Transcript Torsion - iMechanica

Saint-Venant Torsion Problem
Finite Element Analysis of the
Saint-Venant Torsion Problem
Using ABAQUS
Overview
Saint-Venant Torsion Problem
 Fully Plastic Torsion
 ABAQUS Model
 Results

Saint-Venant Torsion Problem
Prismatic Bar
 Longitudinal Axis: 3-axis
 Cross Section: Closed Curve C
in the 1-2-plane

2
1
L
3
Saint-Venant Torsion Problem
Bar is in a State of Torsion
 No Tractions on the
Lateral Surface
 Rotation at x3=0 is 0
 Relative Rotation
at x3=L is θL

3
2
1
L
Saint-Venant Torsion Problem

Boundary Conditions
2
 u1= u2= 0, σ33= 0 @ x3= 0
 u1= -θLx2, u2= θLx1, σ33= 0
@ x3= L
 Ti= σijnj= σiαnα=
0,
where n1= dx2/ds, n2= -dx1/ds
on C, 0<x3<L
3
1
L
Saint-Venant Torsion Problem

Stress Assumptions
 σ11= σ22= σ33= σ12=
2
0
→ τ1 and τ2 are the only non-zero stresses
 Equilibrium



Equations
1
For α= 1,2
τα,3= 0 → τ1, τ2 ≠ f(x3)
τα,α= 0 → φ(x1, x2)

τ1= φ,2 and τ2= φ,1
L
3
Saint-Venant Torsion Problem

Satisfy Boundary Conditions
 ταnα= φ,α
2
dxα/ds|C= dφ/ds|C= 0
→ φ is Constant on C

1
Torque, T
 T=
-∫A xαφ,α dA= ∫A φ dA
L
3
Fully Plastic Torsion

Equivalent to the Mathematical Problem
 |φ|=
k in A
 φ = 0 on C

This Problem has a Unique Solution
Denoted φp
 φp(x1, x2)=k ∙ distance from (x1, x2) to C

Fully Plastic Torsion

Ridge Point
 (x1, x2) has More than One
 Plastic Strain Rates Vanish

Nearest Point on C
Ridge Lines
 Line
Consisting of Ridge Points
Fully Plastic Torsion

Regular Polygons

Irregular Polygons
ABAQUS Model
3D Analytical Rigid
3D Deformable
ABAQUS Model

Torsion: Imposed Boundary Conditions
 Fixed
at Origin
 Impose Rotation about 3-axis
Fixed Plate
Rotated Plate
ABAQUS Model

Bar Cross Sections
 Triangle
 Rectangle
 Square
L
 Circle
 Square
Tube
ABAQUS Model

Material Properties
 Steel

Elastic-Isotropic
Young’s Modulus: 210 GPa
 Poisson’s Ratio: 0.3


Plastic-Isotropic

Yield Stress: 250 MPa
Results: Triangle
Results: Triangle
Results: Square
Results: Circle
Results: Circle
Results: Rectangle
Results: L
Results: Square Tube
Results

ABAQUS Issues
 Time/Processing
 Bar
Mesh Size
Power
A More Complicated Problem
References



[1] W. Wagner, F. Gruttmann, “Finite Element Analysis of Saint-Venant
Torsion Problem with Exact Integration of the Elastic-Plastic Constitutive
Equations,” Baustatik, Mitteilung 3, 1999.
[2] J. Lubliner, Plasticity Theory, New York: Macmillan Publishing Company,
1990.
[3] F. Alouges, A. Desimone, “Plastic Torsion and Related Problems,” Journal
of Elasticity 55: 231–237, 1999.