MECHANICS OF FIBROUS COMPOSITES

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British Library Cataloguing in Publication Data

Datoo, Mahmood HuseinMechanics of fibrous composites.1. Composite Materials I. Title620.118

ISBN 1-85166-600-1

Library of Congress Cataloging-in-Publication Data

Datoo, Mahmood Husein.Mechanics of fibrous composites/Mahmood Husein Datoo.

p. cm.Includes bibliographical references and index.ISBN 1-85166-600-11. Fiber reinforced plastics. I. Title.

TA455.P55D27 1991620.1'923-dc20 91-2204

CIP

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Preface

Contents

Preface ............................................................................................................ vii 1 Introduction .................................................................................................. 1

1.1 Fibrous Composites ........................................................................................................ 1 1.2 FRP Constituents ............................................................................................................ 2 1.3 Reinforcement Types ...................................................................................................... 6 1.4 Types of Materials .......................................................................................................... 9 1.5 Terminology ................................................................................................................. 11

2 Ply Stiffness Analysis ................................................................................. 14

2.1 Isotropic Ply .................................................................................................................. 16 2.2 Specially Orthotropic Ply .............................................................................................. 24 2.3 Generally Orthotropic Ply ............................................................................................. 34 2.4 Transformation of Elastic Constants ............................................................................. 45 2.5 Typical Elastic Properties ............................................................................................. 66

3 Ply Strength Analysis ................................................................................... 68

3.1 Isotropic Ply .................................................................................................................. 68 3.2 Orthotropic Ply ............................................................................................................. 74 3.3 Failure Criteria .............................................................................................................. 75 3.4 Sign of Shear Stresses ................................................................................................... 92 3.5 Choice of Failure Criterion ........................................................................................... 96 3.6 Typical Strength Properties ........................................................................................... 97

4 Layered Laminate ...................................................................................... 99

4.1 Beam Constitutive Equation ......................................................................................... 99 4.2 Laminate Constitutive Equation .................................................................................. 113 4.3 Laminate Notation ...................................................................................................... 129 4.4 Equivalent Elastic Constants ....................................................................................... 136

xi

xii Contents

5 Laminate Stiffness Analysis ................................................................... 149 5.1 Stiffness Formulation Procedure .................................................... 149 5.2 Laminate Configuration Types ...................................................... 153 5.3 Symmetric SOPL: Isotropic Plies .................................................. 155 5.4 Symmetric SOPL: Specially Orthotropic Plies .............................. 163 5.5 Symmetric SOPL: Generally Orthotropic Plies ............................. 170 5.6 Symmetric MOPL: Isotropic Plies ................................................. 177 5.7 Symmetric MOPL: Generally Orthotropic Plies ........................... 185 5.8 Symmetric MOPL: Cross-ply ......................................................... 191 5.9 Symmetric MOPL: Angle-ply ........................................................ 201 5.10 Symmetric MOPL: Quasi-isotropic ............................................. 216 5.11 Antisymmetric MOPL: Cross-ply ................................................ 225 5.12 Antisymmetric MOPL: Angle-ply ............................................... 231 5.13 Estimated Membrane Elastic Constants ....................................... 236 5.14 Laminate Stiffness: Summary ...................................................... 245

6 Laminate Strength Analysis .................................................................. 248

6.1 First-Ply-Failure (FPF) Procedure ................................................. 248 6.2 FPF: Symmetric Laminate—Membrane Load ............................... 253 6.3 FPF: Symmetric Laminate—Bending Load .................................. 260 6.4 FPF: Unsymmetric Laminate—Membrane Load .......................... 265 6.5 Last-Ply-Failure (LPF) Procedure .................................................. 269 6.6 LPF: Complete Ply Failure—Membrane Load .............................. 273 6.7 LPF: Partial Ply Failure—Membrane Load ................................... 302 6.8 LPF: Complete Ply Failure—Bending Load .................................. 333 6.9 Estimated Laminate Strength ......................................................... 355 6.10 Laminate Strength: Summary ....................................................... 364

