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M6104 Mechanics of Materials
Selected Problems and Answers
Lecturer Dr. David Shu Dongwei/MPE/NTU
Disclaimer
1.
The questions are slelected randomly, whenever such problems are
available.
2.
The limited answers are casually worked out, not verified. If you have
worked through it. Do share your work with others and me the lecturer
too.
3.
I have no Further answers and solutions to these.
1 Lamina
P1.1 What is composite material ? Is fibre reinforced plastics a macrocomposite
or a microcomposite ?
P1.2 Why FRP have been used most widely in aerospace industry ?
P1.3 List five major open mould composites manufacturing processes, and six closed
mould manufacturing routes for composites.
P1.4 Outline the manufacturing process by filament winding, and the typical products
coming out from such manufacturing route.
P1.5 Replace filament winding by other manufacturing routes and answer the same
questions.
Fibres
P2.1 For brittle material in tension, explain that fibres have higher strength than bulk
material, and that short fibres have higher strength than long fibres.
P2.2 Explain the function of plastics matrix material in a fibre reinforced plastics
material.
P2.3 Derive the flexibility of fibres in terms of its physical and geometrical attributes.
Explain why thinner fibre is more flexible.
P2.4 Describe the main features of the carbon fibre. Explain how these features influence
the
properties of the carbon fibres. Which two important properties of carbon fibres
are anisotropic ?
P2.5 List the main characteristics of Carbon, explain briefly the production process.
P2.6 Describe the microstructures of Glass fibres, and the resultant properties of glass
fibres.
P2.7 Compare the physical as well as mechanical properties of metals and ceramics.
Matrix
P3.1 List the main types of polymers and their structures.
P3.2 What are thermosets and thermoplastics ? What are their properties and their
influences on the productions of composites materials.
Geometry
P5.1 A graphite/epoxy composite specimen has dimensions of 2.54cm, 2.54cm, and
0.3cm and a weight of 2.98g. After "resin digestion" in acid solution the remaining
graphite fibres weigh 1.863g. From independent tests, the densities of the graphite fibres
and epoxy matrix materials are found to be 1.9g/cm
3
and 1.2 g/cm
3
, respectively.
Determine the volume fractions of fibres, epoxy matrix, and voids in the specimen.
2
P5.2 If the volume fraction is 50% and the fibres radius is 7
µ
m, determine the average
spacing (distance between neighbor fibre centers) for assumed
1) square packing, or
2) hexagonal packing.
Then determine what is the theoretical maximum volume fraction for the two packing
order.
2 Laminate
P6.1 Derive the expressions of E
1
, E
2
,
ν
12
, and G
12
for a composite lamina with 2 fibres
and one matrix materials, in terms of the Young's modulus and Poisson's ratios of the
fibres and matrix. Use f1 and f2 as indices for the two fibres and m for the matrix.
Follow the same assumptions in the lecture notes 6.1-4.
Unidirectional Laminate Stress-Strain Relation
P9.1* Prove relation
ν
12
/E
1
=
ν
21
/E
2
, using elastic energy principle.
P9.2* Express stiffness components Q
ij
in terms of engineering constants E
1
, E
2
,
ν
12
, and
G
12
.
P9.3* Derive the stress and strain transformation between the principal materials
directions 1, 2, 3 and a random one x, y, z.
P9.4 Plot E
x
/E
2
, Gxy/G
12
,
ν
xy
/
ν
12
, G
xy
/G
12
, against
θ
from 0 to 90
o
, for the following
two composite materials:
a) glass/epoxy, E
1
/E
2
=3, G
12
/E
2
=0.5,
ν
12
=0.25;
b) boron/epoxy, E
1
/E
2
=10, G
12
/E
2
=1/3,
ν
12
=0.3;
c) graphite/epoxy, E
1
/E
2
=40, G
12
/E
2
=0.5,
ν
12
=0.25.
Discuss the results obtained.
Laminate under pure bending
P11.1
Determine the unit width flexural rigidity EI of E-glass/epoxy laminated beams
having stacking sequences of [0/90/0]s and [90/0/90]s. The ply modules are E
1
=34.48
GPa, and E
2
=10.34 GPa, and the plies all have the same thickness. Sketch the distribution
of normal and shear stresses through the thickness of the beam. Assume a ply thickness
of 0.254mm.
Classic Laminate Bending Theory
P12.1*
Derive equation (12.1).
P12.2*
From equation (resultant-moment), derive the extension stiffness, bending
stiffness, and
coupling stiffness for isotropic plate. Discuss the results.
P12.3* Determine the extension, coupling, and bending stiffness of an equal thickness
bimetallic strip with properties (E
1
and
ν
1
, E
2
and
ν
2
).
P12.4* A composite plate is made by four plies [0,90,90,0] with equal thickness t=2mm,
the 90
o
ply is glass/epoxy and the 0 plies are graphite/epoxy. direction is designated
along the 0 plies, while direction 2 is along 90 plies. The properties of the two
composites are given as
a) glass/epoxy, E
1
=38GPa, E
2
=10GPa, G
12
=5GPa,
ν
12
=0.25;
b) graphite/epoxy, E
1
=220GPa, E
2
=40GPa, G
12
=20GPa,
ν
12
=0.25;
1) derive the stress and strain relationship in coordinate 1-2 for material a);
2) derive the stress and strain relationship in coordinate 1-2 for material b);
3) derive the resultant in-plane forces and moments in terms of the mid plane stretching
and the mid plane curvatures, using the classic beam theory;
3
4) what loading are necessary to produce a mid plane stretch
ε
1
=0.01 for a small
rectangular elements of the laminate?
