Introduction to Composite Materials Design

Introduction to Composite Materials Design
اسم المؤلف
Ever J. Barbero
التاريخ
3 أكتوبر 2020
المشاهدات
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Introduction to Composite Materials Design
Ever J. Barbero
Second Edition
Contents
Acknowledgment xv
Preface xvii
List of Figures xxv
List of Tables xxxi
List of Symbols xxxiii
List of Examples xli
1 Introduction 1
1.1 Basic Concepts 1
1.2 The Design Process 5
1.3 Composites Design Methods . 8
1.4 Design for Reliability . 9
1.4.1 Stochastic Representation 11
1.4.2 Reliability-based Design . 12
1.4.3 Load and Resistance Factor Design . 15
1.4.4 Determination of Resistance Factors 16
1.4.5 Determination of Load Factors . 17
1.4.6 Limit States Design . 18
1.5 Fracture Mechanics 18
Exercises 21
2 Materials 27
2.1 Fiber Reinforcements . 28
2.2 Fiber Types 29
2.2.1 Glass Fibers 29
2.2.2 Silica and Quartz Fibers . 30
2.2.3 Carbon Fibers 31
2.2.4 Carbon Nanotubes 33
2.2.5 Organic Fibers 35
2.2.6 Boron Fibers . 36
viiviii Introduction to Composite Materials Design
2.2.7 Ceramic Fibers 36
2.2.8 Basalt Fibers . 36
2.2.9 Metallic Fibers 37
2.3 Fiber–Matrix Compatibility . 37
2.4 Fiber Forms 38
2.4.1 Continuous and Discontinuous Fibers . 38
2.4.2 1D Textiles: Strand, Tow, End, Yarn, and Roving 40
2.4.3 2D Textiles: Fabrics . 42
2.5 Matrix Materials . 45
2.6 Thermoset Matrices . 48
2.6.1 Polyester Resins . 49
2.6.2 Vinyl Ester Resins 51
2.6.3 Epoxy Resins . 51
2.6.4 Phenolic Resins 52
2.7 Thermoplastic Matrices . 53
2.8 Creep, Temperature, and Moisture . 54
2.9 Corrosion Resistance . 57
2.10 Flammability . 58
Exercises 59
3 Manufacturing Processes 71
3.1 Hand Lay-up . 72
3.2 Prepreg Lay-up 74
3.3 Bag Molding . 75
3.4 Autoclave Processing . 76
3.5 Compression Molding 78
3.6 Resin Transfer Molding . 79
3.7 Vacuum Assisted Resin Transfer Molding . 81
3.8 Pultrusion . 83
3.9 Filament Winding 86
Exercises 88
4 Micromechanics 91
4.1 Basic Concepts 92
4.1.1 Volume and Mass Fractions . 92
4.1.2 Representative Volume Element . 94
4.1.3 Heterogeneous Material . 95
4.1.4 Anisotropic Material . 95
4.1.5 Orthotropic Material . 96
4.1.6 Transversely Isotropic Material . 97
4.1.7 Isotropic Material 97
4.2 Stiffness 97
4.2.1 Longitudinal Modulus 98
4.2.2 Transverse Modulus . 100
4.2.3 In-plane Poisson’s Ratio . 102Table of Contents ix
4.2.4 In-plane Shear Modulus . 103
4.2.5 Intralaminar Shear Modulus . 106
4.2.6 Restrictions on the Elastic Constants 106
4.2.7 Stress Partitioning Parameter 107
4.2.8 Periodic Microstructure . 109
4.3 Moisture and Thermal Expansion 112
4.3.1 Thermal Expansion . 113
4.3.2 Moisture Expansion . 114
4.3.3 Transport Properties . 115
4.4 Strength 116
4.4.1 Longitudinal Tensile Strength 117
4.4.2 Longitudinal Compressive Strength . 120
4.4.3 Fiber Microbuckling . 121
4.4.4 Transverse Tensile Strength . 128
4.4.5 Mode I Fracture Toughness . 129
4.4.6 Transverse Compressive Strength 131
4.4.7 Mohr-Coulomb 131
4.4.8 In-plane Shear Strength . 136
4.4.9 Mode II Fracture Toughness . 138
4.4.10 Intralaminar Shear Strength . 139
Exercises 140
5 Ply Mechanics 143
5.1 Coordinate Systems . 143
5.2 Stress and Strain . 143
5.2.1 Stress . 144
5.2.2 Strain . 147
5.3 Stress–Strain Equations . 149
5.4 Off-axis Stiffness . 153
5.4.1 Coordinate Transformations . 154
5.4.2 Stress and Strain Transformations . 155
5.4.3 Stiffness and Compliance Transformations . 159
5.4.4 Specially Orthotropic Lamina 162
Exercises 164
6 Macromechanics 167
6.1 Plate Stiffness and Compliance . 168
6.1.1 Assumptions . 168
6.1.2 Strains . 170
6.1.3 Stress Resultants . 173
6.1.4 Plate Stiffness and Compliance . 174
6.2 Computation of Stresses . 180
6.3 Common Laminate Types 181
6.3.1 Laminate Description 182