7 Residual Stresses ...................................................................................... 367

7.1 Compound Isotropic Beam ............................................................. 367 7.2 Laminate Thermal Stresses ............................................................ 374 7.3 Determining Thermal Residual Stresses ........................................ 389 7.4 Constant Temperature Distribution ................................................ 391 7.5 Hygroscopic Diffusion Mechanism ............................................... 422 7.6 Laminate Hygroscopic Stresses ..................................................... 426 7.7 Constant and Linear Moisture Distribution ................................... 431 7.8 Influence on Laminate Strength ..................................................... 471 7.9 Typical Physical Properties ............................................................ 482

8 Thin-Walled Composite Sections ........................................................... 483

8.1 Assumptions and Axes Systems ..................................................... 484 8.2 Axial Force ..................................................................................... 485 8.3 Symmetrical Bending ..................................................................... 497 8.4 Unsymmetrical Bending ................................................................. 508 8.5 Shear Forces ................................................................................... 517

Contents xiii

8.6 Torsion of Open Sections ........................................................................................... 533 8.7 Torsion of Closed Sections ........................................................................................ 547

9 Interlaminar Stresses ........................................................................557

9.1 Free Edge Effects ....................................................................................................... 557 9.2 Interlaminar Stress: fxz .............................................................................................................558 9.3 Interlaminar Stresses: fyz and fZ ............................................................................................562 9.4 Prediction Methods .................................................................................................... 567 9.5 Effect of Different Stacking Sequences ..................................................................... 572

10 Miscellaneous Topics ......................................................................581

10.1 Holes in Laminates .................................................................................................. 581 10.2 Buckling of Laminates ............................................................................................. 588 10.3 Computer Aided Laminate Analysis ........................................................................ 597 10.4 FE Method: Line Elements ...................................................................................... 602 10.5 FE Method: Shell Elements ...................................................................................... 610 10.6 Test Methods ............................................................................................................ 619

Index .......................................................................................................................... 631

1 Introduction

2 Ply Stiffness Analysis

3 Ply Strength Analysis

4 Layered Laminate

5 Laminate Stiffness Analysis

6 Laminate Strength Analysis

7 Residual Stresses

8 Thin-walled Composite Sections

9 Interlaminar Stresses

10 Miscellaneous Topics

Index

631

Angle-ply antisymmetric MOPL,231—6

Angle-ply electric constants, 238—41Angle-ply laminate elastic constants,238—41

Angle-ply symmetric laminates, 246—7Angle-ply symmetric MOPL, 201—16Anisotropic bodies, 10—11Antisymmetric laminate

configuration, 154Antisymmetric MOPL, 225—36

angle-ply, 231—6cross-ply, 225—31

Aramid. See KevlarAxes, 12—13

material, 12—13reference, 12—13

Axes systems, thin-walled compositesections, 484—5

Axial forces, thin-walled compositesections, 485—97

Beam constitutive equation, 99—113symmetric compound, isotropic,104—9

unsymmetric compound, isotropic,109—13

Bending equivalent constants, 144—8Bending load, ply failure, and,333—55

first ply, 334—9second ply, 339—51third ply, 351—5

Bending .orthotropy, 589—91absence of, 591—7

Buckling, laminates, of, 588—97bending orthotropy, with, 589—91bending orthotropy, without, 591

7

Carbon fibres, 3Chopped strand mat, 7Chopped strands, 6—8Closed sections, torsion, and, 547—56Complete ply failure—bending load,

333—55first ply, 334—9second ply, 339—51third ply, 351—5

Complete ply failure—membrane load (LPF), 273—302

first ply, 274—80, 291—3second ply, 280—5, 293—4

third ply, 285—90, 298—302Composite beams, line elements,

606—10 Composite,definition of, 1 Composite properties,4—8 Compound isotropic beams,104—13,

