Forces that cause a member to twist about a longitudinal axis are called torsional loads. Simple torsion is produced only by a couple, or moment, in a plane perpendicular to the axis.

If a couple lies in a nonperpendicular plane, it can be resolved into a torsional moment, in a plane perpendicular to the axis, and bending moments, in planes through the axis.

## Shear Center

The point in each normal section of a member through which the axis passes and about which the section twists is called the share center. The location of the shear center depends on the shape and dimensions of the cross section. If the loads on a beam do not pass through the shear center, they cause the beam to twist. See also Art. 5.5.19.

If a beam has an axis of symmetry, the shear center lies on it. In doubly symmetrical beams, the share center lies at the intersection of the two axes of symmetry and hence coincides with the centroid.

For any section composed of two narrow rectangles, such as a T beam or an angle, the shear center may be taken as the intersection of the longitudinal center lines of the rectangles.

For a channel section with one axis of symmetry, the shear center is outside the section at a distance from the centroid equal to e(1 h2A/4I ), where e is the distance from the centroid to the center of the web, h is the depth of the channel, A the cross-sectional area, and I the moment of inertia about the axis of symmetry.

(The web lies between the shear center and the centroid.)

Locations of shear centers for several other sections are given in Friedrich Bleich, Buckling Strength of Metal Structures, Chap. III, McGraw-Hill Publishing Company, New York.

## Stresses Due to Torsion

Simple torsion is resisted by internal shearing stresses. These can be resolved into radial and tangential shearing stresses, which being normal to each other also are equal (see Art. 5.3.2). Furthermore, on planes that bisect the angles between the planes on which the shearing stresses act, there also occur compressive and tensile stresses. The magnitude of these normal stresses is equal to that of the shear. Therefore, when torsional loading is combined with other types of loading, the maximum stresses occur on inclined planes and can be computed by the methods of Arts.

5.3.3 and 5.3.6.

Circular Sections. If a circular shaft (hollow or solid) is twisted, a section that is plane before twisting remains plane after twisting. Within the proportional limit, the shearing unit stress at any point in a transverse section varies with the distance from the center of the section. The maximum shear, psi, occurs at the circumference and is given by

where G is the shearing modulus of elasticity (see Art. 5.2.4).

Noncircular Sections. If a shaft is not circular, a plane transverse section before twisting does not remain plane after twisting. The resulting warping increases the shearing stresses in some parts of the section and decreases them in others, compared wit the sharing stresses that would occur if the section remained plane. Consequently, shearing stresses in a noncircular section are not proportional to distances from the share center. In elliptical and rectangular sections, for example, maximum shear occurs on the circumference at a point nearest the shear center.

For a solid rectangular section, this maximum may be expressed in the following form:

Hollow Tubes. If a thin-shell hollow tube is twisted, the shearing force per unit of length on a cross section (shear flow) is given approximately by

where b is the thickness of the web or the flange of the member. Maximum shear will occur at the center of one of the long sides of the rectangular part that has the greatest thickness. (A. P. Boresi, O. Sidebottom, F. B. Seely, and J. O. Smith, Advanced Mechanics of Materials, 3d ed., John Wiley & Sons, Inc., New York.)

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