6.3.2 Symmetric Laminates 182x Introduction to Composite Materials Design
6.3.3 Antisymmetric Laminate 184
6.3.4 Balanced Laminate 184
6.3.5 Quasi-isotropic Laminates . 184
6.3.6 Cross-ply Laminate . 189
6.3.7 Angle-ply Laminate . 190
6.3.8 Specially Orthotropic Laminate 190
6.4 Laminate Moduli . 192
6.5 Design Using Carpet Plots 194
6.5.1 Stiffness Controlled Design . 196
6.5.2 Design for Bending 201
6.6 Hygrothermal Stresses 204
Exercises 213
7 Strength 217
7.1 Lamina Failure Criteria . 220
7.1.1 Strength Ratio 220
7.1.2 Maximum Stress Criterion 221
7.1.3 Maximum Strain Criterion . 223
7.1.4 Interacting Failure Criterion . 226
7.1.5 Hygrothermal Failure 232
7.2 Laminate First Ply Failure 233
7.2.1 In-Situ Strength . 237
7.3 Laminate Strength 243
7.3.1 Ply Discount . 243
7.3.2 Truncated-Maximum-Strain Criterion . 246
7.4 Strength Design Using Carpet Plots 252
7.5 Stress Concentrations 258
7.5.1 Notched Plate Under In-plane Load 260
Exercises 265
8 Damage 267
8.1 Continuum Damage Mechanics . 267
8.2 Longitudinal Tensile Damage 269
8.3 Longitudinal Compressive Damage . 273
8.4 Transverse Tension and In-plane Shear . 277
8.4.1 Limitations 279
8.4.2 Approximations 280
8.4.3 Displacement Field 280
8.4.4 Strain Field 281
8.4.5 Laminate Reduced Stiffness . 282
8.4.6 Lamina Reduced Stiffness 282
8.4.7 Fracture Energy . 283
8.4.8 Solution Algorithm 285
8.4.9 Lamina Iterations 285
8.4.10 Laminate Iterations . 285Table of Contents xi
Exercises 286
9 Fabric-reinforced Composites 289
9.1 Weave Pattern Description . 289
9.2 Analysis 293
9.3 Tow Properties 297
9.4 Element Stiffness and Constitutive Relationship 301
9.4.1 Bending-Restrained Model . 302
9.4.2 Bending-Allowed Model . 303
9.5 Laminate Properties . 305
9.5.1 Elastic Constants . 305
9.5.2 Thermal and Moisture Expansion Coefficients . 306
9.6 Failure Analysis 308
9.6.1 Stress Analysis 308
9.6.2 Damage Initiation, Evolution, and Fracture 310
9.6.3 Cross-ply Approximation 314
9.7 Woven Fabrics with Gap . 316
9.8 Twill and Satin 318
9.8.1 Twill Weave with n
g > 2, ns = 1, ni = 1 318
9.8.2 Twill Weave with ns = 1 . 321
9.8.3 Satin Weave with ni = 1 . 323
9.8.4 Twill and Satin Thermo-elastic Properties . 325
9.9 Randomly Oriented Reinforcement . 326
9.9.1 Elastic Moduli 326
9.9.2 Strength 328
Exercises 329
10 Beams 339
10.1 Preliminary Design 340
10.1.1 Design for Deflections 341
10.1.2 Design for Strength . 343
10.1.3 Design for Buckling . 345
10.1.4 Column Behavior . 346
10.2 Thin-Walled Beams . 348
10.2.1 Wall Constitutive Equations . 352
10.2.2 Neutral Axis of Bending and Torsion 354
10.2.3 Axial Stiffness 356
10.2.4 Mechanical Center of Gravity 357
10.2.5 Bending Stiffness . 358
10.2.6 Torsional Stiffness 364
10.2.7 Shear of Open Sections . 367
10.2.8 Shear of Single-Cell Closed Section . 376
10.2.9 Beam Deformations . 377
10.2.10 Segment Deformations and Stresses 378
10.2.11 Restrained Warping of Open Sections . 382xii Introduction to Composite Materials Design
Exercises 383
11 Plates and Stiffened Panels 385
11.1 Plate Bending . 386
11.2 Plate Buckling 387
11.2.1 All Edges Simply Supported . 388
11.2.2 All Sides Clamped 391
11.2.3 One Free Edge 392
11.2.4 Biaxial Loading 392
11.2.5 Fixed Unloaded Edges 393
11.3 Stiffened Panels 393
11.3.1 Stiffened Panels under Bending Loads . 394
11.3.2 Stiffened Panel under In-plane Loads 399
Exercises 406
12 Shells 409
12.1 Shells of Revolution . 411
12.1.1 Symmetric Loading 412
12.2 Cylindrical Shells with General Loading 421
Exercises 427
13 Strengthening of Reinforced Concrete 429
13.1 Strengthening Design . 431
13.2 Materials . 433
13.2.1 Concrete . 434
13.2.2 Steel Reinforcement . 434
13.2.3 FRP 435
13.3 Flexural Strengthening of RC Beams 435
13.3.1 Unstrengthened Behavior 436
13.3.2 Strengthened Behavior 436
