lecture 6 planar problems - west virginia...
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MAE 456 Finite Element Analysis
Planar Problems
Structural problems that can be described in 2 dimensions.
MAE 456 Finite Element Analysis
Stresses on an Arbitrary Plane
• Example:
• Using Mohr’s Circle
2
MAE 456 Finite Element Analysis
Stresses on an Arbitrary Plane
ji
jnin yx
ˆ sinˆ cos
ˆˆ
φφ +=
+=n
( ) yxyxyyxyxxn
yyyxxyxxn
nnnnnn
nnnn
στστ
στσσ
+−+−=
++=
22
22 2
( ) ( )
( ) )2cos()2sin(2
1
)2sin()2cos(2
1
2
1
φτφσστ
φτφσσσσσ
xyyxn
xyyxyxn
+−−=
+−++=
or
σy
σx
τxy
τyx x
y
σn
τn
n
φ
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MAE 456 Finite Element Analysis
Principal Stresses
• Principal stresses occur on planes that have zero shear stress.
• For planar problems, this condition is:
( )
yx
xy
yxyxyyxyxx nnnnnn
σσ
τφ
στσ
−=
=+−+−
2)2tan(
0 22
)sin(
)cos(
φ
φ
=
=
y
x
n
n
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MAE 456 Finite Element Analysis
Principal Stresses
• Maximum normal stress at a point
• Minimum normal stress at a point
• Maximum shear stress at a point
=1σ
=2σ
=maxτ5
MAE 456 Finite Element Analysis
Theories of Failure
• Ductile Material
– Maximum Shear Stress Theory
– Maximum Distortion Energy Theory
• von Mises Stress
• Brittle Material
– Maximum Normal Stress Theory
– Coulomb-Mohr Theory
( ) ( ) ( )2
32
2
31
2
212
1σσσσσσσ −+−+−
=e
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MAE 456 Finite Element Analysis
Theories of Failure
Image from J.A. Collins, “Failure of Materials in Mechanical Design,” Wiley
fs
2σ
fs
1σ
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MAE 456 Finite Element Analysis
Theories of Failure
Image from J.A. Collins, “Failure of Materials in Mechanical Design,” Wiley
ys
1σ
ys
2σ
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MAE 456 Finite Element Analysis
Theories of Failure
Image from J.A. Collins, “Failure of Materials in Mechanical Design,” Wiley
us
1σ
us
2σ
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MAE 456 Finite Element Analysis
Stress-Strain Relations
• The relationship between stress and strain is given by:
where:
σσσσ0 is a pre-stress, whereand εεεε0 is a pre-strain (usually zero; but would be
non-zero for thermal strain, strain hardening, etc.).
00Eεσ −=
=
xy
y
x
τ
σ
σ
σ
=
xy
y
x
γ
ε
ε
ε
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MAE 456 Finite Element Analysis
Stress-Strain Relations
• The matrix E depends on what you assume
is happening in the z direction.
• Two situations can be assumed:
– plane stress (σz= 0), or
– plane strain (εz= 0).
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MAE 456 Finite Element Analysis
Plane Stress Problem
• A plane stress problem is one in which the cross-section is free to move in the z direction.
• The thickness in the z direction is normally small compared to the profile.
• σz= τyz = τzx = 0
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MAE 456 Finite Element Analysis
Plane Stress Problem
• The inverse relationship (between strain and stress for
plane stress) is given by:
( )υ+=
12
EG
0
1 εσε += −E
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MAE 456 Finite Element Analysis
Plane Strain Problem
• A plane strain problem is one in which the cross-
section is restrained from moving in the z direction.
• The thickness in the z direction is normally very large
compared to the cross section.
• εz= γyz = γzx = 0
• but ( )yxz σσνσ +=0== zxyz γγ
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MAE 456 Finite Element Analysis
Plane Strain Problem
• The inverse relationship (between strain and stress for
plane strain) is given by:
( )υ+=
12
EG
0
1 εσε += −E
−−−
−−−
=−
G
EE
EE
100
01
01
22
22
1 ννν
ννν
E
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MAE 456 Finite Element Analysis
Strain-Displacement Relations
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