Mechanics of Composites

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Transcript of Mechanics of Composites
`
i
B B VL Deepak
M V A Raju Bahubalendruni
Mechanics of Composites
Using MsExcel
(Completely solved)
(
(a
No prior
Knowledge is
required
`
ii
Mechanics of Composite materials using MsExcel
Composite materials are occupying major role in several industries like Aerospace,
Automotive, Construction and Industrial engineering applications because of their high
strength to weight ratio and many other distinctive properties. So the Engineers deal with
the structures in any domain with knowledge on composites will be in a great demand.
Although much literature is available on basic mechanics of composites and composite
structures, the calculations part comprise of complex trigonometric equations & matrices
operation which demands a computing tool to perform efficiently. This book addresses that
need by offering the computation methodology using simple and most available soft tool
named MsExcel. The derivations and the related assumptions are presented in lucid
manner. The focus of the current work is on fiber reinforced composite materials composed
of fibers embedded in a matrix material and the laminates made of fiber reinforced
composites.
The micro mechanics of the composite materials based on different models, laminate
analysis to obtain the midplane strains using classical laminate theory, Macro mechanics of
composites to obtain detailed ply level stresses and strains in local coordinated system and
implementation of various failure theories (limit failure theories, semiinteractive failure
theories and interactive failure theories) are clearly explained with detailed examples.
Classification of laminates and their selection procedure is also briefly described. The effect
of coupling elements in the behaviour of the laminate is explained.
The code explained in this book can be readily applied on a practical design problem. More
than 20 solved example problems are discussed in this book. This book doesn’t demand any
prior cognition on matrix and trigonometric operations. This book will be of great practical
use to students, researchers and practicing engineers seeking to acquire knowledge on
implementation and solving the mechanics of composites.
B B V L Deepak
`
ii
B B V L Deepak
Mechanics of
Composite materials
using MsExcel
Dedicated with love to my family members
B B V L Deepak
`
iii
PREFACE
This is a book is specially designed for people, who are interested in mechanics of composite
materials. In this book, how to use MsExcel as a calculator to perform numerical
calculations needed in basic mechanics of composite materials is explained. The steps
involved in the mechanical calculations at micro and macro level will be accentuated in this
book. A detailed stepbystep solving methodology for mechanics of composite material
problems are illustrated using MsExcel. Along with these the user can find readymade
spread sheets for instant usability.
The book is divided into 5 major chapters followed with introduction
Introduction: This gives a short introduction to MsExcel functions and applications.
Chapter1 describes the computations involved in micro mechanics of composite materials
based on rule of mixtures and Chamis equations with detailed examples.
Chapter2 is focused on the mechanical behavior of lamina at macro level, which is essential
to understand the performance of laminated fiberreinforced structures. Procedure involved
in obtaining ply level stresses and strains in local coordinated systems are clearly explained
with detailed examples.
Chapter3 is focused on implementation of Limit failure theories and Semi interactive failure
theories on the ply level stresses and strains to evaluate the strength of the plies are clearly
explained with detailed examples.
Chapter4 is focused on implementation of interactive failure theories on the ply level
stresses and strains to evaluate the strength of the plies are clearly explained with detailed
examples.
Chapter5 is focused on laminated composites, where types of laminates and their
behaviour, Effect of coupling elements and their elimination procedure is clearly explained
and obtaining Global stiffness matrix , compliance matrix and Effective laminate constants
are clearly explained with detailed examples.
Master example is solved, which covers from Micro mechanics to macro mechanics and the
strength is evaluated using different failure criterions.
B B V L Deepak
`
iv
CONTENTS
List of figures vi
List of tables vi
Introduction to MsExcel 01
Types of Operators
Arithmetic, Comparison, Text concatenation & Reference operators
Calculation order & Operator precedence
General Math functions
Trigonometric functions
Matrix functions
ChapterI Micro Mechanics of composite materials 11
A. Basic Definitions
B. Rule of Mixtures Equations for engineering constants
C. HalpinTsai Equations for engineering constants
D. Chamis Equations for engineering constants
E. Rule of Mixtures Equations for hygro and thermal expansion coefficients
F. Chamis Equations for hygro and thermal expansion coefficients
G. Rule of Mixtures and Chamis Equations for hygro and thermal coefficients
H. Chamis Equations for lamina strengths
ChapterII Macro Mechanics of composite materials 21
A. Basic Definitions
B. StressStrain relations for orthotropic material
C. StressStrain relations in an angled ply
ChapterIII Failure theoriesI 31
A. Failure theories – Definition
B. Maximum stress failure theory
C. Maximum strain failure theory
D. Maximums strain failure theory expressed in Stresses
E. StowellLin Failure theory
F. Kelly Davies Failure theory
G. Prager failure theory
H. Hashin Rotem Failure theory
I. Hashin Failure theory
J. Modified Puck failure theory
`
v
ChapterIV Failure theoriesII 41
A. Tsai Hill failure theory
B. Tsai Wu failure theory
C. Hoffman failure theory
D. Norris Mckinnon failure theory
E. Yamada Sun Failure theory
F. Puppo  Evensen failure theory
G. Norris failure theory
H. Fischer failure
I. Puck Schneider failure theory
ChapterV Laminated composites 51
A. Classical laminate theory
B. Influence of Coupling Elements on Laminate
C. Classification of Laminates
D. Laminate selection criteria
E. Effective Laminate Constants
Master Example 61
References
Appendix
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vi
List of Tables Page No.