13.3.3 Analysis . 437
13.3.4 Strong Strengthening Configuration (SSC) . 439
13.3.5 Weak Strengthening Configuration (WSC) . 441
13.3.6 Balanced Strengthening Configuration (BSC) . 443
13.3.7 Serviceability Limit States . 445
13.3.8 Summary Design Procedure: Bending . 451
13.4 Shear Strengthening . 456
13.4.1 Summary Design Procedure: Shear . 460
13.5 Beam-column . 463
13.5.1 Column: Pure Axial Compression . 465
13.5.2 Summary Design Procedure: Column . 468
13.5.3 Beam-column: Combined Axial Compression and Bending 471
13.5.4 Summary Verification Procedure: Beam-column . 477
Exercises 487Table of Contents xiii
Appendix A Gauss Distribution 489
Appendix B Weibull Distribution 491
Appendix C Conversion of Units 495
Bibliography 497
Index 50
List of Symbols
overline ( ) Transformed, usually to laminate coordinates
widetilde ( ) f Undamaged (virgin), or effective quantity
( ) c Average
α Load factor. Also, fiber misalignment
ασ Standard deviation of fiber misalignment
α0 Angle of the fracture plane
α1, α2 Longitudinal and transverse coefficient of thermal expansion (CTE)
αA, αT Axial and transverse CTE of fibers
[α] Membrane compliance of a laminate
α, β, γ Thickness ratio used with carpet plots
[α] In-plane compliance of a laminate
[β] Bending-extension compliance of a laminate
[δ] Bending compliance of a laminate
θk Orientation of lamina k in a laminate
β1, β2 Longitudinal and transverse coefficient of moisture expansion
δb, δs Bending and shear deflections of a beam
ϵ1t Ultimate longitudinal tensile strain
ϵ2t Ultimate transverse tensile strain
ϵ1c Ultimate longitudinal compressive strain
ϵ2c Ultimate transverse compressive strain
ϵfu Ultimate fiber strain (tensile)
ϵmu Ultimate matrix strain (tensile)
ϵ Strain tensor
εij Strain components in tensor notation
ϵα Strain components in contracted notation
ϵe
α Elastic strain
ϵ

Plastic strain
ϵ0
x, ϵ0 y, γxy 0 Strain components at the midsurface of a shell
eϵ Effective strain in contracted notation (ϵ6 = γ6)
εe Effective strain in tensor notation (ϵ6 = γ6/2)
γ6u Ultimate shear strain
γxy 0 In-plane shear strain
κ0
x, κ0 y, κ0 xy Curvatures of the midsurface of a shell
xxxiiixxxiv Introduction to Composite Materials Design
λ Lame constant. Also, crack density
µ Mean value of a distribution
η Eigenvalues
ηL, ηT Coefficients of influence, longitudinal, transverse
η2, η4, η6 Stress partitioning parameters
ηi = Ei/E Modular ratio in the transformed section method
ϕ Resistance factor. Also, angle of internal friction
ϕ(z) Standard PDF
ϕx, ϕy Rotations of the normal to the midsurface of a shell
ϖ Standard deviation
ρ Density
ρf, ρm, ρc Density of fiber, matrix, and composite
ψ Load combination factor
σ Stress tensor
σ0 Weibull scale parameter
σij Stress components in tensor notation
σα Stress components in contracted notation
σe Effective stress
τL, τT Longitudinal and transverse shear stress
ν Poisson’s ratio
ν12 In-plane Poisson’s ratio
ν23, ν13 Intralaminar Poisson’s ratios
ν
xy Laminate Poisson ratio x-y
νA Axial Poisson’s ratio of fibers
Γ Gamma function
Λ0
22, Λ0 44 Dvorak parameters
Ω = √I-D Integrity tensor
2a0 Representative crack size
g Damage activation function
df Degradation factor
kf Fiber stress concentration factor
m Weibull shape parameter
n1, n2, n3 Components of the vector normal to a surface
p(z) Probability density function (PDF)
r1(φ), r2(φ) Radii of curvature of a shell
tk Thickness of lamina k in a laminate
t1, t2, t3 Projection components of a vector on the coordinate axes 1, 2, 3
tt = 4a0 Transition thickness
u, v, w Components of the displacement along the directions x, y, z
u0, v0, w0 Components of the displacement at the midsurface of a shell
w Fabric weight per unit area
z Standard variable
