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Stress-Strain Relationship
(Strength of Materials)
Dave Morgan
Stress-Strain Relationship p. 1/21
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Stress-Strain Relationship
The tension test:
Stress-Strain Relationship p. 2/21
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Stress-Strain Relationship
The tension test:
Is a common standardised test that can be performed on
a material
Stress-Strain Relationship p. 2/21
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Stress-Strain Relationship
The tension test:
Is a common standardised test that can be performed on
a materialProvides information about how the material responds toan applied tensile force
Stress-Strain Relationship p. 2/21
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Stress-Strain Relationship
The tension test:
Is a common standardised test that can be performed on
a materialProvides information about how the material responds toan applied tensile force
We can find:
The proportional limit
E
, the modulus of elasticity (Youngs Modulus)The yield strength
The ultimate strength
Stress-Strain Relationship p. 2/21
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Stress-Strain Relationship
In the tension test:
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recorded
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recordedOne end of the sample is fixed
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recordedOne end of the sample is fixed
A tensile load is applied to the other end
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recordedOne end of the sample is fixed
A tensile load is applied to the other end
The tensile stress caused by the load induces a strain(deformation) in the sample
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recordedOne end of the sample is fixed
A tensile load is applied to the other end
The tensile stress caused by the load induces a strain(deformation) in the sample
For a range of loads, load and the correspondingdeformation are recorded
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
In the tension test:
The length, L, and cross-sectional area, A, of the sample
are measured and recordedOne end of the sample is fixed
A tensile load is applied to the other end
The tensile stress caused by the load induces a strain(deformation) in the sample
For a range of loads, load and the correspondingdeformation are recorded
Load, P, and deformation, , are plotted on a graph
Stress-Strain Relationship p. 3/21
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Stress-Strain Relationship
Deformation,
Load, P
Stress-Strain Relationship p. 4/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
=
P/A/L
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
=
P/A/L
=LA
P
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
=
P/A/L
=LA
P
L and A are recorded at the beginning of the test and the
ratio LA remains constant throughout the test
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
=
P/A/L
=LA
P
L and A are recorded at the beginning of the test and the
ratio LA remains constant throughout the test
E is proportional to P
Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material:
Stress is proportional to strain and Hookes Law applies:
E =
=
P/A/L
=LA
P
L and A are recorded at the beginning of the test and the
ratio LA remains constant throughout the test
E is proportional to P
We can find E from L, A and the slope of the plotted
graph (which is P
)Stress-Strain Relationship p. 5/21
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Stress-Strain Relationship
Up to the proportional limit for the material, the graph is astraight line
Deformation,
Load,P
Stress-Strain Relationship p. 6/21
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Stress-Strain Relationship
Deformation,
Load, P
Load is proportional todeformation
Stress-Strain Relationship p. 7/21
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Stress-Strain Relationship
Deformation,
Load, P
Load is proportional todeformation
Stress is proportional to strain
Stress-Strain Relationship p. 7/21
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Stress-Strain Relationship
Deformation,
Load, P
Load is proportional todeformation
Stress is proportional to strainMaterial behaves elastically
Stress-Strain Relationship p. 7/21
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Stress-Strain Relationship
Deformation,
Load, P
Load is proportional todeformation
Stress is proportional to strainMaterial behaves elastically
There is no permanent
change to the material;when the load is removed,the material resumes its
original shape
Stress-Strain Relationship p. 7/21
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Stress-Strain Relationship
At the proportional limit, the graph changes from a straightline
Deformation,
Load, P
Proportional Limit
Stress-Strain Relationship p. 8/21
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Stress-Strain Relationship
The elastic limit is the point after which the sample will notreturn to its original shape when the load is released.
Deformation,
Load, P
Proportional Limit
Elastic Limit
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Stress-Strain Relationship
The elastic limit is the point after which the sample will notreturn to its original shape when the load is released.
Deformation,
Load, P
Proportional Limit
Elastic Limit
The proportional limit andthe elastic limit are veryclose. For most purposes,
we may consider them tobe the same point.
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Stress-Strain Relationship
The elastic limit is the point after which the sample will notreturn to its original shape when the load is released.
Deformation,
Load, P
Proportional Limit
Elastic Limit
The proportional limit andthe elastic limit are veryclose. For most purposes,
we may consider them tobe the same point.
There is permanent
change to the structure ofthe material.
Stress-Strain Relationship p. 9/21
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Stress-Strain Relationship
There may be a region of increased deformation withoutincreased load
Deformation,
Load, P
Proportional Limite
lastic
Stress-Strain Relationship p. 10/21
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Stress-Strain Relationship
There may be a region of increased deformation withoutincreased load
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
This point is known as the
yield point
Stress-Strain Relationship p. 10/21
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Stress-Strain Relationship
There may be a region of increased deformation withoutincreased load
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
This point is known as the
yield pointThe stress at this point iscalled the yield strength.
Stress-Strain Relationship p. 10/21
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Stress-Strain Relationship
There may be a region of increased deformation withoutincreased load
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic
This point is known as the
yield pointThe stress at this point iscalled the yield strength.
