Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using...
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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
L
2
1
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 θLL
2
1
3
Saint-Venant Torsion Problem Boundary Conditions
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 L
2
1
3
Saint-Venant Torsion Problem Stress Assumptions
σ11= σ22= σ33= σ12= 0
→ τ1 and τ2 are the only non-zero stresses
Equilibrium Equations 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
2
1
3
Saint-Venant Torsion Problem
L
2
1
3
Satisfy Boundary Conditions ταnα= φ,α dxα/ds|C= dφ/ds|C= 0
→ φ is Constant on C Torque, T
T= -∫A xαφ,α dA= ∫A φ dA
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 Nearest
Point on CPlastic Strain Rates Vanish
Ridge LinesLine Consisting of Ridge Points
Fully Plastic Torsion
Regular Polygons
Irregular Polygons
ABAQUS Model3D Analytical Rigid
3D Deformable
ABAQUS Model
Torsion: Imposed Boundary ConditionsFixed at OriginImpose Rotation about 3-axis
Fixed Plate
Rotated Plate
ABAQUS Model
Bar Cross SectionsTriangle
Square
Circle
Rectangle
L
Square Tube
ABAQUS Model
Material PropertiesSteel
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 IssuesTime/Processing PowerBar Mesh Size
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.