[A] Membrane stiffness of a laminate with components Aij ; i, j = 1, 2, 6
[B] Bending-extension coupling stiffness of a laminate BijList of Symbols xxxv
Cij 3D stiffness matrix
CDF Cumulative distribution function (CDF)
CF , Cσ Strength and load coefficient of variance (COV)
CTE Coefficient of thermal expansion
D Damage tensor
[D] Bending stiffness of a laminate with components Dij ; i, j = 1, 2, 6
Ex, Ey, Gxy Laminate moduli
E Young’s modulus
E1, E2 Longitudinal and transverse moduli
EA, ET Axial and transverse moduli of fibers
F Resistance (material strength)
F1t Longitudinal tensile strength
F2t Transverse tensile strength
F6 In-plane shear strength
F1c Longitudinal compressive strength
F2c Transverse compressive strength
F4 Intralaminar shear strength
Fft Apparent fiber tensile strength
Fmt Apparent matrix tensile strength
Fcsm−t Tensile strength of a random-reinforced lamina
Fb
x, Fyb, Fxy b Flexural strength
G Shear modulus
G12 In-plane shear modulus
G23 Transverse shear modulus
GA, GT Axial and transverse shear modulus of fibers
GIc, GIIc Fracture toughness mode I and II
[H] Transverse shear stiffness of a laminate Hij ; i, j = 4…5
HDT Heat distortion temperature
I Moment of inertia of the cross-section of a beam
IF Failure Index
M Bending moment applied to a beam
Mx, My, Mxy Bending moments per unit length at the midsurface of a shell
MT
x , MyT , Mxy T Thermal moments per unit length
Nx, Ny, Nxy Membrane forces per unit length at the midsurface of a shell
NT , ϵT Membrane thermal force and strain caused by thermal expansion
NT
x , NyT , Nxy T Thermal forces per unit length
Q Reduced stiffness matrix in lamina coordinates x1, x2, x3
Q∗ Intralaminar reduced stiffness matrix
Q Reduced stiffness matrix in laminate coordinates X, Y, Z
Qe Undamaged reduced stiffness matrix in lamina coordinates
Qe Undamaged reduced stiffness matrix in laminate coordinates
QCSM Reduced stiffness of a random-reinforced lamina
R Strength ratio, safety factor
[R] Reuter matrixxxxvi Introduction to Composite Materials Design
Re = 1 − CDF Reliability
Sij 3D compliance matrix
S∗ Intralaminar components of the compliance matrix S
SF T Stress-free temperature [◦C]
Tg
Glass transition temperature
T = T(θ) Stress transformation matrix from laminate to lamina coordinates
T−1 = T(−θ) Stress transformation matrix from lamina to laminate coordinates
Vx, Vy Transverse shear forces per unit length at the midsurface of a shell
Vf, Vm Fiber and matrix volume fraction
Vv Void content (volume fraction)
Wf, Wm Fiber and matrix weight fraction
Symbols Related to Fabric-reinforced Composites
θf, θw Undulation angle of the fill and gap tows, respectively
Θf, Θw Coordinate transformation matrix for fill and gap
af, aw Width of the fill and gap tows, respectively
gf, gw Width of the gap along the fill and gap directions, respectively
hf, hw, hm Thickness of the fill and gap tows, and matrix region, respectively
n
g Harness
ns Number of subcells between consecutive interlacings
ni Number of subcells in the interlacing region
zf(x), zw(y) Undulation of the fill and gap tows, respectively
Afill, Awarp Cross-section area of fill and warp tows
Ffa Apparent tensile strength of the fiber
Fmta, Fmsa Apparent tensile and shear strength of the matrix
Lfill, Lwarp Developed length of fill and warp tows
Tf, Tw Stress transformation matrix for fill and gap
V o
f Overall fiber volume fraction in a fabric-reinforced composite
Vmeso Volume fraction of composite tow in a fabric-reinforced composite
V f
f , Vfw Volume fraction of fiber in the fill and warp tows
w Weight per unit area of fabric
Symbols Related to Beams
β Rate of twist
ϕ Angle of twist
λ2 Dimensionless buckling load
ωs Sectorial area
ω Principal sectorial area
ηc, ζc Mechanical shear center
