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Saint-Venant Torsion Problem Finite Element Analysis of the Saint-Venant Torsion Problem Using ABAQUS
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Overview Saint-Venant Torsion Problem Fully Plastic Torsion ABAQUS Model Results
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Saint-Venant Torsion Problem Prismatic Bar Longitudinal Axis: 3-axis Cross Section: Closed Curve C in the 1-2-plane L 2 1 3
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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
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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
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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
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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
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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
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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
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Fully Plastic Torsion Regular Polygons Irregular Polygons
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ABAQUS Model 3D Analytical Rigid 3D Deformable
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ABAQUS Model Torsion: Imposed Boundary Conditions Fixed at Origin Impose Rotation about 3-axis Fixed Plate Rotated Plate
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ABAQUS Model Bar Cross Sections Triangle Square Circle Rectangle L Square Tube
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ABAQUS Model Material Properties Steel Elastic-Isotropic Young’s Modulus: 210 GPa Poisson’s Ratio: 0.3 Plastic-Isotropic Yield Stress: 250 MPa
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Results: Triangle
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Results: Square
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Results: Circle
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Results: Rectangle
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Results: L
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Results: Square Tube
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Results ABAQUS Issues Time/Processing Power Bar Mesh Size
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A More Complicated Problem
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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.
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