Table 1.1 Rule of mixtures equations for lamina elastic properties 17
Table 1.2 Graphite and Epoxy 934 Material properties 18
Table 1.3 HalpinTsai Equations for E2, G12 & G23 115
Table 1.4 Chamis Equations for E2, G12 & G23 115
Table 1.5 Hygro thermal coefficients of composites based on Rule
of Mixture equations. 119
Table 1.6 Hygro thermal coefficients of composites based on
Chamis equations. 119
Table 1.7 Hygro thermal coefficients of composites based on Rule
of Mixture equations and Chamis equations. 122
Table 1.8. Chamis equations for lamina strength(s). 123
Table 2.1 Graphite Epoxy Material properties 24
Table 6.1 AS4 fiber properties 61
Table 6.2 35016 Epoxy matrix properties 61
List of Figures
Fig. 2.1 Schematic representation of lamina 21
Fig. 2.2 Isotropic Material Vs Orthotropic material 22
Fig. 2.3a Schematic representation of loads acting on an
Orthotropic lamina 28
Fig. 2.3b Schematic representation of loads acting on an
Orthotropic lamina 29
Fig. 5.1 Normal displacements along x and y axes 51
Fig. 5.2 Shear displacement along x and y axes 51
Fig. 5.3 Bending displacement along z direction 51
Fig. 5.4 Representation of bending slopes in a laminated
structure 52
Fig. 5.5a Laminated composite structure 53
Fig. 5.5b Loads on Laminated composite structure 53
Fig 5.6 Undeformed and Deformed cross sections 53
Fig 5.7 Laminate configuration 58
`
The main book is available at
http://pothi.com/pothi/book/bbvldeepakmechanicscompositematerialsusingmsexcel
`
APPENDIX
A
Abs 03, 39.
Angel ply laminate 517.
Angled ply 28.
Anti symmetric laminate 517.
Arithmetic operators 01.
B
Balanced laminate 517.
Bending stiffness matrix [D] 511.
C
Calculation order 02.
Chamis equations 115, 119,
122, 123.
Classical laminate theory 51.
Classification of laminate 517.
Coefficient of moisture expansion 15, 119.
Coefficient of thermal expansion 14, 119.
Comparison operators 01
Compliance matrix 23, 518.
Coupling elements 516 , 518.
Cross ply laminate 517.
D
Degrees 04,210.
Density 11,17.
E
Effective laminate constants 520,522,61
Extension stiffness matrix [A] 511
Extensionbending coupling
Stiffness matrix [B] 511
F
Failure 31.
Failure index 31.
Failure theories 31.
Fiber failure 31,317,
319,325.
Fischer failure 425.
Flexural inplane constants 522.
H
HalpinTsai equations 115.
Hashin failure theory 325.
Hashin Rotem failure theory 321.
Hoffman failure theory 41.
Hooke's law 22, 312.
I
Interactive failure theory 41.
Invariants 220.
Inverse transformation matrix 210.
K
Kelly Davies failure theory 319.
L
Lamina 21.
Laminate selection 518.
M
Matrix failure 31,32,318,
320,326.
Maximum strain failure theory 36.
`
Maximum stress failure theory 31.
Micromechanics 17.
Minverse 06,519.
Mmult 06,26,610.
Modified puck failure theory 327.
Moisture diffusivity 16,122.
N
Norris failure theory 423.
Norris Mckinnon failure theory 416.
O
Operator precedence 02
P
Poisson's ratio 12,17.
Power 04.
Prager failure theory 320.
Puck  Schneider failure theory 426.
Puppo  Evensen failure theory 424.
Q
Q matrix 23.
Q bar matrix 214,219.
Quotient 04.
R
Radians 04.
Rand 04.
Reference operators 01.
Reserve factor 31.
Round 05.
Rule of mixtures equations 17, 19,
122.
S
Shear matrix cracking 32, 313.
Shear modulus 12, 17,
115,521,522.
Specific heat 15, 122.
Sqrt 05, 44.
Stiffness matrix 23, 214,
216, 511.
StowellLin failure theory 317.
Strain failure theory in stresses 312.
Stressstrain relation 22,28.
Sum 02, 5
16.Symmetric laminate 517,518.
T
Text concatenation operator 02.
Thermal conductivity 16,122.
Transformation matrix 210.
Transpose 05.
Transverse matrix cracking 31.
Tsai Hill failure theory 41.
Tsai Wu failure theory 46.
U
Ultimate compressive strength 13,123.
Ultimate shear strength 14,123.
Ultimate strains 36.
Ultimate tensile strength 13.
Unsymmetric laminate 518.
Unbalanced laminate 517.
Unidirection laminate 517.
V
Volume fraction 18.
Y
Yamada sun failure theory 421
Young's modulus 11,17,
115,521,522