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 in2 .
Determine the tensile stress σt 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
70
60
proportional limit
Stress, MPa
50
yield stress
40
Modulus of elasticity = 2.35 GPa
30
Proportional limit = 47 MPa
20
Yield stress = 52.5
10
0
0.00
0.01
0.02
0.03
0.04
0.05
Strain
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
Stress-strain diagrams illustrating
(a) elastic behavior, and (b) partially elastic
behavior
FIG. 1-18
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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
• 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
Reloading of a material and raising
of the elastic and proportional limits
FIG. 1-19
Copyright 2005 by Nelson, a division of Thomson Canada Limited
1.4: Creep
• 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 t0, the bar gradually lengthens
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
• Relaxation is a process at which, after
time t0 , 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
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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 stressstrain curve begin with a straight line passing through origin (line OA)
Stress-strain diagram for a typical
structural steel in tension (not to scale)
FIG. 1-10
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Linear elastic materials are useful for designing structures and machines
when permanent deformations, due to yielding, must be avoided
1.5: Hooke’s law
The linear relationship between stress and strain for a bar
in simple tension or compression is expressed by:
σ is axial stress
ε is axial strain
E is modulus of elasticity
σ=Eε
Hooke’s law
Robert Hooke
(1635-1703)
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).
Esteel = 190 - 210 GPa
• Plastics have lower E values than metals.
Epolyethylene = 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
• When a prismatic bar is loaded in tension the
axial elongation is accompanied by lateral
contraction
longitudinal
extension
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
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.)
FIG. 1-22
Copyright 2005 by Nelson, a division of Thomson Canada Limited
• The ratio of the above two strains is known
as Poisson’s ratio (ν)
ν = - (lateral strain / axial strain = - (ε’ / ε )
1.5: Poisson’s ratio
• 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
Simeon Denis Poisson
(1781-1840)
• 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
Siméon-Denis Poisson was a French mathematician, geometer, and
physicist.
1.5: Limitations
• 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
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.)
FIG. 1-22
Copyright 2005 by Nelson, a division of Thomson Canada Limited
Good luck with your homework
Deadline: 28 January 2008