1. Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS

2. Overview Saint-Venant Torsion Problem Fully Plastic Torsion ABAQUS Model Results

3. Saint-Venant Torsion Problem Prismatic Bar Longitudinal Axis: 3-axis Cross Section: Closed Curve C in the 1-2-plane L 2 1 3

4. Saint-Venant Torsion Problem Bar is in a State of Torsion No Tractions on the Lateral Surface Rotation at x 3 =0 is 0 Relative Rotation at x 3 =L is θL L 2 1 3

5. Saint-Venant Torsion Problem Boundary Conditions  u 1 = u 2 = 0, σ 33 = 0 @ x 3 = 0  u 1 = -θLx 2, u 2 = θLx 1, σ 33 = 0 @ x 3 = L  T i = σ ij n j = σ iα n α = 0, where n 1 = dx 2 /ds, n 2 = -dx 1 /ds on C, 0<x 3 <L L 2 1 3

6. Saint-Venant Torsion Problem Stress Assumptions  σ 11 = σ 22 = σ 33 = σ 12 = 0 → τ 1 and τ 2 are the only non-zero stresses  Equilibrium Equations For α= 1,2 τ α,3 = 0 → τ 1, τ 2 ≠ f(x 3 ) τ α,α = 0 → φ(x 1, x 2 )  τ 1 = φ,2 and τ 2 = φ,1 L 2 1 3

7. Saint-Venant Torsion Problem L 2 1 3 Satisfy Boundary Conditions  τ α n α = φ,α dx α /ds| C = dφ/ds| C = 0 → φ is Constant on C Torque, T  T= -∫ A x α φ,α dA= ∫ A φ dA

8. Fully Plastic Torsion Equivalent to the Mathematical Problem  |φ|= k in A  φ = 0 on C This Problem has a Unique Solution  Denoted φ p  φ p (x 1, x 2 )=k ∙ distance from (x 1, x 2 ) to C

9. Fully Plastic Torsion Ridge Point  (x 1, x 2 ) has More than One Nearest Point on C  Plastic Strain Rates Vanish Ridge Lines  Line Consisting of Ridge Points

10. Fully Plastic Torsion Regular Polygons Irregular Polygons

11. ABAQUS Model 3D Analytical Rigid 3D Deformable

12. ABAQUS Model Torsion: Imposed Boundary Conditions  Fixed at Origin  Impose Rotation about 3-axis Fixed Plate Rotated Plate

13. ABAQUS Model Bar Cross Sections  Triangle  Square  Circle  Rectangle  L  Square Tube

14. ABAQUS Model Material Properties  Steel Elastic-Isotropic  Young’s Modulus: 210 GPa  Poisson’s Ratio: 0.3 Plastic-Isotropic  Yield Stress: 250 MPa

15. Results: Triangle

17. Results: Square

18. Results: Circle

20. Results: Rectangle

21. Results: L

22. Results: Square Tube

23. Results ABAQUS Issues  Time/Processing Power  Bar Mesh Size

24. A More Complicated Problem

25. 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.