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i

B B VL Deepak

M V A Raju Bahubalendruni

Mechanics of Composites

Using Ms-Excel

(Completely solved)

(

(a

No prior

Knowledge is

required

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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

composites.

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

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ii

B B V L Deepak

Mechanics of

Composite materials

using Ms-Excel

Dedicated with love to my family members

B B V L Deepak

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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 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

examples.

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

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CONTENTS

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

Trigonometric functions

Matrix 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

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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

References

Appendix

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vi

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

structure 5-2

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

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The main book is available at

http://pothi.com/pothi/book/b-b-v-l-deepak-mechanics-composite-materials-using-ms-excel

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APPENDIX

A

Abs 0-3, 3-9.

Angel ply laminate 5-17.

Angled ply 2-8.

Anti symmetric laminate 5-17.

Arithmetic operators 0-1.

B

Balanced laminate 5-17.

Bending stiffness matrix [D] 5-11.

C

Calculation order 0-2.

Chamis equations 1-15, 1-19,

1-22, 1-23.

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.

D

Degrees 0-4,2-10.

Density 1-1,1-7.

E

Effective laminate constants 5-20,5-22,6-1

Extension stiffness matrix [A] 5-11

Extension-bending coupling-

Stiffness matrix [B] 5-11

F

Failure 3-1.

Failure index 3-1.

Failure theories 3-1.

Fiber failure 3-1,3-17,

3-19,3-25.

Fischer failure 4-25.

Flexural in-plane constants 5-22.

H

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.

I

Interactive failure theory 4-1.

Invariants 2-20.

Inverse transformation matrix 2-10.

K

Kelly- Davies failure theory 3-19.

L

Lamina 2-1.

Laminate selection 5-18.

M

Matrix failure 3-1,3-2,3-18,

3-20,3-26.

Maximum strain failure theory 3-6.

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Maximum stress failure theory 3-1.

Micromechanics 1-7.

Minverse 0-6,5-19.

Mmult 0-6,2-6,6-10.

Modified puck failure theory 3-27.

Moisture diffusivity 1-6,1-22.

N

Norris failure theory 4-23.

Norris- Mckinnon failure theory 4-16.

O

Operator precedence 0-2

P

Poisson's ratio 1-2,1-7.

Power 0-4.

Prager failure theory 3-20.

Puck - Schneider failure theory 4-26.

Puppo - Evensen failure theory 4-24.

Q

Q matrix 2-3.

Q bar matrix 2-14,2-19.

Quotient 0-4.

R

Radians 0-4.

Rand 0-4.

Reference operators 0-1.

Reserve factor 3-1.

Round 0-5.

Rule of mixtures equations 1-7, 1-9,

1-22.

S

Shear matrix cracking 3-2, 3-13.

Shear modulus 1-2, 1-7,

1-15,5-21,5-22.

Specific heat 1-5, 1-22.

Sqrt 0-5, 4-4.

Stiffness matrix 2-3, 2-14,

2-16, 5-11.

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.

T

Text concatenation operator 0-2.

Thermal conductivity 1-6,1-22.

Transformation matrix 2-10.

Transpose 0-5.

Transverse matrix cracking 3-1.

Tsai Hill failure theory 4-1.

Tsai Wu failure theory 4-6.

U

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.

V

Volume fraction 1-8.

Y

Yamada sun failure theory 4-21

Young's modulus 1-1,1-7,

1-15,5-21,5-22