MECHANICS OF FIBROUS COMPOSITES
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ELSEVIER SCIENCE PUBLISHING CO., INC. 655 Avenue of the Americas, New York, NY 10010, USA WITH 85 TABLES AND 241 ILLUSTRATIONS © 1991 ELSEVIER SCIENCE PUBLISHERS LTD British Library Cataloguing in Publication Data Datoo, Mahmood Husein Mechanics of fibrous composites. 1. Composite Materials I. Title 620.118 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 1-85166-600-1 1. Fiber reinforced plastics. I. Title. TA455.P55D27 1991 620.1'923-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
Angle-ply antisymmetric MOPL, 231—6 Angle-ply electric constants, 238—41 Angle-ply laminate elastic constants, 238—41 Angle-ply symmetric laminates, 246—7 Angle-ply symmetric MOPL, 201—16 Anisotropic bodies, 10—11 Antisymmetric laminate configuration, 154 Antisymmetric MOPL, 225—36 angle-ply, 231—6 cross-ply, 225—31 Aramid. See Kevlar Axes, 12—13 material, 12—13 reference, 12—13 Axes systems, thin-walled composite sections, 484—5 Axial forces, thin-walled composite sections, 485—97
Buckling, laminates, of, 588—97 bending orthotropy, with, 589—91 bending orthotropy, without, 591 7 Carbon fibres, 3 Chopped strand mat, 7 Chopped strands, 6—8 Closed sections, torsion, and, 547—56 Complete ply failure—bending load, 333—55 first ply, 334—9 second ply, 339—51 third ply, 351—5 Complete ply failure—membrane load ( LPF), 273—302 first ply, 274—80, 291—3 second ply, 280—5, 293—4 third ply, 285—90, 298—302 Composite beams, line elements, 606—10 Composite, definition of, 1 Composite properties, 4—8 Compound isotropic beams, 104—13, 367—74 symmetric, 104—8 unsymmetric, 109—13 Compression tests, uniaxial, 624—5 Compressive strength longitudinal, 358—9, 362 transverse, 359 Computer-aided laminate analysis, 597—602 Constant moisture distribution, 431 71 Constant temperature distribution, 391—422
Beam constitutive equation, 99—113 symmetric compound, isotropic, 104—9 unsymmetric compound, isotropic, 109—13 Bending equivalent constants, 144—8 Bending load, ply failure, and, 333—55 first ply, 334—9 second ply, 339—51 third ply, 351—5 Bending .orthotropy, 589—91 absence of, 591—7 631
632 Constitutive equations, layered laminate, 99129 beam, 99113 laminate, 11329 Cross-ply antisymmetric MOPL, 22531 Cross-ply laminate elastic constants, 2378 Cross-ply symmetric laminates, 246 Cross-ply symmetric MOPL, 191201 E-glass, 2 Elastic constants, 13648 laminates, 23645 angle-ply, 23841 cross-ply, 2378 quasi-isotropic, 241-5 transformation, 4566 Elastic properties, 667 Electrical glass. See E-glass Element deformation, 1722 Epoxy resins, 4 Equivalent electric constants, 13648 Estimated laminate strength, 35565 Estimated membrane elastic constants, 23645 angle-ply laminates, 23841 cross-ply laminates, 2378 quasiisotropic laminates, 2415 Failure criterion choice, 967 Failure criteria, ply stength analysis, and, 7592 Fibre properties, 28 Fibre-reinforced plastic (FRP), 2-6 Fibrous composite failure criteria, 7692 Fibrous composites, 12 Finite element method, 60219 line elements, 60210 shell elements, 61019 First ply failures, 27480, 2913, 303, 31621, 3349 complete, 27480, 2913 bending load, 3349 membrane load, 27480, 2913 mechanical/hygrothermal, 47682 mechanical/thermal residual, 472 6 partial, membrane load, 303, 31621
Index
First ply failures—contd. procedure, 24853 symmetric laminate, 25365 bending load, 2605 membrane load, 25360 unsymmetric laminate, membrane load, 2659 residual stress, 47282 FPF. See First ply failures Free edge effects, 5578 FRP. See Fibre-reinforced plastic Generally orthotropic ply, 3445 symmetric MOPL, 18591 symmetric SOPL, 1707 Glass fibre types, 2 Heterogeneous bodies, 9 Hoffman theory, 789 Holes, laminates, in, 5818 membrane orthotropy, 5868 specially orthotropic ply, 5826 Homogeneous bodies, 9 Hookes law, 25, 26, 46, 99, 100, 142 Hygroscopic diffusion, 4226 Hygroscopic stress, laminates, 42631 In-plane shear tests, 6268 Interlaminar stresses, 55780 55862 fYZ, 5627 fz, 5627 free edge effects, 5578 interlaminar stress, 55867 prediction method, 56772 stacking sequence, 5729 Isotropic beam line elements, 6036 Isotropic beams, 10013, 36774 axial deformation, 100 bending deformation, 101 strain/stress distribution, 1014 symmetric compound, 1049 unsymmetric compound, 10913 Isotropic bodies, 910 Isotropic ply, 1625, 6874 general, and, 26 maximum principal theories, 70 strain, 70 stress, 70 shear strain energy theory, 71
