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.