367—74symmetric, 104—8unsymmetric, 109—13

Compression tests, uniaxial, 624—5Compressive strength

longitudinal, 358—9, 362transverse, 359

Computer-aided laminate analysis,597—602

Constant moisture distribution, 43171

Constant temperature distribution,391—422

Index632

Constitutive equations, layeredlaminate, 99129

beam, 99113laminate, 11329

Cross-ply antisymmetric MOPL, 22531Cross-ply laminate elastic constants,

2378Cross-ply symmetric laminates, 246Cross-ply symmetric MOPL, 191201

E-glass, 2Elastic constants, 13648

laminates, 23645angle-ply, 23841cross-ply, 2378quasi-isotropic, 241-5

transformation, 4566Elastic properties, 667Electrical glass. See E-glassElement deformation, 1722Epoxy resins, 4Equivalent electric constants, 13648Estimated laminate strength, 35565Estimated membrane elastic

constants, 23645angle-ply laminates, 23841cross-ply laminates, 2378 quasi-isotropic laminates, 2415

Failure criterion choice, 967Failure criteria, ply stength analysis,

and, 7592Fibre properties, 28Fibre-reinforced plastic (FRP), 2-6

Fibrous composite failure criteria,7692

Fibrous composites, 12Finite element method, 60219

line elements, 60210shell elements, 61019

First ply failures, 27480, 2913, 303,31621, 3349

complete, 27480, 2913bending load, 3349membrane load, 27480, 2913

mechanical/hygrothermal, 47682mechanical/thermal residual, 472

6partial, membrane load, 303, 31621

First ply failures—contd. procedure, 24853symmetric laminate, 25365

bending load, 2605membrane load, 25360

unsymmetric laminate,membrane load, 2659

residual stress, 47282FPF. See First ply failuresFree edge effects, 5578FRP. See Fibre-reinforced plastic

Generally orthotropic ply, 3445symmetric MOPL, 18591symmetric SOPL, 1707

Glass fibre types, 2

Heterogeneous bodies, 9Hoffman theory, 789Holes, laminates, in, 5818

membrane orthotropy, 5868specially orthotropic ply, 5826

Homogeneous bodies, 9Hookes law, 25, 26, 46, 99, 100, 142Hygroscopic diffusion, 4226Hygroscopic stress, laminates, 42631

In-plane shear tests, 6268Interlaminar stresses, 5578055862

fYZ, 5627fz, 5627

free edge effects, 5578interlaminar stress, 55867prediction method, 56772stacking sequence, 5729

Isotropic beam line elements, 6036Isotropic beams, 10013, 36774

axial deformation, 100 bendingdeformation, 101 strain/stressdistribution, 1014 symmetriccompound, 1049 unsymmetriccompound, 10913

Isotropic bodies, 910Isotropic ply, 1625, 6874

general, and, 26maximum principal theories, 70

strain, 70stress, 70

shear strain energy theory, 71

633Index

Isotropic sections, torsion, and, 534–8, 547–51

Isotropic symmetric MOPL plies,177–85

Isotropic symmetric plies, 155–63

Kevlar, 1, 3

Lamina, definition, 12–13Laminate analysis, computer-aided,

597–602Laminate configuration types, 153–5

antisymmetric, 154symmetric, 153unsymmetric, 154–5

Laminate constitutive equation, 113–29Laminate equivalent hygroscopic

coefficients, 430–1Laminate equivalent thermal

coefficients, 388–9Laminate hygroscopic stresses, 426–31

constant moisture distribution,427–8

equivalent coefficients, 430–1linearly varying moisture

distribution, 429–30residual stress determination, 427

Laminate notation, 129–36Laminate positive system, axes, of,

114–17Laminate stiffness analysis, 149–247Laminate strength, 355–65Laminate strength analysis, 248–366

longitudinal compressive strength,358–9

longitudinal tensile strength, 357–8, 361

maximum stress theory, 356–7negative shear strength, 360positive shear strength, 360transverse compressive strength,

359transverse tensile strength, 362–3

Laminate strength residual stresses,472–82

FPF, 472–82mechanical/hygrothermal, 476–82

mechanical/thermal, 472–6

Laminate thermal stresses, 374–89equivalent thermal coefficients,388–9

linearly varying temperaturedistribution, 385–8

residual determination, 381–5temperature distribution, 381–5Laminates, holes in, 581–8

