NASA Technical Reports Server (NTRS) 20050180642: Analysis of a Circular Composite Disk Subjected to Edge Rotations and Hydrostatic Pressure

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ANALYSIS OF A CIRCULAR COMPOSITE DISK SUBJECTED TO EDGE
ROTATIONS AND HYDROSTATIC PRESSURE
by
STANLEY T. OLn7ER
A THESIS
Submitted in partial fulfillment of the requirements
of the degree of Master of Science
of
The School of Mechanical and Aerospace Engineering
of
The University of Alabama in Huntsville
HUNTSVILLE, ALABAMA
2004
In presenting this thesis in partial fulfillment of the requirements for a master’s degree
from The University of Alabama in Huntsville, I agree that the Library of this University
shall make it fieely available for inspection. I further agree that permission for extensive
copying of scholarly purposes may be granted by my advisor or, in hisher absence, by
the Chair of the Department (Director of Program) or the Dean of the School of Graduate
Studies. It is also understood that due recognition shall be given to me and to The
University of Alabama in Huntsville in any scholarly use which may be made of any
makri-a4i m- u14.s h &s.
(student signature) (date)
ll
THESIS APPROVAL FORM
Submitted by Stanley Oliver in partial fulfillment of the requirements for the degree of
Master of Science in Engineering in Mechanical and Aerospace Engineering and
accepted on behalf of the Faculty of the School of Graduate Studies by the thesis
committee.
We, the undersigned members of the Graduate Faculty of The University of Alabama in
Huntsville, certify that we have advised andor supervi'sed the candidate on the work
described in this thesis. We further certify that we have reviewed the thesis manuscript
and approve it in partial fulfillment of the requirements of the degree of Master of
Science in Mechanical and Aerospace Engineering.
Committee Chair
Pate)
Department Chair
College Dean
Graduate Dean
...
ll1
Abstract
The structural analysis results for a graphite/epoxy quasi-isotropic circular plate subjected
to 2 hrcecl rntzttinn at the boundary and pressure is presented. The analysis is to support
a specialized material characterization test for composite cryogenic tanks. Finite element
models were used to ensure panel integrity and determine the pressure necessary to
achieve a predetermined equal biaxial strain value. The displacement results due to the
forced rotation at the boundary led to a detailed study of the bending stifhess matrix [D].
The variation of the bending stifhess terms as a function of angular position is presented
graphically, as well as, an illustrative technique of considering the laminate as an I-beam.
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ACKNOWLEDGEMENTS
The work described in this thesis.. . . . .. . ...
V
TABLE OF CONTENTS
...
Lis? of Figures .................................................................................................................. vi11
List of Tables .................................................................................................................... ix
List c?f Smhnls ................................................................................................................... x
-d---
~
Chapter
1. INTRODUCTION ................................................................................................... 1
1.1. Background ........................................................................................... 1
1.2. Cryogenic Biaxial Permeability Apparatus ........................................ 2
1.2.1 CBPA Test .............................................................................. 3
1.2.2 CBPA Design .......................................................................... 4
1.3. Objective ............................................................................................... 6
2 . REVIEW OF LITERATURE .................................................................................. 7
2.1 Introduction ............................................................................................ 7
2.2 Test Methods .......................................................................................... 7
2.3 Mechanics of Composites ..................................................................... -9
2.4 Finite Element Model Convergence .................................................... 11
3 . BACKGROUND THEORY .................................................................................. 13
3.1 Introduction .......................................................................................... 13
3.2 Lamina Mechanics ............................................................................... 13
3.3 Laminate Mechanics ............................................................................ 17
3.4 Laminate Stifhess ............................................................................... 20
4 . APPROACH .......................................................................................................... 22
4.1 Introduction .......................................................................................... 22
vi
4.2 Panel Design Standarization ................................................................ 22
. 4.3 Finite Element Model .......................................................................... 23
4.4 Analysis ................................................................................................ 24
4.5 Bending Stiffiess Matrix [D] ............................................................... 26
4.6 I-Beam Analogy ................................................................................... 26
5 . RESULTS .............................................................................................................. 28
5.1 Introduction .......................................................................................... 28
5.2 Finite Element Model .......................................................................... 28
5.3 Convergence Study .............................................................................. 30
5.4 Confidence Study ................................................................................. 38
5.5 Analysis of Test Apparatus .................................................................. 39
5.5.1 Taper Angle Optimization .................................................... 38
5.5.2 Equal Biaxial Strain Region .................................................. 40
5.6 Displacements ...................................................................................... 44
5.7 Bending Stiffness [D] Study ................................................................ 4.'-/
5.8 I-Beam Analogy ................................................................................... 52
6 . CONCLUSIONS. ................................................................................................... 57
APPENDIX A: Stiffness Matrix Calculations in MathCAD ............................................ xx
APPENDIX B: MathCAD program using invariants ....................................................... xx
APPENDIX C:N ASTR4.N Case Control and Important Bulk Data Entries ................... xx
.................................................................................................................
