Solutions Manual for ANALYSIS AND PERFORMANCE OF FIBER COMPOSITES

147 Pages·2017·2.836 MB·English

by  Bhagwan D. Agarwal, Lawrence J. Broutman, K. Chandrashekhara

Solutions Manual for
ANALYSIS AND
PERFORMANCE OF FIBER
COMPOSITES

FOURTH EDITION

Bhagwan D. Agarwal
Consultant
Lombard, Illinois, USA
Lawrence J. Broutman
Consultant
Chicago, Illinois, USA
K. Chandrashekhara
Missouri University of Science and Technology
Rolla, Missouri, USA
1
C
ONTENTS

Chapter 1. Introduction 3
Chapter 3. Micromechanics of Unidirectional Composites 6
Chapter 4. Short-Fiber Composites 26
Chapter 5. Macromechanics Analysis of an Orthotropic Lamina 33
Chapter 6. Analysis of Laminated Composites 46
Chapter 7. Analysis of Laminated Plates and Beams 109
Chapter 8. Advanced Topics in Fiber Composites 124
Chapter 9. Performance of Fiber Composites: Fatigue,
Impact, and Environmental Effects 135
Chapter 10. Experimental Characterization of Composites 137
2
Chapter 1
Introduction
1.3

3
1.4

My 1 M h 6M
, I bh3,
x I 12 max I 2 bh2
1
(a-1) “EI” of steel beam 210 GPa 0.1 0.0063m4 378 N m2
12
1
“EI” of 2024-T4 aluminum beam 73 GPa 0.1 h3m4 378 N m2
12
h = beam thickness for equivalent “EI” = 8.5 mm

6M
(a-2) Bending moment on steel beam:
ULT steel bh2
1
M 0.83 GPa 0.1 0.0062m3 498 N m
6
6M 6 498 N m
Aluminum beam 0.41 GPa
ULT Al 0.1h2 0.1h2m
Beam thickness for equivalent strength: h 8.54 10 3m 8.54 mm

(b-1) Weight of steel beam per unit length:
bh 10 cm 0.6 cm 7.8 gm/cm3 46.8 gm/cm3
Weight of aluminum beam for equivalent “EI”:
bh from part(a-1) 10 0.85 2.7 23.0 gm/cm
Al

4
(b-2) Weight of steel beam = 46.8 gm/cm
Weight of aluminum beam with equivalent strength:
bh from part(a-2) 10 0.854 2.7 23.1 gm/cm
Al
(c-1) Weight of steel beam = 46.8 gm/cm
Weight of aluminum beam = 46.8gm/cm bh
Al
46.8
h 1.73 cm
10 2.7
“EI” of aluminum beam with same weight as steel beam:
1
EI 73 GPa 0.1 0.01733m4 299 N m2
12
(c-2) Steel beam:
1 1
Bending moment = M bh2 0.83 GPa 0.1 0.0062m3 498 N m
6 ULT 6
Bending moment for aluminum beam with same weight as steel beam
1
M b h from part(c-1)
6 ULT Al
1
0.41 0.1 0.0172 GN m
6
1,970 N m
(b-1) ( b -2 )
wt/unit wt/unit
(a-1) (a-2) length le n g t h (c - 1 ) ( c-2)
h(mm) h(mm) (Normalized) (Normalized) EI Moment
Material Stiffness Strength Stiffness S trength Normalized Normalized
Steel 6 6 1 1 1.0 1.0
Al-2024-T4 8.5 8.5 0.49 0.49 7.9 4.0
Al-6061-T6 8.7 10.7 0.50 0.62 7.5 2.5
E-glass epoxy
12.8 7.2 0.54 0.30 6.6 11.0
(V = 0.57)
f
Kevlar-49/epoxy
10.4 6.8 0.31 0.20 31.7 23.7
(V = 0.6)
f
Carbon
fiber/epoxy 8.2 8.9 0.27 0.29 49.5 11.4
(V = 0.58)
f
Boron fiber/epoxy
7.5 8.9 0.32 0.38 28.3 6.73
(V = 0.60)
f

5
Chapter 3
Micromechanics of
Unidirectional Composites
3.1
A
V f
f A
Square array:
1 d2 d2 d2
A 4 ,A s2, V
f 4 4 4 f 4s2