Last-ply failure procedure, 269–73complete ply failure, 273–301,333–55

bending load, 333–55membrane load, 273–301partial failure—membrane load,302–33

Layered laminates, 99–148terminology, 11–13types of materials, 9–11

Line elements, 602–10composite beam, 606–10isotropic beam, 603–6

Linear moisture distribution, 431–71Linearly varying temperature

distribution, 385–9Longitudinal stength, 357–3

compressive, 358–9, 362tensile, 357–8, 361–3

Longitudinal stress, 16–17

Macromechanics, 11–12Material axes, 12–13Materials properties comparisons,

2–8Materials classification, 9–11

anisotropic, 10–11heterogeneous, 9homogeneous, 9 isotropic,9–10 orthotropic, 10

Matrixdefinition of, 1functions of, 2

Maximum principal stress theory,70

Maximum shear stress theory, 71Maximal stress theory, 76–8, 356–7Maxwell's reciprocal theorem, 50,

142Membrane elastic constants, 140–4,

236–45Membrane load, ply failure, and,

273–333complete, 273–302

634 Index

Membrane load, ply failure, andcontd.

first ply, 27480, 2913, 303,31621

partial, 30233second ply, 2805, 2934, 3039,

3215third ply, 285-90, 298302, 309

16, 32533Membrane orthotropy, 5868Micromechanics, 11Moisture, 42271

distributionconstant, 4279linearly varying, 42930

Moisture content profiles, 4701MOPL

antisymmetric, 22536angle-ply, 2316cross-ply, 22531

symmetric, 177225angle-ply, 20116cross-ply, 191200generally orthotropic plies, 18591isotropic plies, 17785quasi-isotropic, 21625

Multioriented ply laminates. SeeMOPL

Negative shear stength, 360Negative shear stress, 9394, 360Negative stress system, 16Notation, laminate, 12936

Open sections, torsion, and, 53347Orthotropic bodies, 10Orthotropic plies

MOPL, 18591SOPL, 16377

generally, 1707specially, 163-70

Orthotropic ply, 2534, 745generally, 3445

Orthotropic ply shear stresses, 924Orthotropy

membrane, 5868specially, 5827bending, with, 58991bending, without, 5917

Partial ply failure—membrane load,30233

first ply, 303, 31621second ply, 3039, 3215third ply, 30916, 32533

Ply, definition of, 1213see also specific types of

Ply angle, 557longitudinal strength variation, 86longitudinal tensile strength, 87positive shear strength, 91 wovenply, 602

elastic constants, 60Ply configuration, laminate notation,

and, 13640Ply failure, 248366Ply laminates

multioriented. See MOPLsingle oriented. See SOPL

Ply, specially orthotropic, 5827Ply stiffness analysis, 1467

assumptions for, 14electric constant transformation,

4566isotropic ply, 1625orthotropic ply, 2545

Ply stength, 978analysis, 6898

failure criteria, 7597 Hoffmantheory, 789 maximal straintheory, 778 maximal stresstheory, 767 shear stress signs,926 TsaiHill theory, 78 TsaiWu strain theory, 7980 TsaiWu stress theory, 79 ultimatestrengths, 745

Point stress, 1516, 70Poissons ratio, 224, 26, 28, 31, 47,

50, 53, 58, 141, 143, 1457 plyangle, 56

Polyester resins, 45Positive shear stress, 923, 360Positive stress system, 16Prepregs, 89

Quasi-isotropic electric constants,2415

Quasi-isotropic laminate elasticconstants, 2415 Quasi-

isotropic laminates, 247 Quasi-isotropic symmetric MOPL,

21625

635Index

Rankine failure criterion, 72Rankine theory, 70Reduced compliances, 2830Reference axes, 1213Reinforcement, definition, 1Reinforcing fibres, 69Residual stresses, 367482

compound isotropic beam, 36774constant temperature distribution,

391422hygroscopic diffusion, 4227laminate hygroscopic, 42631laminate stength, 47182moisture distribution, 43171thermal, determination, 38991

