 and y = b/2 Ply2 top
0.0
0.5
1.0
1.5
2.0
2.5
0.2 0.4 0.6 0.8 1
h/a
txz
/q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-28: Stress ( τxz/q) at x = 0 and y = b/2 Ply2 bottom
Appendix A
235
0
1
2
3
4
5
6
7
8
0.2 0.4 0.6 0.8 1
h/a
txz
/q
Exactsolid C3D8solid C3D20shell S4shell S8Rshell SC8R
Figure A-29: Stress ( τxz/q) at x = 0 and y = b/2 Ply3 top