Torsion - iMechanica
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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.