Mechanics of Composites
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B B VL Deepak
M V A Raju Bahubalendruni
Mechanics of Composites
Mechanics of Composite materials using Ms-Excel
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 Ms-Excel. 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
The micro mechanics of the composite materials based on different models, laminate
analysis to obtain the mid-plane 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, semi-interactive 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
B B V L Deepak
Dedicated with love to my family members
B B V L Deepak
This is a book is specially designed for people, who are interested in mechanics of composite
materials. In this book, how to use Ms-Excel 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 step-by-step solving methodology for mechanics of composite material
problems are illustrated using Ms-Excel. 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 Ms-Excel functions and applications.
Chapter-1 describes the computations involved in micro mechanics of composite materials
based on rule of mixtures and Chamis equations with detailed examples.
Chapter-2 is focused on the mechanical behavior of lamina at macro level, which is essential
to understand the performance of laminated fiber-reinforced structures. Procedure involved
in obtaining ply level stresses and strains in local coordinated systems are clearly explained
with detailed examples.
Chapter-3 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.
Chapter-4 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
Chapter-5 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
List of figures vi
List of tables vi
Introduction to Ms-Excel 0-1
Types of Operators
Arithmetic, Comparison, Text concatenation & Reference operators
Calculation order & Operator precedence
General Math functions
Chapter-I Micro Mechanics of composite materials 1-1
A. Basic Definitions
B. Rule of Mixtures Equations for engineering constants
C. Halpin-Tsai 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
Chapter-II Macro Mechanics of composite materials 2-1
A. Basic Definitions
B. Stress-Strain relations for orthotropic material
C. Stress-Strain relations in an angled ply
Chapter-III Failure theories-I 3-1
A. Failure theories – Definition
B. Maximum stress failure theory
C. Maximum strain failure theory
D. Maximums strain failure theory expressed in Stresses
E. Stowell-Lin 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
Chapter-IV Failure theories-II 4-1
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
Chapter-V Laminated composites 5-1
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 6-1
List of Tables Page No.
Table 1.1 Rule of mixtures equations for lamina elastic properties 1-7
Table 1.2 Graphite and Epoxy 934 Material properties 1-8
Table 1.3 Halpin-Tsai Equations for E2, G12 & G23 1-15
Table 1.4 Chamis Equations for E2, G12 & G23 1-15
Table 1.5 Hygro thermal coefficients of composites based on Rule
of Mixture equations. 1-19
Table 1.6 Hygro thermal coefficients of composites based on
Chamis equations. 1-19
Table 1.7 Hygro thermal coefficients of composites based on Rule
of Mixture equations and Chamis equations. 1-22
Table 1.8. Chamis equations for lamina strength(s). 1-23
Table 2.1 Graphite Epoxy Material properties 2-4
Table 6.1 AS4 fiber properties 6-1
Table 6.2 3501-6 Epoxy matrix properties 6-1
List of Figures
Fig. 2.1 Schematic representation of lamina 2-1
Fig. 2.2 Isotropic Material Vs Orthotropic material 2-2
Fig. 2.3a Schematic representation of loads acting on an
Orthotropic lamina 2-8
Fig. 2.3b Schematic representation of loads acting on an
Orthotropic lamina 2-9
Fig. 5.1 Normal displacements along x and y axes 5-1
Fig. 5.2 Shear displacement along x and y axes 5-1
Fig. 5.3 Bending displacement along z direction 5-1
Fig. 5.4 Representation of bending slopes in a laminated
Fig. 5.5a Laminated composite structure 5-3
Fig. 5.5b Loads on Laminated composite structure 5-3
Fig 5.6 Un-deformed and Deformed cross sections 5-3
Fig 5.7 Laminate configuration 5-8
The main book is available at
Abs 0-3, 3-9.
Angel ply laminate 5-17.
Angled ply 2-8.
Anti symmetric laminate 5-17.
Arithmetic operators 0-1.
Balanced laminate 5-17.
Bending stiffness matrix [D] 5-11.
Calculation order 0-2.
Chamis equations 1-15, 1-19,
Classical laminate theory 5-1.
Classification of laminate 5-17.
Coefficient of moisture expansion 1-5, 1-19.
Coefficient of thermal expansion 1-4, 1-19.
Comparison operators 0-1
Compliance matrix 2-3, 5-18.
Coupling elements 5-16 , 5-18.
Cross ply laminate 5-17.
Effective laminate constants 5-20,5-22,6-1
Extension stiffness matrix [A] 5-11
Stiffness matrix [B] 5-11
Failure index 3-1.
Failure theories 3-1.
Fiber failure 3-1,3-17,
Fischer failure 4-25.
Flexural in-plane constants 5-22.
Halpin-Tsai equations 1-15.
Hashin failure theory 3-25.
Hashin Rotem failure theory 3-21.
Hoffman failure theory 4-1.
Hooke's law 2-2, 3-12.
Interactive failure theory 4-1.
Inverse transformation matrix 2-10.
Kelly- Davies failure theory 3-19.
Laminate selection 5-18.
Matrix failure 3-1,3-2,3-18,
Maximum strain failure theory 3-6.
Maximum stress failure theory 3-1.
Modified puck failure theory 3-27.
Moisture diffusivity 1-6,1-22.
Norris failure theory 4-23.
Norris- Mckinnon failure theory 4-16.
Operator precedence 0-2
Poisson's ratio 1-2,1-7.
Prager failure theory 3-20.
Puck - Schneider failure theory 4-26.
Puppo - Evensen failure theory 4-24.
Q matrix 2-3.
Q bar matrix 2-14,2-19.
Reference operators 0-1.
Reserve factor 3-1.
Rule of mixtures equations 1-7, 1-9,
Shear matrix cracking 3-2, 3-13.
Shear modulus 1-2, 1-7,
Specific heat 1-5, 1-22.
Sqrt 0-5, 4-4.
Stiffness matrix 2-3, 2-14,
Stowell-Lin failure theory 3-17.
Strain failure theory in stresses 3-12.
Stress-strain relation 2-2,2-8.
Sum 0-2, 5-
16.Symmetric laminate 5-17,5-18.
Text concatenation operator 0-2.
Thermal conductivity 1-6,1-22.
Transformation matrix 2-10.
Transverse matrix cracking 3-1.
Tsai Hill failure theory 4-1.
Tsai Wu failure theory 4-6.
Ultimate compressive strength 1-3,1-23.
Ultimate shear strength 1-4,1-23.
Ultimate strains 3-6.
Ultimate tensile strength 1-3.
Un-symmetric laminate 5-18.
Unbalanced laminate 5-17.
Uni-direction laminate 5-17.
Volume fraction 1-8.
Yamada sun failure theory 4-21
Young's modulus 1-1,1-7,