5) what loading are necessary to produce a mid plane bending of
κ
2
=0.5 for a small
rectangular elements of the laminate?
6) what mid plane deformation will be produced with in-plane tension
σ
1
=1.9GPa for a
small rectangular elements of the laminate?
7) what mid plane deformation will be produced with unit width bending moment
Μ
2
=190MN for a small rectangular elements of the laminate?
P12.5* Show that equations (12.1) and (12.2) clasps into the classic beam theory for
isotropic beams.
3 Strength and Failure
Unidirectional Laminate Strength
P7.1 In 7.1 for the case that matrix has lower ductility than fibres,
1) discuss the sequence of events as the stress increases leading to failure;
2) calculate the critical volume fraction which divides the two different sequences of
failures of the composites;
3) derive the longitudinal tensile strength of the composite.
P7.2 In 10.1 for the case that matrix has higher ductility than fibres, and the lamina is
subjected to longitudinal stress along the fibres direction.
1) discuss the sequence of events as the stress increases leading to failure;
2) calculate the critical volume fraction which divides the two different sequences of
failures of the composites;
3) derive the longitudinal tensile strength of the composite.
P7.3 Glass fibres and polyester resins have the following properties,
Glass: Young's modulus 86GN/m
2
, Tensile strength 2100MPa, elongation to fracture
2.5%; Polyester: Young's modulus 9GN/m
2
, Tensile strength 72MPa, elongation to
fracture 2%; There are two glass/polyester laminae with volume fraction of (a) V
f
=0.50
and (b) V
f
=0.05 respectively, both are subjected to longitudinal stress along the fibres
direction,
1) discuss the sequence of failures as the stress increases leading to final failure of the
lamina;
2) calculate the critical volume fraction which divides the two different sequences of
failures of the lamina for this combination of fibre/resin;
3) derive the longitudinal tensile strength of the composite.
P7.4* Type I carbon and epoxy resin
have the following properties,
Carbon I: Young's modulus 220MN/m
2
, Tensile strength 2.0GN/m
2
, elongation to
fracture 0.5%; Epoxy: Young's modules 5.3MN/m
2
, Tensile strength 80MN/m
2
,
elongation to fracture 2%; For each of the two carbon I/epoxy (V
f
=0.60, 0.015) lamina
subjected to longitudinal stress along the fibres direction.
1) discuss the sequence of failures as the stress increases leading to final failure;
2) calculate the critical volume fraction which divides the two different sequences of
failures of the lamina for this combination of fibre/resin;
3) derive the longitudinal tensile strength of the composite.
P7.5* Some glass fibres and a flexible polyester resin have the following properties,
Glass: Young's modules 76MN/m
2
, Tensile strength 2100MN/m
2
, elongation to fracture
2.5%; Polyester: Young's modules 9MN/m
2
, Tensile strength 65MN/m
2
, stress at strain
0.025 is 52MN/m
2
, elongation to fracture 3.5%; For each of the two glass/polyester
laminae with volume fraction V
f
=0.50 and 0.005 respectively, and are subjected to
longitudinal stress along the fibres direction,
1) discuss the sequence of failures as the stress increases leading to final failure of the
lamina;
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2) calculate the critical volume fraction which divides the two different sequences of
failures of the lamina for this combination of fibre/resin;
3) derive the longitudinal tensile strength of the composite.
Multi-axial Failure
8.1 What are the major criteria used in predicting laminate failure ?
8.2 Which criteria are easy to use ? Which has the most accuracy ?
8.3 Problem 4.1 on Gibson p126
8.4 Problem 4.2 on Gibson p126-127
8.5 Problem 4.11 on Gibon p128
ANSWERS AND SOLUTIONS
S2.4
1) carbon hexagonal 2-D;
2) folded sheets of 2-D (turbostratic);
3) aligned in axial direction;
4) some irregularity.
S5.1
ρ
c
=1540Kg/m
3
, V
v
=1.22%, V
f
=50.6%, and V
m
=48.2%.
S7.1
See the notes 10.1 or Hull p128.
S7.2
See the notes 10.1 or Hull p130.
S7.3
σ
'
f
=1520MN/m2, and V'
f
=0.11;
σ
0.5
=1050MPa,
σ
0.05
= 144 MPa.
S7.4
σ
'
m
=26.5MN/m2, and V'
f
=0.026.
S7.5
V'
f
=0.006.
S10.4
a) b) in Jones p55-56.
S11.1
Gibson p199.
S12.2
Jones p156
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SOLUTION
Conceptualize Look back at the Example "A Grand Entrance," where we analyzed a cable used to support an actor as he swung
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If it is INCOMPLETE SOLUTION and there are SHORTCUTS OF SOLUTION, I will rate you with “DISLIKE/DOWNVOTE.”
THANK YOU FOR YOUR HELP.
PS: If you have answered this already, don’t answer it again; give chance to other experts to answer it. I want to verify if all of you will arrive in the same final answer; thats why I ask it multiple times. If you answer it again, i'll dislike all your entries/answers.
arrow_forward
Topics:
Statics of Rigid Bodies, Equilibrium of Rigid Bodies, Equilibrium in Two Dimensions, etc.
I will rate you with “LIKE/UPVOTE," if it is COMPLETE STEP-BY-STEP SOLUTION.
If it is INCOMPLETE SOLUTION and there are SHORTCUTS OF SOLUTION, I will rate you with “DISLIKE/DOWNVOTE.”
THANK YOU FOR YOUR HELP.
PS: If you have answered this already, don’t answer it again; give chance to other experts to answer it. I want to verify if all of you will arrive in the same final answer; thats why I ask it multiple times. If you answer it again, i'll dislike all your entries/answers.
arrow_forward
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