REFERENCES xx
Vii
LIST OF FIGURES
Figure Page
1.1 A cross section of the CBPA ................................................................................... 3
1.2 A schematic of the pre-assembly configuration ....................................................... 5
3.1 Lamina coordinate system relative to structural coordinate system ...................... 15
3.2 Nomenclature of an n-layered laminate ................................................................. 17
5.1 Models used in convergence study ........................................................................ 31
5.2 Outer fiber ,E strains along structural x-axis ........................................................ 34
5.3 Four circular plate problems used in confidence study ......................................... 35
5.4 Optimization of taper angle to minimize stress in composite panel ...................... 39
5.5 Outer fiber strains at the center of the panel .......................................................... 41
5.6 Outer fiber strains in the material coordinate system ............................................ 42
5.7 Outer fiber strains along the structural x-axis ........................................................ 43
5.8 O~tefri ber biaxial strain ratio ................................................................................ 43
5.9 Out-of-plane displacement of the panel ................................................................. 44
5.10 Plot of normal displacements. ................................................................................ 45
5.11 The normal displacements along the structural x-y axes due to assembly and taper
angle of twelve degrees ...................................................................................................... 49
5.12 The stiffness terms of [D] in cyIindrica1 coordinates ............................................. 49
a,
5.13 Normalization of for eachply at 0.degrees. along the structural x-axis ......... 53
5.14 Normalization of for eachply at 9O.degrees. along the structural x-axis .......5 4
5.15 D11 and D22 at 60 degrees. ...................................................................................... 56
...
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LIST OF TABLES
Table Page
3.1 lM7/8552 GraphiteEpoxy material properties ...................................................... 20
..
5.1 Comparison of finite element models .................................................................... j z
5.2 E Mp ercent difference for outer fiber stress ........................................................ 32
5.3 Convergence study results ..................................................................................... 33
5.4 Displacement and edge moment comparisons between theory and FEM .............3 8
5.5 Contribution of each ply to the overall stifhess value .......................................... 54
5.6 Contribution of each ply to the overall stifhess value .......................................... 55
5.7 Comparison between D11 and D22 at angle of60 degrees ................................... 56
ix
LIST OF SYMBOLS
General Abbreviations
laminate extensional stifhess matrix
CAI
rR1 laminate bending-stretching coupling matrix
L- J
laminate bending stifhess matrix
[Dl
E Young's modulus - isotropic
El, E2, E3 axial moduli - principal material directions
Ex axial modulus
EY transverse modulus
G shear modulus - isotropic
[MI moments per unit length
in-plane forces per unit length
[Nl
CQ1 reduced stiffness
Reuter matrix
CRI
[TI . transformation matrix
structural axes
X7Y
%Y,Z global coordinates
17293 material principal coordinates
CBPA Cryogenic Biaxial Permeability Apparatus
CLT Classical Lamination Theory
CTE coefficient of thermal expansion
GrEp graphi te/epoxy
F" degree Fahrenheit
LH2 liquid hydrogen
MSFC Marshall Space Flight Center
NASTRAN NASA Structural Analysis
NASA National Aeronautics and Space Administration
a distance fiom center to outer edge of circular plate
b radial position
P concentrated load
r radial position
upward normal of plate
Z
distributed pressure
90
in inch
psi pound force per square inch
msi . one million pound force per square inch = 1* 1O 6 psi
*
ksi one thousand pound force per square inch = 1 1O 3 psi
lbf pound force
plate normal direction
Z
X