Hexagonal array:
d2 1 d2 d2 d2 3 d2
A 6
f 4 3 4 4 2 4
1 3s 3s2 3 3
A 6 s 6 s2
2 2 4 2
d2
V
f 2 3s2
6
Alternatively, consider triangular array:
1 d2 d2
A 3
f 6 4 8
1 3s 3 d2
A s s2 V
2 2 4 f 2 3s2

When adjacent fibers touch each other, s = d
V for square array=78.5%
max 4

= for hexagonal array=90.7%
2 3
3.2
49.4476 47.6504
W 0.71
f 50.1817 47.6504
W 1 0.71 0.29
m
1 1
1.902 g/cm3
c W W 0.71 0.29
f m
2.5 1.2
f m
7
ρ 1.902
V = c W = ×0.71=0.54
f ρ f 2.5
f
1.902
V = ×0.29=0.46
m 1.2
3.3
w
v =v +v +v = c (1)
c f m v ρ
ce
w
v +v = c (2)
f m ρ
ct
(1)-(2) gives:
w w w  ρ 
v = c − c = c 1− ce 
v ρ ρ ρ ρ
 
ce ct ce ct
ρ −ρ 
=v  ct ce 
c ρ
 
ct
v ρ −ρ
V = v = ct ce
v v ρ
c ct
3.4
From Problem 3.3, ρ =1.902
ct
1.902−1.86
From Problem 3.3, V = =0.0221 or 2.21%
v 1.902
3.5
(a) Weight of composite: w =9.301 g
c
Weight of carbon fibers: w =6.428 g
f
Weight of epoxy: w =w −w =9.301−6.428=2.873 g
m c f
w 6.428
Weight fraction of carbon fibers: W = f = =69.11%
f w 9.301
c
8
w 2.873
Weight fraction of epoxy: W m 30.89%
m w 9.301
c
1 1
Density of the composite: 1.559 g/cm3
c W W 0.6911 0.3089
f m
1.8 1.2
f m
W 0.6911 1.559
Volume fraction of carbon fibers: V f c 59.87%
f 1.8
f
W 0.3089 1.559
Volume fraction of epoxy: V m c 40.13%
m 1.2
m
(b) Theoretical density of the composite: 1.559 g/cm3
ct c
Experimental density of the composite:
9.301
1.546 1 1.546 g/cm3
ce 9.301 3.285 water
The volume fraction of voids in the composite:
1.559 1.546
V ct ce 0.834%
v 1.559
ct
3.6
E 400
f f 125 for all V.
f
E 3.2
m m
E
V E (GPa) f f
f c E
c c
10 42.88 9.33
25 102.40 3.91
50 201.60 1.98
75 300.80 1.33
3.7
The following values are obtained by using rule of mixtures for E and and
L νLT
the Halpin-Tsai equations for E and G :
T LT
9
V Property
f
System (%) E (GPa) E (GPa) G (GPa)
L T LT LT
Glass/Epoxy 25 20.13 6.39 2.07 0.313
50 36.75 11.46 3.48 0.275
75 53.38 22.81 6.96 0.238
Graphite/Epoxy 25 65.13 6.81 2.13 0.313
50 126.75 13.18 3.76 0.275
75 188.38 30.41 8.36 0.238
Kevlar/Epoxy 25 37.63 6.67 2.11 0.313
50 71.75 12.60 3.67 0.275
75 105.88 27.59 7.88 0.238
Boron/Aluminum 25 140.00 105.00 37.37 0.298
50 210.00 154.00 54.3 0.265
75 280.00 275.50 83.49 0.233

3.8
1
1.74
c 0.35 0.45 0.20
1.3 2.5 1.6
Volume fractions:
1.74
V 0.35 0.47
m 1.3
1.74
V 0.45 0.31
fA 2.5
1.74
V 0.20 0.22
fB 1.6
Fracture stains:
0.06
100 1.71%
m 3.5
1.4
100 2.0%
fA 70
0.45
100 7.5%
fB 6
Failure sequence will be binder, fiber A, fiber B.
(a) Composite stress at fracture strain of binder:
0.0171(3.5 0.47 70 0.31 6 0.22) 0.422 GPa
c
10