Deformations of In-plane Loaded Unsymmetrically Laminated Composite Plates

294 Pages·2005·8.64 MB·English

Deformations of In-plane Loaded Unsymmetrically Laminated
Composite Plates
Majed A. Majeed
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mechanical Engineering
Michael W. Hyer, Co-Chair
Daniel J. Inman, Co-Chair
Mehdi Ahmadian
Donald J. Leo
Eric Johnson
February 17, 2005
Blacksburg, Virginia
Keywords: Stability, Geometrically Nonlinear, Bifurcation Buckling, Unsymmetric
Cross-ply Laminates, Antisymmetric Angle-Ply Laminates, ABAQUS FEA
Copyright 2005, Majed A. Majeed
Abstract
Deformations of In-plane Loaded Unsymmetrically Laminated
Composite Plates
by
Majed A. Majeed
This study focuses on the response of flat unsymmetric laminates to an inplane
compressive loading that for symmetric laminates are of sufficient magnitude to cause
bifurcation buckling, postbuckling, and secondary buckling behavior. In particular,
the purpose of this study is to investigate whether or not the concept of bifurcation
buckling is applicable to unsymmetric laminates. Past work by other researchers has
suggested that such a concept is applicable for certain boundary conditions. The
study also has as an objective the determination of the response of flat unsymmet-
ric laminates if bifurcation buckling does not occur. The finite-element program
ABAQUS is used to obtain results, and a portion of the study is devoted to be-
coming familiar with the way ABAQUS handles such highly geometrically nonlinear
problems, particularly for composite materials and particularly when instabilities and
dynamic behavior are involved. Familiarity with the problem, in general, and with
the use of ABAQUS, in particular, is partially gained by considering semi-infinite
unsymmetrically laminated cross- and angle-ply plates, a one-dimensional problem
that can be solve in closed form and with ABAQUS by making the appropriate ap-
proximations for the infinite geometry. In this portion of the study it is found that
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semi-infinite cross-ply laminates with clamped boundary conditions and semi-infinite
angle-ply plates with simple-support boundary conditions remain flat under a com-
pressive load until the load magnitude reaches a certain level, at which time the
out-of-plane deflection become indeterminate, essentially an eigenvalue problem as
encountered with classic bifurcation buckling analyses. Obviously, a linear analy-
sis of such problems would not reveal this behavior and, in fact, there are other
revealed significant differences between the predictions of linear and nonlinear analy-
ses. Transversely-loaded and inplane-loaded finite isotropic plates are studied by
way of semi-closed form Rayleigh-Ritz-based solutions and ABAQUS in a step to
approaching the problem with unsymmetric laminates. A method to investigate the
unloading behavior of postbuckled finite isotropic plates is developed that reveal mul-
tiple plate configurations in the postbuckled region of the response, and this method
is then extended to the study of finite inplane-loaded unsymmetric laminates. To
that end, two specific laminates, a [0 /90 ] laminate and a [0 /90 ] laminate, and
2 2 S 4 4 T
a variety of boundary conditions are used to study the response of inplane-loaded
unsymmetric laminates. The symmetric laminate is included to provide a famil-
iar baseline case and a means of comparison. Plates with all four edges clamped
and a variety of inplane boundary conditions are studied. Of course the [0 /90 ]
2 2 S
laminate exhibits bifurcation behavior, and when the tangential displacement on the
loaded edges and the normal displacement on the unloaded edges are restrained, sec-
ondary buckling behavior occurs. For the [0 /90 ] laminate, bifurcation buckling
4 4 T
behavior does not occur unless the tangential displacement on the loaded edges and
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the normal displacement on the unloaded edges are restrained, or the tangential dis-
placement on the loaded edges and the normal displacement on the unloaded edges
are free. If either of these conditions are not satisfied, the [0 /90 ] laminate ex-
4 4 T
hibits what could be termed ’near-bifurcation’ behavior. In all cases rather complex
behavior occurs for high levels of inplane load, including asymmetric postbuckling
and secondary buckling behavior. For clamped loaded edges and simply-supported
unloaded edges, bifurcation buckling behavior does not occur unless the tangential
displacement on the loaded edges and the normal displacement on the unloaded edges
are restrained. For this case, rather unusual asymmetric bifurcation and associated
limit point behavior occur, as well as secondary buckling. This is a very interesting
boundary condition case and is studied further for other unsymmetric cross-ply lami-
nates, including the use of a Rayleigh-Ritz-based solution in attempt to quantify the
problem parameters responsible for the asymmetric response. The overall results of
the study have led to an increased understanding of the role of laminate asymmetry
and boundary conditions on the potential for bifurcation behavior, on the response
of the laminate for loads beyond that level.
This work received support from the Kuwait University, Mechanical Engineering De-
partment and the Department of Engineering Science and Mechanics at Virginia
Tech.
Dedication
This dissertation is dedicated to the memories of my father Adulrazzq H. Majeed who
inspired me to pursue my studies and was always supportive. My father taught me
the appreciation of knowledge and education, where his love for education served as
my compass and directed me towards this goal. He was a great source of guidance
and I am grateful form him; may Allah (the Lord) provides his soul with peace and
reside him in heaven.
I would also like to dedicate this work to my mother and brothers and sisters.
