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.