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Prof. Ramesh Singh
Outline
• Principal stresses• Mohr’s circle in 3D• Strain tensor• Principal strains
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Prof. Ramesh Singh
Principal Stresses in 3D• 3-D Stresses can be represented by in usual
notation
We will use a concept from continuum mechanics
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é
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stttsttts
Tn!" =×s
Stress TensorUnit normal vector
Traction vectorForce/area
![Page 3: Outline - Welcome to IIT Mechanical | IIT Mechanicalramesh/courses/ME423/Mechanics2.pdf · • Mohr’s circle in 3D • Strain tensor • Principal strains . Prof. Ramesh Singh Principal](https://reader033.vdocument.in/reader033/viewer/2022050401/5f7fa8064fc43a707d2990b3/html5/thumbnails/3.jpg)
Prof. Ramesh Singh
Principal Stresses
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Prof. Ramesh Singh
Principal Stresses in 3D
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Prof. Ramesh Singh
Principal Stresses in 3D
0
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=-
--
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Prof. Ramesh Singh
3D Stress – Principal Stresses
3 21 2 3 0I I Is s s- + - =
The three principal stresses are obtained as the three real roots of the following equation:
where
1
2 2 22
2 2 23 2
x y z
x y x z y z xy xz yz
x y z xy xz yz x yz y xz z xy
I
I
I
s s s
s s s s s s t t t
s s s t t t s t s t s t
= + +
= + + - - -
= + - - -
I1, I2, and I3 are known as stress invariants as they do not change in value when the axes are rotated to new positions.
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Prof. Ramesh Singh
Principal Stress
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In[3]:= Eigensystem[{{0, -240, 0}, {-240, 200, 0}, {0, 0, -280}}]Out[3]= {{360, -280, -160}, {{-2, 3, 0}, {0, 0, 1}, {3, 2, 0}}}
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Prof. Ramesh Singh
Principal Stresses in 3-D
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Prof. Ramesh Singh
Linear Strains
Dx
Du
Linear strain formulation:
zw
yv
xu
xu
zyx
x
¶¶
=¶¶
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=
DD
=
eee
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as, drepresente becan it limits Taking
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Prof. Ramesh Singh
Shear Strain
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21
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Dx
Dy
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Prof. Ramesh Singh
Strain Tensor
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Prof. Ramesh Singh
Strain Transformation
www.efunda.com
xyxy ge21
=
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Prof. Ramesh Singh
Mohr’s Circle for Strain
xyxy ge21
=
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Prof. Ramesh Singh
Principal Strains
xyxy
where
ge21,
=