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![Page 1: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/1.jpg)
Saint-Venant Torsion Problem
Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS
![Page 2: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/2.jpg)
Overview
Saint-Venant Torsion Problem Fully Plastic Torsion ABAQUS Model Results
![Page 3: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/3.jpg)
Saint-Venant Torsion Problem Prismatic Bar Longitudinal Axis: 3-axis Cross Section: Closed Curve C
in the 1-2-plane
L
2
1
3
![Page 4: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/4.jpg)
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
![Page 5: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/5.jpg)
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
![Page 6: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/6.jpg)
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
![Page 7: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/7.jpg)
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
![Page 8: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/8.jpg)
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
![Page 9: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/9.jpg)
Fully Plastic Torsion
Ridge Point(x1, x2) has More than One Nearest
Point on CPlastic Strain Rates Vanish
Ridge LinesLine Consisting of Ridge Points
![Page 10: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/10.jpg)
Fully Plastic Torsion
Regular Polygons
Irregular Polygons
![Page 11: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/11.jpg)
ABAQUS Model3D Analytical Rigid
3D Deformable
![Page 12: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/12.jpg)
ABAQUS Model
Torsion: Imposed Boundary ConditionsFixed at OriginImpose Rotation about 3-axis
Fixed Plate
Rotated Plate
![Page 13: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/13.jpg)
ABAQUS Model
Bar Cross SectionsTriangle
Square
Circle
Rectangle
L
Square Tube
![Page 14: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/14.jpg)
ABAQUS Model
Material PropertiesSteel
Elastic-Isotropic Young’s Modulus: 210 GPa Poisson’s Ratio: 0.3
Plastic-Isotropic Yield Stress: 250 MPa
![Page 15: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/15.jpg)
Results: Triangle
![Page 16: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/16.jpg)
Results: Triangle
![Page 17: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/17.jpg)
Results: Square
![Page 18: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/18.jpg)
Results: Circle
![Page 19: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/19.jpg)
Results: Circle
![Page 20: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/20.jpg)
Results: Rectangle
![Page 21: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/21.jpg)
Results: L
![Page 22: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/22.jpg)
Results: Square Tube
![Page 23: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/23.jpg)
Results
ABAQUS IssuesTime/Processing PowerBar Mesh Size
![Page 24: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/24.jpg)
A More Complicated Problem
![Page 25: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/25.jpg)
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.
![Page 26: Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS.](https://reader033.fdocuments.in/reader033/viewer/2022061400/56649ebf5503460f94bc936d/html5/thumbnails/26.jpg)