The material behavesplastically (when the loadis removed, the sample
does not return to itsoriginal shape)
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Stress-Strain Relationship
After the yield point, there may be a region of whereincreased load is necessary for increased deformation
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic
Stress-Strain Relationship p. 11/21
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Stress-Strain Relationship
After the yield point, there may be a region of whereincreased load is necessary for increased deformation
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic harde
ning
This is the strain-hardening
region
Stress-Strain Relationship p. 11/21
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Stress-Strain Relationship
After the yield point, there may be a region of whereincreased load is necessary for increased deformation
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic
Ultimate Strength
harden
ing
This is the strain-hardening
region
Load (stress) rises to a
maximum; this is the ultimate
strength of the material
Stress-Strain Relationship p. 11/21
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Stress-Strain Relationship
After the yield point, there may be a region of whereincreased load is necessary for increased deformation
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic
Failure Point
Ultimate Strength
harden
ing
This is the strain-hardening
region
Load (stress) rises to a
maximum; this is the ultimate
strength of the material
Load required for further
deformation is reduced as thefailure or breaking point is
approached.
Stress-Strain Relationship p. 11/21
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Stress-Strain Relationship
Deformation,
Load, P
Proportional Limit
Yield Point
ela
stic
plastic
Failure Point
Ultimate Strength
harden
ing
Stress-Strain Relationship p. 12/21
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Stress-Strain Relationship
Each material has its own stress-strain curve, with differentcharacteristics:
Low Carbon Steel Aluminum, Brass (Ductile) Cast Iron (Brittle)
Stress-Strain Relationship p. 13/21
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Stress-Strain Relationship
Ductile Materials:
Can sustain large deformation before failure
Deformation is an indication of impending failure
Commonly used in construction
Some steels, wood
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Stress-Strain Relationship
Brittle Materials:
Small deformation before failure
Little indication of impending failure
Cast iron, non-reinforced concrete
Stress-Strain Relationship p. 15/21
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Stress-Strain Relationship
It is not always easy to identify the yield point from thestress-strain (load-deformation) curve. In these cases theoffset method is used.
Deformation,
Load, P
Stress-Strain Relationship p. 16/21
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Stress-Strain Relationship
It is not always easy to identify the yield point from thestress-strain (load-deformation) curve. In these cases theoffset method is used.
Deformation,
Load, P
Offset
An offset for the material isgiven. It is marked on the
deformation (strain) axis
Stress-Strain Relationship p. 16/21
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Stress-Strain Relationship
It is not always easy to identify the yield point from thestress-strain (load-deformation) curve. In these cases theoffset method is used.
Deformation,
Load, P
Offset
An offset for the material isgiven. It is marked on the
deformation (strain) axis
A line through the offset point,parallel to the straight
(proportional) part of the curve,
is drawn
Stress-Strain Relationship p. 16/21
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Stress-Strain Relationship
It is not always easy to identify the yield point from thestress-strain (load-deformation) curve. In these cases theoffset method is used.
Deformation,
Load, P
Yield Point
Offset
An offset for the material isgiven. It is marked on the
deformation (strain) axis
A line through the offset point,parallel to the straight
(proportional) part of the curve,
is drawnThe intersection of the line with
the stress-strain curve is taken
to be the yield point
Stress-Strain Relationship p. 16/21
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Stress-Strain Relationship
For the tension and compression labs, we use the followingoffsets:
Deformation,
Load, P
For steel, use 0.2% strain
For brass, use 0.35% strain
For cast iron, use 0.05% strain
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Tension and Compression Labs
Tension and Compression Labs #1You shall be provided with graphs showing the tension andcompression curves from the lab
1. Draw horizontal and vertical axes
2. Mark each 10mm division
3. Each division along the horizontal axis represents a deformation of0.04mm. Mark the divisions 0, 0.04, 0.08, etc.
Note that the scale is 250 : 1 (that is, 1 0 : 0.04)
4. Load is on the vertical axis, with 10 in = 120 kN
(120/10) (1/25.4) = 0.472 kN/mm
Each division along the vertical axis represents a change in load of4.72 kN. Mark the divisions 0, 4.72, 9.44, etc.
Stress-Strain Relationship p. 18/21
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Tension and Compression Labs
Tension and Compression Labs #2Compute the location of the proportional limit:
1. Draw a vertical line through = 0.08 (20mm division). The proportional
limit is where this line crosses the curve. (This does not follow ourtheoretical description but is a common practice to find PL)
2. Calculate the strain at the PL
3. Calculate PL in MPa
4. Calculate E, the modulus of elasticity, in MPa
Stress-Strain Relationship p. 19/21
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Tension and Compression Labs
Tension and Compression Labs #3
1. Determine the yield point using the appropriate offset
2. Determine the deformation at the yield point
3. Calculate the strain at the yield point ( = /Loriginal)
4. Calculate the ultimate strength
5. Calculate the stress at failure
6. Calculate the % elongation at failure
7. Calculate the % reduction in area at failure
Stress-Strain Relationship p. 20/21
S S i R l i hi
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Stress-Strain Relationship
Created by Dave Morgan using LATEX 2and Prosper on February 24, 2005