Index
Isotropic sections, torsion, and, 534– 8, 547–51 Isotropic symmetric MOPL plies, 177–85 Isotropic symmetric plies, 155–63
Kevlar, 1, 3
Lamina, definition, 12–13 Laminate analysis, computer-aided, 597–602 Laminate configuration types, 153–5 antisymmetric, 154 symmetric, 153 unsymmetric, 154–5 Laminate constitutive equation, 113–29 Laminate equivalent hygroscopic coefficients, 430–1 Laminate equivalent thermal coefficients, 388–9 Laminate hygroscopic stresses, 426–31 constant moisture distribution, 427–8 equivalent coefficients, 430–1 linearly varying moisture distribution, 429–30 residual stress determination, 427 Laminate notation, 129–36 Laminate positive system, axes, of, 114–17 Laminate stiffness analysis, 149–247 Laminate strength, 355–65 Laminate strength analysis, 248–366 longitudinal compressive strength, 358–9 longitudinal tensile strength, 357– 8, 361 maximum stress theory, 356–7 negative shear strength, 360 positive shear strength, 360 transverse compressive strength, 359 transverse tensile strength, 362–3 Laminate strength residual stresses, 472–82 FPF, 472–82 mechanical/hygrothermal, 476– 82 mechanical/thermal, 472–6
633 Laminate thermal stresses, 374–89 equivalent thermal coefficients, 388–9 linearly varying temperature distribution, 385–8 residual determination, 381–5 temperature distribution, 381–5 Laminates, holes in, 581–8 Last-ply failure procedure, 269–73 complete ply failure, 273–301, 333–55 bending load, 333–55 membrane load, 273–301 partial failure—membrane load, 302–33 Layered laminates, 99–148 terminology, 11 – 13 types of materials, 9–11 Line elements, 602–10 composite beam, 606–10 isotropic beam, 603–6 Linear moisture distribution, 431–71 Linearly varying temperature distribution, 385–9 Longitudinal stength, 357–3 compressive, 358–9, 362 tensile, 357–8, 361–3 Longitudinal stress, 16–17
Macromechanics, 11–12 Material axes, 12–13 Materials properties comparisons, 2–8 Materials classification, 9–11 anisotropic, 10–11 heterogeneous, 9 homogeneous, 9 isotropic, 9–10 orthotropic, 10 Matrix definition of, 1 functions of, 2 Maximum principal stress theory, 70 Maximum shear stress theory, 71 Maximal stress theory, 76–8, 356–7 Maxwell's reciprocal theorem, 50, 142 Membrane elastic constants, 140–4, 236–45 Membrane load, ply failure, and, 273–333 complete, 273–302
634
Index
Membrane load, ply failure, and contd.
first ply, 27480, 2913, 303, 31621 partial, 30233 second ply, 2805, 2934, 3039, 3215 third ply, 285-90, 298302, 309 16, 32533 Membrane orthotropy, 5868 Micromechanics, 11 Moisture, 42271 distribution constant, 4279 linearly varying, 42930 Moisture content profiles, 4701 MOPL antisymmetric, 22536 angle-ply, 2316 cross-ply, 22531 symmetric, 177225 angle-ply, 20116 cross-ply, 191200 generally orthotropic plies, 18591 isotropic plies, 17785 quasi-isotropic, 21625 Multioriented ply laminates. See MOPL Negative shear stength, 360 Negative shear stress, 9394, 360 Negative stress system, 16 Notation, laminate, 12936 Open sections, torsion, and, 53347 Orthotropic bodies, 10 Orthotropic plies MOPL, 18591 SOPL, 16377 generally, 1707 specially, 163-70 Orthotropic ply, 2534, 745 generally, 3445 Orthotropic ply shear stresses, 924 Orthotropy membrane, 5868 specially, 5827 bending, with, 58991 bending, without, 5917
Partial ply failure—membrane load, 30233 first ply, 303, 31621 second ply, 3039, 3215 third ply, 30916, 32533 Ply, definition of, 1213 see also specific types of
Ply angle, 557 longitudinal strength variation, 86 longitudinal tensile strength, 87 positive shear strength, 91 woven ply, 602 elastic constants, 60 Ply configuration, laminate notation, and, 13640 Ply failure, 248366 Ply laminates multioriented. See MOPL single oriented. See SOPL Ply, specially orthotropic, 5827 Ply stiffness analysis, 1467 assumptions for, 14 electric constant transformation, 4566 isotropic ply, 1625 orthotropic ply, 2545 Ply stength, 978 analysis, 6898 failure criteria, 7597 Hoffman theory, 789 maximal strain theory, 778 maximal stress theory, 767 shear stress signs, 926 TsaiHill theory, 78 Tsai Wu strain theory, 7980 Tsai Wu stress theory, 79 ultimate strengths, 745 Point stress, 1516, 70 Poissons ratio, 224, 26, 28, 31, 47, 50, 53, 58, 141, 143, 1457 ply angle, 56 Polyester resins, 45 Positive shear stress, 923, 360 Positive stress system, 16 Prepregs, 89 Quasi-isotropic electric constants, 2415 Quasi-isotropic laminate elastic constants, 2415 Quasiisotropic laminates, 247 Quasiisotropic symmetric MOPL, 21625