Resins. See specific types ofRoving, 7

Second ply failures, 2805, 2938,3039, 3215, 33951

complete, 2805, 293, 33951bending load, 33951membrane load, 2805, 2934

S-glass, 2Shear, 71

Shear forces, thin-walled compositesections, 51733

closed section, 52733Shear modulus, ply angle, and, 56Shear strain energy theory, 71Shear strength, 360

negative, 360positive, 360

Shear stress signs, ply stress analysis, 92Shear tests, in-plane, 6268Shell elements, FE method, 61019

symmetry conditions, 61617validation tests, 61719 Silica-

glass. See S-glass Singly-orientedply laminates. See

SOPLSOPL

generally orthotropic, 1707isotropic plies, 15563specially orthotropic plies, 16370symmetric, 15577

Specially orthotropic ply, 2434,5827

symmetric SOPL, 16370Specific modulus, 26

Stacking sequences, interlaminarstresses, 5729

Stiffness formulation, 149Strain theory, 70, 778TsuWu, 7980Strength. See Laminate strength

analysisproperties, 978

Stressresidual, 367482unsymmetric compound isotropic

beam, 113Stressstrain relationship, 2930Stress theory, 70, 767 Stresses,interlaminar. See

Interlaminar stressesSymmetric angle-ply laminates, 2467Symmetric compound isotropic

beams, 1049Symmetric cross-ply laminates, 246Symmetric laminate—bending load

(FPF), 2605Symmetric laminate configuration

types, 153Symmetric laminate—membrane

load, 25360Symmetric MOPL, 177225

angle-ply, 20116 cross-ply, 191201generally orthotropic plies, 185

91isotropic plies, 17785quasi-isotropic, 21625

Symmetric quasi-isotropic laminates,247

Symmetric SOPL, 15577generally orthotropic, 1707isotropic plies, 15563specially orthOtropic, 16370

Symmetrical bending, thin-walledcomposite sections, 497508

Temperature distribution constants,3815, 391422

Temperature distribution, linearlyvarying, 3859

Temperature distribution residualstresses, 391422

Tensile strength, 28longitudinal, 3578, 361

transverse, 359, 3623Tension tests, uniaxial, 6223Test coupons, 6212Test methods, 61928

coupons, 621

636 Index

Test methods—contd.in-plane shear, 6268standards, 621uniaxial compression, 6246uniaxial tension, 6223

Thermal coefficients, 3889Thermal stresses, 37489

laminates, 37489Thermal residual strength

determination, 38991Thermoplastic, 34

Thin-walled composite sections, 483556

assumptions for, 484-5axial force, 48597shear forces, 51733symmetrical bending, 497508torsion, 53356

closed sections, of, 54756%open sections, of, 53347 Third ply

failure, 28590, 298302,30916, 32533, 3515

complete, 28590, 298302, 3515bending load, 3515membrane load, 28590, 298

302parital, membrane load, 30916,

32533Torsion, 53356

composite sections, 53947, 5516closed, 54756open, 53347

isotropic sections, 5348, 54751Transverse compressive strength, 359Transverse isotropy, 3134Transverse strength, compressive,

Transverse stress elementdeformation, 26-27

Transverse tensile strength, 3623Tresca failure criterion, 72Tresca theory, 71TsaiHill theory, 78TsaiWu strain theory, 7980TsaiWu stress theory, 79

Uniaxial compression tests, 6245Uniaxial tension tests, 622-3Unidirection ply elastic constants, ply

angle and, 55Unsymmetric compound isotropic

beam, 10913Unsymmetric laminate configuration,

1545Unsymmetric laminate—membrane

load (FPF), 2659Unsymmetrical bending

thin-walled composite sections, in,50817

Von Mises failure criterion, 72Von Mises theory, 71

Woven ply, 602, 89

Youngs modulus, 25, 52, 55, 58, 100,1367,1412,1457