Lecture 4 - Mechanical and Aerospace Engineering

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Mechanics of Materials


MAE 243 (Section 002)

Spring 2008

Dr. Konstantinos A. Sierros

Problem 1.2
-
11

A reinforced concrete slab 8.0 ft square and 9.0 in. thick is lifted by four cables
attached to the corners, as shown in the figure. The cables are attached to a
hook at a point 5.0 ft above the top of the slab. Each cable has an effective
cross
-
sectional area
A
=
0.12 in
2

.

Determine the tensile stress σ
t

in the cables due to the weight of the concrete
slab. (See Table H
-
1, Appendix H, for the weight density of reinforced concrete.)

Problem 1.3
-
3

Three different materials, designated
A, B,
and
C
, are tested in tension using
test specimens having diameters of 0.505 in. and gage lengths of 2.0 in. (see
figure).

At failure, the distances between the gage marks are found to be 2.13, 2.48,
and 2.78 in., respectively. Also, at the failure cross sections the diameters are
found to be0.484, 0.398, and 0.253 in., respectively. Determine the percent
elongation and percent reduction in area of each specimen, and then, using
your own judgment, classify each material as brittle or ductile.

Problem 1.3
-
6

A specimen of a methacrylate plastic is tested in tension at room temperature
(see figure), producing the stress
-
strain data listed in the accompanying table.
Plot the stress
-
strain curve and determine the proportional limit, modulus of
elasticity (i.e., the slope of the initial part of the stress
-
strain curve), and yield
stress at 0.2% offset. Is the material ductile or brittle?


Solution to Problem 1.3
-
6

0.00
0.01
0.02
0.03
0.04
0.05
0
10
20
30
40
50
60
70
Yield stress = 52.5


Proportional limit = 47 MPa
Stress, MPa
Strain
proportional limit
Modulus of elasticity = 2.35 GPa
yield stress
Material is
brittle
, because the strain after the proportional limit is
exceeded is relatively small.

1.4: Elasticity



What happens when the load is
removed (i.e. the material is unloaded)?



Tensile load is applied from O to A (fig
1.18a) and when load is removed the
material follows the same curve back.
This property is called
elasticity



If we load the same material from O to
B (fig 1.18b) and then unloading occurs,
the material follows the line BC. Line
OC represents the
residual

or
permanent

strain. Line CD represents the elastic
recovery of the material. During
unloading the material is partially elastic

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
18
Stress
-
strain diagrams illustrating
(a) elastic behavior, and (b) partially elastic
behavior
1.4: Plasticity


Plasticity

is the characteristic of a material which undergoes inelastic
strains beyond the strain at the elastic limit ”

When large deformations occur in a ductile material loaded in the
plastic region
, the material is undergoing
plastic flow

1.4: Reloading of a material

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
19
Reloading of a material and raising
of the elastic and proportional limits


If the material is in the elastic range, it
can be loaded, unloaded and loaded again
without significantly changing the
behaviour



When loaded in the plastic range, the
internal structure of the material is altered
and the properties change



If the material is reloaded (fig 1
-
19), CB
is a linearly elastic region with the same
slope as the slope of the tangent to the
original loading curve at origin O



By stretching steel or aluminium into
the plastic range, the properties of the
material are changed

1.4: Creep

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
20
Creep in a bar under constant load
Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
21
Relaxation of
stress in a wire
under constant
strain


When loaded for periods of time, some
materials develop additional strains and
are said to
creep



Even though the load P remains constant
after

time t
0
, the bar gradually lengthens



Relaxation is a process at which, after
time t
0
, the stress in the wire gradually
diminishes and eventually is reaching a
constant value



Creep is more important at high
temperatures and has to be considered in
the design of engines and furnaces

1.5: Hooke’s law

Many structural materials such as metals, wood, plastics and ceramics
behave both elastically and linearly when first loaded and their stress
-
strain curve begin with a straight line passing through origin (line OA)

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
10
Stress
-
strain diagram for a typical
structural steel in tension (not to scale)
Linear elastic materials are useful for designing structures and machines
when permanent deformations, due to yielding, must be avoided

1.5: Hooke’s law

Robert Hooke

(1635
-
1703)

The linear relationship between stress and strain for a bar
in simple tension or compression is expressed by:

σ = E ε


σ
is axial stress


ε
is axial strain


E
is modulus of elasticity

Hooke’s law

The above equation is a limited version of Hooke’s Law relating only the
longitudinal stresses and strains that are developed during the uniaxial
loading of a prismatic bar

Robert Hooke

was an English inventor, microscopist, physicist,
surveyor, astronomer, biologist and artist, who played an important role
in the scientific revolution, through both theoretical and experimental
work.


1.5: Modulus of elasticity



E is called modulus of elasticity or Young’s modulus and


is a constant



It is the slope of the stress


strain curve in the linearly
elastic region



Units of E are the same as the units of stress (i.e. psi for
USCS and Pa for SI units)



For stiff materials E is large (i.e. structural metals).

E
steel
= 190
-

210 GPa



Plastics have lower E values than metals.

E
polyethylene

= 0.7


1.4 GPa



Appendix H, Table H
-
2 contains values of E for materials



Thomas Young

was an English polymath, contributing to the scientific
understanding of vision, light, solid mechanics, energy, physiology, and
Egyptology.


1.5: Poisson’s ratio

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
22
Axial elongation and lateral
contraction of a prismatic bar in tension:
(a) bar before loading, and (b) bar after
loading. (The deformations of the bar are
highly exaggerated.)


When a prismatic bar is loaded in tension the
axial elongation is accompanied by lateral
contraction






The lateral strain ε


at any point in a bar is
proportional to the axial strain ε at the same
point if the material is linearly elastic



The ratio of the above two strains is known

as
Poisson’s ratio

(ν)

longitudinal
extension

lateral
contraction

ν =
-

(lateral strain / axial strain =
-




/ ε )

1.5: Poisson’s ratio

Simeon Denis Poisson

(1781
-
1840)

Siméon
-
Denis Poisson

was a French mathematician, geometer, and
physicist.




The minus sign in the equation is because the lateral
strain is negative (width of the bar decreases) and the
axial tensile strain is positive. Therefore, the Poisson’s
ratio will have a positive value.



When using the Poisson’s ratio equation we need to
know that it applies only to a prismatic bar in uniaxial
stress



Poisson’s value of concrete = 0.1


0.2



Poisson’s value of rubber = 0.5



Appendix H, Table H
-
2 contains values of ν for
various materials

1.5: Limitations

Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 1
-
22
Axial elongation and lateral
contraction of a prismatic bar in tension:
(a) bar before loading, and (b) bar after
loading. (The deformations of the bar are
highly exaggerated.)


Poisson’s ratio is constant in the
linearly elastic range



Material must be homogeneous (same
composition at every point)



Materials having the same properties in
all directions are called isotropic



If the properties differ in various
directions the materials called anisotropic

Good luck with your homework

Deadline: 28 January 2008