3.11 Recitation #11
November 25, 2003
If there’s anything you’d like covered, please let me know.
Also—please let me know good times to hold a review for the final. I was
thinking perhaps Thursday or Friday after classes end. Let me know if this
would be appropriate. By email is fine.
Today:
Review vocabulary for mechanical properties of Materials.
Go over stress-strain relationships, plasticity
Example Problems, if time.
Stress-strain diagrams
The relationship between loads and deflection/stress-strain in a
structure of a member can be obtained from experimental loaddeflection/stress-strain curves
1
P
dL/2
P/2
P/2
P
P
δ
L
Bending test
Shear Test
P
dL/2
T
T
P
Tension test
Torsion test
P
Compression test
The most common tests are tension test for ductile materials (steel)
& compression test for brittle materials (concrete)
2
Tension Test
σ
Yield Stress σ y :
Stress at which a
slight increase in
stress will result in
appreciably increas
in strain without
increase in stress
σ ′f
C'
True Stress-strain using
actual area to calculate
B'
Ultimate stress σ u
Failure stress σ f
Elastic Limit:
The upper
stress level
at which the
material
behaves
elastically
C
Stress-strain using original
area to calculate
A
Proportional Limitσ PL
: The upper stress
limit that strain
varies linearly with
stress. Material
follows Hooke's Law
Elastic
Necking
Yield strain ε y :
10 - 40 times
elastic strain
Yielding
Strain Hardening:
Material can
resist more load
increase
ε
Necking
Plastic Behaviour: Material will deform
permanently and will NOT return to its
orginal shape upon unloading. The
Elastic Behaviour: Material
deformation that occurs is called
will return to its orginal
plastic deformation
shape if material is loaded
and unloaded within this
range
Stress-strain diagram for ductile materials
Hooke’s Law:
σ = Eε
E is the modulus of elasticity
Esteel = 200 GPa
Econcrete = 29 GPa (21 – 29 GPa)
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Ductile Materials
Materials that can be subjected to large strains before rupture
Have high percent elongation
 L − Lo 
Percent elongation =  f
 × 100
 Lo 
Have high percent reduction in area
 A − Af 
Percent reduction in area =  o
 × 100
 Ao 
Have capacity to absorb energy
If structure made of ductile materials is overloaded, it will present
large deformation before failing
Some ductile materials do not exhibit a well-defined yield point, we
will use offset method to define a yield strength
Some ductile materials do not have linear relationship between stress
and strain, we call them nonlinear materials
σ
σ
σ = f (ε )
σy
ε
ε
0.002 or 0.2% offset
Elastic-plastic Materials
Stress-strain for structural steel will consist of elastic and perfectly
plastic region. We call this kind of material elastoplastic material
Analysis of structures on the basis of elastoplastic diagram is called
elastoplastic analysis or plastic analysis
4
Bilinear stress-strain diagram having different slopes is sometimes
used to approximate the general nonlinear diagrams. This will include
the strain hardening.
σ
σ
Perfectly plastic
σy
ning
rd e
a
h
ain
Str
Nonlinear
Linearly elastic
Linearly elastic
ε
εy
ε
Brittle Materials
Materials that do not exhibit yielding before failure
Some materials will show both ductile and brittle behaviours, e.g. steel
with high carbon content will demonstrate brittle behaviours while
steel with low carbon content will be ductile or steel subjects to low
temperature will be brittle while those in the high temperature
environment will be ductile
Creep
Deformation which increases with time under constant load (examples:
rubber band; concrete bridge deck: sagging between supports due to
self weight therefore the deck is constructed with an upward camber)
δ
δo
δ
to
t
P
In several situations, creep will associate with high temperature
If creep becomes important, creep strength will be used in design
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Relaxation
Loss of stress with time under constant strain
Another manifestation of creep
σ
σo
Prestressed
wire
Creep
strength
t
Cyclic loading and fatigue
Fracture after many cycles of loading
If material is loaded into the plastic region, upon unloading elastic
strain will be recovered but plastic strain remains
σ
Elastic
region
Loading
Unloading
ε
Permanent
set
Elastic
recovery
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Strain energy
Energy stored internally throughout the volume of a material which is
deformed by an external load
k
x
F
F
Fo
Work
x
xo
Consider a linear spring having stiffness k
If we apply a force F , the spring will stretch x . The relationship
between F and x is
F = kx
If we apply a force from zero to Fo and the spring stretches to the
amount of xo , the work done is the average force magnitude times the
displacement, i.e.
1 
W =  Fo  xo
2 
From the conservation of energy, this work done must be equivalent to
the internal work or strain energy stored within the spring when it is
deformed
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σ
If an infinitesimal element of
elastic material is subjected to a
normal stress σ , then tensile
force on the element will be
dz
dF = σ dxdy
The change in its length is ε dz
dy
dx
σ
The work done, which equals to the strain energy stored in the
element, is
1
1
dU = (σ dxdy )( ε dz ) or dU = σε dV
2
2
The total strain energy stored in a material will be
U = ∫ σε dV
V
The strain energy per unit volume or the strain energy density is
u=
dU 1
= σε
dV 2
If the material is linear elastic ( σ = Eε , Hooke’s Law holds), the strain
energy density will be
1 σ  1 σ 2
u= σ =
2 E 2 E
σ
If the stress σ reaches the
proportional limit, the strain
energy density is called the
modulus of resilience ur
σ PL
1
ur = σ PL ε PL
2
ur
ε PL
ε
8
σ
The total strain energy
density which stored in the
material just before it fails
is called the modulus of
toughness ut
ut
ε
Two Example Problems:
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Answer to 1.
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Answer to 2.
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More on crazing and shear deformations and zones next time. (Phenomena in
amorphous polymers, as discussed yesterday)
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