SaintVenant Torsion Problem Finite Element Analysis of the
Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS
Overview Saint-Venant Torsion Problem n Fully Plastic Torsion n ABAQUS Model n Results n
Saint-Venant Torsion Problem Prismatic Bar n Longitudinal Axis: 3 -axis n Cross Section: Closed Curve C in the 1 -2 -plane n 2 1 L 3
Saint-Venant Torsion Problem Bar is in a State of Torsion n No Tractions on the Lateral Surface n Rotation at x 3=0 is 0 n Relative Rotation at x 3=L is θL n 3 2 1 L
Saint-Venant Torsion Problem n Boundary Conditions 2 ¨ u 1= u 2= 0, σ33= 0 @ x 3= 0 ¨ u 1= -θLx 2, u 2= θLx 1, σ33= 0 @ x 3= L ¨ Ti= σijnj= σiαnα= 0, where n 1= dx 2/ds, n 2= -dx 1/ds on C, 0<x 3<L 3 1 L
Saint-Venant Torsion Problem n Stress Assumptions ¨ σ11= σ22= σ33= σ12= 2 0 → τ1 and τ2 are the only non-zero stresses ¨ Equilibrium n n n Equations 1 For α= 1, 2 τα, 3= 0 → τ1, τ2 ≠ f(x 3) τα, α= 0 → φ(x 1, x 2) ¨ τ1= φ, 2 and τ2= φ, 1 L 3
Saint-Venant Torsion Problem n Satisfy Boundary Conditions 2 ¨ ταnα= φ, α dxα/ds|C= dφ/ds|C= 0 → φ is Constant on C n 1 Torque, T ¨ T= -∫A xαφ, α d. A= ∫A φ d. A L 3
Fully Plastic Torsion n Equivalent to the Mathematical Problem ¨ |φ|= k in A ¨ φ = 0 on C n This Problem has a Unique Solution Denoted φp ¨ φp(x 1, x 2)=k ∙ distance from (x 1, x 2) to C ¨
Fully Plastic Torsion n Ridge Point ¨ (x 1, x 2) has More than One Nearest ¨ Plastic Strain Rates Vanish n Ridge Lines ¨ Line Consisting of Ridge Points Point on C
Fully Plastic Torsion n Regular Polygons n Irregular Polygons
ABAQUS Model 3 D Analytical Rigid 3 D Deformable
ABAQUS Model n Torsion: Imposed Boundary Conditions ¨ Fixed at Origin ¨ Impose Rotation about 3 -axis Fixed Plate Rotated Plate
ABAQUS Model n Bar Cross Sections ¨ Triangle ¨ Rectangle ¨ Square ¨L ¨ Circle ¨ Square Tube
ABAQUS Model n Material Properties ¨ Steel n Elastic-Isotropic Young’s Modulus: 210 GPa ¨ Poisson’s Ratio: 0. 3 ¨ n Plastic-Isotropic ¨ Yield Stress: 250 MPa
Results: Triangle
Results: Triangle
Results: Square
Results: Circle
Results: Circle
Results: Rectangle
Results: L
Results: Square Tube
Results n ABAQUS Issues ¨ Time/Processing ¨ Bar Mesh Size Power
A More Complicated Problem
References n n n [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.
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