A bar is under torsional stress when it is held fast at one end, and a force acts at the other end to twist the bar. In a round bar (Fig. 4.9) with a constant force acting, the straight-line ab becomes the helix ad, and a radial line in the cross-section, ob, moves to the position ad. The angle bad remains constant while the angle bod increases with the length of the bar. Each cross section of the bar tends to shear off the one adjacent to it, and in any cross section the shearing stress at any point is normal to a radial line drawn through the point. Within the shearing proportional limit, a radial line of the cross section remains straight after the twisting force has been applied, and the unit shearing stress at any point is proportional to its distance from the axis.
The twisting moment, T, is equal to the product of the resultant, P, of the twisting forces, multiplied by its distance from the axis, p. Resisting moment, T,, in torsion, is equal to the sum of the moments of the unit
shearing stresses acting along a cross section with respect to the axis of the bar. If d A is an elementary area of the section at a distance of z units from the axis of a circular shaft [Fig. 4.9 (b)], and c is the distance from
the axis to the outside of the cross section where the unit shearing stress is Z, then the unit shearing stress acting on dA is (ZZ/C) dA, its moment with respect to the axis is (zz2/c) dA, and the sum of all the moments
of the unit shearing stresses on the cross section is f (rz2/c) dA. In this expression the factor fz2 dA is the polar moment of inertia of the section with respect to the axis. Denoting this by J the resisting moment may be written zJ/c.
The polar moment of inertia of a surface about an axis through its center of gravity and perpendicular to the surface is the sum of the products obtained by multiplying each elementary area by the square of its distance from the center of gravity of its surface; it is equal to the sum of the moments of inertia taken with respect to two axes in the plane of the surface at right angles to each other passing through the center of gravity section of a round shaft.
The analysis of torsional shearing stress distribution along noncircular cross sections of bars under torsion is complex. By drawing two lines at right angles through the center of gravity of a section before twisting,
and observing the angular distortion after twisting, it has been found from many experiments that in noncircular sections the shearing unit stresses are not proportional to their distances from the axis. Thus in a rectangular bar there is no shearing stress at the comers of the sections, and the stress at the middle of the wide side is greater than at the middle of the narrow side. In an elliptical bar the shearing stress is greater along the flat side than at the round side.
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Dominick Rosato and
Donald Rosato
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