Last but not least, I would like to dedicate this work to my lovely wife Masoumah
and my two daughters Zahraa and Zainab.
v
Acknowledgments
Iwouldliketothankmyco-advisors, ProfessorsMichaelW.HyerandDanielJ.Inman
in addition to my Ph.D. committee members, Professors Eric Johnson, Donald Leo,
and Mehdi Ahmedian. Additional support from Prof. Raymond Plaut, Dr. Ziyad
Masoud, Dr. Khaled Alhazza, Marc Schultz, Sontipee Somenia, Mohammed Daqaq,
Christopher Glenn, Sadeq Alnasur, Saifuddin Rayyan, Sameer Arabasi, Sultan Al-
daihani, and my academic advisor at Kuwait Cultural Office Mr. Aderson Exume is
gratefully acknowledged. I am grateful for the patience and support of my family,
especially My wife Masoumah who made all efforts creating student-friendly envi-
ronment. This study was funded by Kuwait University, Mechanical Engineering
Department and supported by the facilities at the department of Engineering Science
and Mechanics at Virginia Tech.
vi
Table of Contents
Abstract ii
Dedication v
Acknowledgments vi
Table of Contents vii
List of Figures xi
List of Tables xxiv
1 Introduction 1
1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of Laminate Nomenclature . . . . . . . . . . . . . . . . . . 15
1.3 Overview of Classical Lamination Theory . . . . . . . . . . . . . . . . 21
1.4 Contributions to the Literature . . . . . . . . . . . . . . . . . . . . . 28
2 Analysis of Unsymmetrically Laminated Semi-Infinite Plates 29
2.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
vii
viii
2.4 Geometrically Nonlinear Analysis of Semi-Infinite Laminated Plates . 34
2.4.1 Unsymmetric Cross-Ply Laminates . . . . . . . . . . . . . . . 42
2.4.1.1 Simply-Supported Boundary Conditions . . . . . . . 44
2.4.1.2 Clamped Boundary Conditions . . . . . . . . . . . . 47
2.4.1.2.1 First Possible Solution . . . . . . . . . . . . 48
2.4.1.2.2 Second Possible Solution . . . . . . . . . . . 50
2.4.1.2.3 Third Possible Solution . . . . . . . . . . . 51
2.4.2 Antisymmetric Angle-Ply Laminates . . . . . . . . . . . . . . 54
2.4.2.1 Simply-Supported Boundary Conditions . . . . . . . 56
2.4.2.1.1 First Possible Solution . . . . . . . . . . . . 58
2.4.2.1.2 Second Possible Solution . . . . . . . . . . . 59
2.4.2.1.3 Third Possible Solution . . . . . . . . . . . 60
2.4.2.2 Clamped Boundary Conditions . . . . . . . . . . . . 62
2.5 Geometrically Linear Analysis of Semi-Infinite Laminated Plates . . . 63
2.5.1 Unsymmetric Cross-Ply Laminates . . . . . . . . . . . . . . . 67
2.5.1.1 Simply-Supported Boundary Conditions . . . . . . . 68
2.5.1.2 Clamped Boundary Conditions . . . . . . . . . . . . 69
2.5.2 Antisymmetric Angle-Ply Laminates . . . . . . . . . . . . . . 70
2.5.2.1 Simply-Supported Boundary Conditions . . . . . . . 71
2.5.2.2 Clamped Boundary Conditions . . . . . . . . . . . . 72
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.7 Strain Results for Selected Plates . . . . . . . . . . . . . . . . . . . . 75
2.7.1 [0 /90 ] Cross-Ply Laminate . . . . . . . . . . . . . . . . . . 78
4 4 T
2.7.1.1 Geometrically Linear Analysis . . . . . . . . . . . . . 79
2.7.1.2 Geometrically Nonlinear Analysis . . . . . . . . . . . 82
2.7.2 [45 /-45 /45 /-45 ] Angle-Ply Laminate . . . . . . . . . . . . 85
2 2 2 2 T
2.8 Validity of Key Assumption Regarding N . . . . . . . . . . . . . . . 90
y
ix
2.9 Deformation Results for Selected Plates . . . . . . . . . . . . . . . . . 92
3 Analysis of Finite Isotropic Plates 116
3.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.3 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.3.1 The Rayleigh-Ritz Technique . . . . . . . . . . . . . . . . . . 130
3.4 Transversely Loaded Plates . . . . . . . . . . . . . . . . . . . . . . . 133
3.4.1 Approximating Functions . . . . . . . . . . . . . . . . . . . . 134
3.4.1.1 Trigonometric Functions . . . . . . . . . . . . . . . . 135
3.4.1.2 Reduced Polynomial Functions . . . . . . . . . . . . 136
3.4.2 Numerical Results for Transversely-Loaded Plates . . . . . . . 138
3.5 Inplane Loaded Plates: Pre- and Postbuckling . . . . . . . . . . . . . 145
3.6 Deep Postbuckling Analysis: Secondary Buckling . . . . . . . . . . . 154
4 Analysis of Finite Laminated Cross-Ply Composite Plates 165
4.1 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.3 Symmetric Laminates: CC-CC [0 /90 ] . . . . . . . . . . . . . . . . 179
2 2 S
4.4 Unsymmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.4.1 [0 /90 ] CC-CC Laminates . . . . . . . . . . . . . . . . . . . 191
4 4 T
4.4.2 [0 /90 ] CC-SS Laminates . . . . . . . . . . . . . . . . . . . 215
4 4 T
4.4.3 [0 /90 ] SS-CC Laminate . . . . . . . . . . . . . . . . . . . . 232
4 4 T
4.4.4 [0 /90 ] SS-SS Laminate . . . . . . . . . . . . . . . . . . . . 235
4 4 T
4.4.5 Comments Regarding Figure 4.2-3 . . . . . . . . . . . . . . . . 237
4.4.6 Effect of Stacking Sequence on the Response of CC-SS Re-
strained Plates . . . . . . . . . . . . . . . . . . . . . . . . . . 238
5 Conclusions and Future Work 253
x
Bibliography 262
Vita 269