Γs′′ Area enclosed by the contour
eb, eq Position of the neutral surface of bending and torsion
q Shear flowList of Symbols xxxvii
s, r Coordinates along the contour and normal to it
yc, zc Mechanical shear center
zG, zρ, zM Geometric, mass, and mechanical center of gravity
(EA) Axial stiffness
(EIyG), (EIzG) Mechanical moment of inertia
(EIyGzG) Mechanical product of inertia
(EIη), (EIζ) Bending stiffness with respect to principal axis of bending
(Eω) Mechanical sectorial static moment
(EI!) Mechanical sectorial moment of inertia
(GA) Shear stiffness
(GJR) Torsional stiffness
(EQ!ζ) Mechanical sectorial linear moment
(EQζ(s)) Mechanical static moment
K Coefficient of restraint
Ni
xs Shear flow in segment i
Le Effective length of a column
T Torque
Z = I/c Section modulus
Symbols Related to Strengthening of Reinforced Concrete
α, αi Load factor, partial load factors
αc, βc Stress-block parameters for confined section
β1 Stress-block parameter for unconfined section
εbi Initial strain at the soffit
εc Strain level in the concrete
εcu Ultimate axial strain of unconfined concrete
εccu Ultimate axial compressive strain of confined concrete
εf Strain level in FRP
εfd FRP debonding strain
εfu, εfe FRP allowable and effective tensile strain
ε∗
fu FRP ultimate strain
εs Strain level in the steel reinforcement
ε
y Steel yield strain
κa FRP efficiency factor in determination of fcc ′
κb FRP efficiency factor in determination of εccu
κv Bond reduction factor
κ” FRP efficiency factor
ϕF Factored capacity
ϕ Strength reduction factor (resistance factor)
ϕ(Pn, Mn) Factored (load, moment) capacity
ϕecc Eccentricity factor
ρf FRP reinforcement ratio
ρg Longitudinal steel reinforcement ratioxxxviii Introduction to Composite Materials Design
ψ Load combination factor
ψf FRP strength reduction factor
c Position of the neutral axis
cb Position of the neutral axis, BSC
bf Width of FRP laminae
b, h Width and height of the beam
d Depth of tensile steel
dfv Depth of FRP shear strengthening
fc;s Compressive stress in concrete at service condition
fc′ Concrete compressive strength, unconfined
fcc ′ Concrete compressive strength, confined
ff;s Stress in the FRP at service condition
fl Confining pressure
ffu ∗ FRP tensile strength
ffu, ffe FRP allowable and effective tensile strength
fs;s Stress in the steel reinforcement at service condition
fy
Steel yield strength
h, b Height and width of the cross-section
n Number of plies of FRP
nf, ns Number of FRP strips and steel bars in shear
rc Radius of edges of a prismatic cross-section confined with FRP
sf, ss FRP and steel spacing in shear
tf Ply thickness of FRP
wf Width of discontinuous shear FRP
Ac Area of concrete in compression
Ae Area of effectively confined concrete
Af Area of FRP
A
g Gross area of the concrete section
Afb Area of FRP, BSC
Asi Area of i-th rebar
As(A′ s) Tensile (compressive) steel reinforcement area
Ast Sum of compressive and tensile steel reinforcement areas
Asv FRP and steel shear area
C Axial compressive force in the concrete
CE Environmental exposure coefficient
D Column diameter
Ec Modulus of concrete
Ef Modulus of FRP
Es Modulus of steel
(EI)cr Bending stiffness of the cracked section
F Nominal capacity (strength)
L Load (applied load, moment, or stress)
Le Active bond length of FRP
Mn Nominal moment capacityList of Symbols xxxix
Mu Required moment capacity
Pn Nominal axial compressive capacity (strength)
Pu Required axial strength
SDL Stress resultant of the dead load
SLL Stress resultant of the live load
Ts, Tf Tensile force in steel and FRP
U Required capacity
Vn Nominal shear capacity
Vu Required shear capacity
Vc, Vf, Vs Nominal shear strength of concrete (C), FRP (F), and steel stirrups (S)
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