Structural Analysis II Section A6.4

### Section I.2 Torsion of Non-Circular Bars

The observations made for torsion of members with circular cross sections do not hold for those with non-circular cross sections. Consider the following facts for members with non-circular cross sections:
1. The shear stress is not constant at a given distance from the axis of rotation. As a result sections perpendicular to the axis of member warp, indicating out-of-plane displacement.
2. The theory of elasticity shows that the shear stress at the corners is zero.
3. Maximum shear strain and stress are not at the farthest distance from the rotational axis of a homogeneous non-circular member.

Members with rectangular cross sections:

For a rectangular member under torsion the corners do not distort; the corner square angles remain square after torque is applied. This indicates that shear strain is zero at the corners since there is no distorsion. This fact is illustrated in this figure.

St. Venant was the first to accurately describe the shear stress distribution on the cross section of a non-circular member using the Theory of Elasticity. For more information click here.

Theory of Elasticity shows that:

• the maximum shear strain and stress occur at the centerline of the long sides of the rectangular cross section
• the shear strain and stress at the corners and center of the rectangular cross section are zero.
• the strain and stress variations on the cross section are primarily nonlinear.
• the preceding statements are demonstrated in the following figure.

The theory of Elasticity has been applied to find analytical solutions for the torsion of rectangular elastic members. The resulting equations for shear stress and angle of twist are as follows:

Here is Table A6.1

NOTE: The equations described in this section can only be used for members having OPEN cross-sectional shapes. For open cross sections composed of multiple thin plates refer to Section I.5.

To Index Page of Pure Torsion To Section I.1 To Section I.3