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Finite Element Method
Chapter 6Part I
Review of the Basic Elasticity Theory
Review of the Basic Elasticity Theory
Stresses & Strains
Review of the Basic Elasticity Theory
Two-dimensional State of Stress
y
xy
xx
y
xy
Equations of Equilibrium:
0
bxyx X
yx
0
byxy Y
xy
022
)1(22
)1(
0
dydy
ydydxdxdx
xdxdy
M
xyxyxy
yxyxyx
C
xyyx
Review of the Basic Elasticity Theory
Differential Equations of Equilibrium for 2D Problems:
xyyx
byyx
byxx
Yyx
Xyx
0
0
Review of the Basic Elasticity Theory
Similarly Equilibrium for 3D Problems:
yzzy
xzzx
xyyx
bzzyzx
bzyyyx
bzxyxx
Zzyx
Yzyx
Xzyx
0
0
0
Review of the Basic Elasticity Theory
Strains for 2D Problem
Review of the Basic Elasticity Theory
Differential element before and after deformatio
1 2
1 2tan tan( / ) ( / )
(1 / ) (1 / )
x
y
x y
x y
uxvy
v x dx u y dydx u x dy v y
u vy x
2 222 2
2 2
222
2 2
22 2
2 2
22 2
2 2
0
Similarly
0
0
x y yx
x yyx
z xz x
y zy z
u vx y x y y x y x y x
or
y x x y
x z z x
z y y z
Review of the Basic Elasticity Theory
Obtain the compatibility equation by differentiating xy with respect to both x and y
Three-dimensional state of stress and strain
yw
zv
xw
zu
xv
yu
zwyvxu
zy
zx
yx
z
y
x
Strains for 3D Problem
Poisson’s Ratio• For a slender bar subjected to axial loading:
0 zyx
x E
• The elongation in the x-direction is accompanied by a contraction in the other directions. Assuming that the material is isotropic (no directional dependence),
0 zy
• Poisson’s ratio is defined as
x
z
x
y
strain axialstrain lateral
Three-dimensional state of stress and strain
Stress/Strain Relationships
yx zx
yx zy
yx zz
E E E
E E E
E E E
, ,
2 (1 )
x y y z z xx y y z z xG G G
EG
Three-dimensional state of stress and strain
xz
zy
yx
z
y
x
xz
zy
yx
z
y
x
E
)1(2000000)1(2000000)1(2000000100010001
1
Stress/Strain Relationships
Three-dimensional state of stress and strain
xz
zy
yx
z
y
x
xz
zy
yx
z
y
x
E
221
0221
00221
000100010001
)21()1(
Stress/Strain Relationships
Two-dimensional state of stress and strainStresses
yx
y
x
equilibrium equations
xyyx
byyx
byxx
Yyx
Xyx
0
0
Two-dimensional state of stress and strain
Principal Stresses
Principal Angle p.
min2
2
2
max2
2
1
22
22
yxyxyx
yxyxyx
yx
yxp
22tan
2D Strains Problem
xv
yu
yv
xu
yx
yx
,
yx
y
x
Plane Stress and Plane Strain
Plane stress is defined to be a state of stress in which the normal stress and shear stresses directed perpendicular to the plane are assumed to be zero.
Generally, members that are thin (z dimension is smallcompared to the in-plane x and y dimensions) and whose loads act only in the x-y plane can be considered to be under plane stress.
Plane Stress and Plane StrainPlane stress
Recall Stress Strain Relationship
xz
zy
yx
z
y
x
xz
zy
yx
z
y
x
E
)1(2000000)1(2000000)1(2000000100010001
1
0 zyzxz
yx
y
x
yx
y
x
E
)1(2000101
1
0,0 zzyzx but
State of Plane stress
yx
y
x
yx
y
x E
2100
01
01
1 2
}{][}{ D
2100
01
01
1][ 2
ED
State of Plane stress
D : is called stress/strain matrix(or constitutive matrix)
yxu
xv
yv
xv
yxv
yu
yu
xu
2
2
2
2
2
2
2
2
2
2
2
2
2
2
21
21
State of Plane stress
Partial differential equations governing the plane stress
Plane Stress and Plane Strain
Plane strain is defined to be a state of strain in which the strain normal to the x-y plane z and the shear strains xz and yz are assumed to be zero.
Constant cross-sectional area subjected to loads that act only in the x and/or y directions and do not vary in the z direction. Dams. retaining walls, and culvert boxes are good examples
Plane Stress and Plane StrainPlane strain
0 zyzxz
xz
zy
yx
z
y
x
xz
zy
yx
z
y
x
E
22100000
02210000
00221000
000100010001
)21()1(
State of Plane Strain
0,0 zzyzx but
{ } [ ] { }
1 0
[ ] 1 0(1 ) (1 2 )
1 20 02
D
ED
yx
y
x
yx
y
x E
22100
01
01
)21()1(
State of Plane Strain
The von Mises Stress
The von Mises stress is the effective or equivalent stress for 2-D and 3-D stress analysis. For a ductile material, the stress level is considered to
be safe, if
Where: σe is the von Mises stress and σY the yield stress of the material.
This is a generalization of the 1-D (axial) result to 2-D and 3-D situations.
e Ys s£
The von Mises Stress
The von Mises stress for 3D problems is defined by
For 2-D problems,
Averaged Stresses:
Stresses are usually averaged at nodes in FEAsoftware packages to provide more accurate stressvalues. This option should be turned off at nodesbetween two materials or other geometrydiscontinuity locations where stress discontinuitydoes exist.