Index
Rankine failure criterion, 72 Rankine theory, 70 Reduced compliances, 2830 Reference axes, 1213 Reinforcement, definition, 1 Reinforcing fibres, 69 Residual stresses, 367482 compound isotropic beam, 36774 constant temperature distribution, 391422 hygroscopic diffusion, 4227 laminate hygroscopic, 42631 laminate stength, 47182 moisture distribution, 43171 thermal, determination, 38991 Resins. See specific types of Roving, 7 Second ply failures, 2805, 2938, 3039, 3215, 33951 complete, 2805, 293, 33951 bending load, 33951 membrane load, 2805, 2934 S-glass, 2 Shear, 71 Shear forces, thin-walled composite sections, 51733 closed section, 52733 Shear modulus, ply angle, and, 56 Shear strain energy theory, 71 Shear strength, 360 negative, 360 positive, 360 Shear stress signs, ply stress analysis, 92 Shear tests, in-plane, 6268 Shell elements, FE method, 61019 symmetry conditions, 61617 validation tests, 61719 Silicaglass. See S-glass Singly-oriented ply laminates. See SOPL SOPL generally orthotropic, 1707 isotropic plies, 15563 specially orthotropic plies, 16370 symmetric, 15577 Specially orthotropic ply, 2434, 5827 symmetric SOPL, 16370 Specific modulus, 26 Stacking sequences, interlaminar stresses, 5729
635 Stiffness formulation, 149 Strain theory, 70, 778 TsuWu, 7980 Strength. See Laminate strength analysis properties, 978 Stress residual, 367482 unsymmetric compound isotropic beam, 113 Stressstrain relationship, 2930 Stress theory, 70, 767 Stresses, interlaminar. See Interlaminar stresses Symmetric angle-ply laminates, 2467 Symmetric compound isotropic beams, 1049 Symmetric cross-ply laminates, 246 Symmetric laminate—bending load (FPF), 2605 Symmetric laminate configuration types, 153 Symmetric laminate—membrane load, 25360 Symmetric MOPL, 177225 angle-ply, 20116 crossply, 191201 generally orthotropic plies, 185 91 isotropic plies, 17785 quasi-isotropic, 21625 Symmetric quasi-isotropic laminates, 247 Symmetric SOPL, 15577 generally orthotropic, 1707 isotropic plies, 15563 specially orthOtropic, 16370 Symmetrical bending, thin-walled composite sections, 497508 Temperature distribution constants, 3815, 391422 Temperature distribution, linearly varying, 3859 Temperature distribution residual stresses, 391422 Tensile strength, 28 longitudinal, 3578, 361 transverse, 359, 3623 Tension tests, uniaxial, 6223 Test coupons, 6212 Test methods, 61928 coupons, 621
636 Test methods—contd. in-plane shear, 6268 standards, 621 uniaxial compression, 6246 uniaxial tension, 6223 Thermal coefficients, 3889 Thermal stresses, 37489 laminates, 37489 Thermal residual strength determination, 38991 Thermoplastic, 34 Thin-walled composite sections, 483 556 assumptions for, 484-5 axial force, 48597 shear forces, 51733 symmetrical bending, 497508 torsion, 53356 closed sections, of, 54756 %open sections, of, 53347 Third ply failure, 28590, 298302, 30916, 32533, 3515 complete, 28590, 298302, 3515 bending load, 3515 membrane load, 28590, 298 302 parital, membrane load, 30916, 32533 Torsion, 53356 composite sections, 53947, 5516 closed, 54756 open, 53347 isotropic sections, 5348, 54751 Transverse compressive strength, 359 Transverse isotropy, 3134 Transverse strength, compressive,
Index
Transverse stress element deformation, 26-27 Transverse tensile strength, 3623 Tresca failure criterion, 72 Tresca theory, 71 TsaiHill theory, 78 TsaiWu strain theory, 7980 TsaiWu stress theory, 79
Uniaxial compression tests, 6245 Uniaxial tension tests, 622-3 Unidirection ply elastic constants, ply angle and, 55 Unsymmetric compound isotropic beam, 10913 Unsymmetric laminate configuration, 1545 Unsymmetric laminate—membrane load (FPF), 2659 Unsymmetrical bending thin-walled composite sections, in, 50817 Von Mises failure criterion, 72 Von Mises theory, 71 Woven ply, 602, 89 Youngs modulus, 25, 52, 55, 58, 100, 1367,1412,1457
