tutorial presentation zhao
TRANSCRIPT
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Definition of normal stress(axial stress)
A
F
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Definition of normal strain
0L
L
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Poissons ratio
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Definition of shear stress
0A
F
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Definition of shear strain
l
x tan
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Tensile Testing
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Stress-Strain Curves
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Stress-Strain Curves
http://www.uoregon.edu/~struct/courseware/461/461_lectures/4
61_lecture24/461_lecture24.html
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Stress-Strain Curve
(ductile material)
http://www.shodor.org/~jingersoll/weave/tutorial/node4.html
http://www.shodor.org/~jingersoll/weave/tutorial/img21.png -
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Stress-Strain Curve
(brittle material)
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Example: stress-strain curve for low-carbon steel
1 - Ultimate Strength
2 - Yield Strength
3 - Rupture
4 - Strain hardening region
5 - Necking region
Hooke's law is only valid for the
portion of the curve between the
origin and the yield point.
http://en.wikipedia.org/wiki/Hooke's_law
http://upload.wikimedia.org/wikipedia/commons/0/00/Stress_v_strain_A36_2.png -
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PLProportional Limit - Stress above which stress is not longer proportional to strain.
ELElastic Limit - The maximum stress that can be applied without resulting in permanent
deformation when unloaded.
YPYield Point - Stress at which there are large increases in strain with little or no increase in
stress. Among common structural materials, only steel exhibits this type of response.
YSYield Strength - The maximum stress that can be applied without exceeding a specified
value of permanent strain (typically .2% = .002 in/in).
OPTI 222 Mechanical Design in Optical Engineering 21
UUltimate Strength - The maximum stress the material can withstand (based on the original
area)
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True stressand true strain
True stressand true strain are based uponinstantaneous values of cross sectional
area and gage length
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The Region of Stress-Strain Curve
Stress Strain Curve
Similar to Pressure-Volume Curve
Area = Work
Volume
Pressure
Volume
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Uni-axial Stress State
Elastic analysis
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Stress-Strain Relationship
EE -- Youngs modulus
GG -- shear modulus
Hookes Law:
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Stresses on Inclined Planes
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Thermal Strain
Straincaused by temperature changes. is amaterial characteristic called the coefficient ofthermal expansion.
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Strains caused by temperature changes and strainscaused by applied loads are essentially independent.
Therefore, the total amount of strain may be expressed as
follows:
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Bi-axial stateelastic analysis
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(1) Plane stress
State of plane stress occurs in a thin plate subjected to forces acting in the mid-plane of the
plate
State of plane stress also occurs on the free surface of a structural element or machine
component, i.e., at any point of the surface not subjected to an external force.
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Transformation of Plane Stress
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Mohrs Circle (Plane Stress)
http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html
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Mohrs Circle (Plane Stress)
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Instruction to drawMohrs Circle1. Determine the point on the body in which the principal stresses are to be
determined.
2. Treating the load cases independently and calculated the stresses for the point
chosen.
3. Choose a set of x-y reference axes and draw a square element centered on the
axes.
4. Identify the stresses x, y, and xy = yx and list them with the proper sign.
5. Draw a set of - coordinate axes with being positive to the right and being
positive in the
upward direction. Choose an appropriate scale for the each axis.
6. Using the rules on the previous page, plot the stresses on the x face of the element
in this coordinate system (point V). Repeat the process for the y face (point H).
7. Draw a line between the two point V and H. The point where this line crosses the axis establishes the center of the circle.
8. Draw the complete circle.
9. The line from the center of the circle to point V identifies the x axis or reference
axis for angle measurements (i.e. = 0).
Note: The angle between the reference axis and the axis is equal to 2p.
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Mohrs Circle (Plane Stress)
http://www.egr.msu.edu/classes/me423/aloos/lecture_notes/lecture_4.pdf
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Principal Stresses
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Maximum shear stress
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http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm
Stress-Strain Relationship
(Plane stress)
))((1
yxzE
xy
y
x
xy
y
xE
2100
01
01
12
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(2) Plane strain
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Coordinate Transformation
The transformation of strains with respect to the {x,y,z} coordinates to
the strains with respect to {x',y',z'} is performed via the equations
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Mohr's Circle (Plane Strain)
(xx' - avg)2
+ ( x'y' / 2 )2
= R2
avg =xx+ yy
2
http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
http://www.shodor.org/~jingersoll/weave4/tutorial/Figures/mc_strain.jpg -
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http://www.efunda.com/formulae/solid_mechani
cs/mat_mechanics/calc_principal_strain.cfm
Principal Strain
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Maximum shear strain
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Stress-Strain Relationship(Plane strain)
)(
211
yxz
E
z
y
x
z
y
xE
)1(2
2100
011
01
1
)21)(1(
)1(
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Tri-axial stress state
elastic analysis
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3D stress at a point
three (3) normal stresses may act on faces of the cube, as well
as, six (6) components of shear stress
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Stress and strain components
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The stress on a inclined plane
))(()2()2(3121
22322232
lnn
))(()2
()2
( 123222132213
mnn
))(()
2
()
2
( 231322212221
nnn
y
x
z
(l, m, n)
p2
3
1
n
n
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3-D Mohrs Circle
* The 3 circles expressed by the 3 equations intersect in point D,
and the value of coordinates of D is the stresses of the inclined
plane
D
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Stress-Strain Relationship
For isotropic materials
Generalized Hookes Law:
0
0
0
1
11
21
2
2100000
0
2
210000
002
21000
0001
0001
0001
)21)(1(
TEE
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x