Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

If a torsion member is subjected to external torques at intermediate points (i.e., points
other than the ends) or if it consists of segments of various diameters or materials, then the
torsion member must be divided into segments that satisfy the three requirements just
listed. For compound torsion members comprising two or more segments, the overall angle
of twist can be determined by algebraically adding the segment twist angles:
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TORSION SIgN CONVENTIONS
TL i i
φ = ∑
(6.14)
JG
i
Here, T i , L i , G i , and J i are the internal torque, length, shear modulus, and polar moment of
inertia, respectively, for individual segments i of the compound torsion member.
The amount of twist in a shaft (or a structural element) is frequently a key consideration
in design. The angle of twist φ determined from Equations (6.12) and (6.13) is applicable to
a constant-diameter shaft segment that is sufficiently removed from sections to which pulleys,
couplings, or other mechanical devices are attached (so that Saint-Venant’s principle is
applicable). However, for practical purposes, it is customary to neglect local distortion at all
connections and compute twist angles as though there were no discontinuities.
Rotation Angles
It is often necessary to determine angular displacements at particular points in a compound
torsional member or within a system of several torsional members. For example, the proper
operation of a system of shafts and gears may require that the angular displacement at a
specific gear not exceed a limiting value. The term angle of twist pertains to the torsional
deformation in shafts or shaft segments. The term rotation angle is used in referring to the
angular displacement at a specific point in the torsion system or at rigid components, such
as pulleys, gears, couplings, and flanges.
i
i
6.6 Torsion Sign conventions
A consistent sign convention is very helpful to us when we analyze torsion members and
assemblies of torsion members. The sign conventions that follow will be used for
• internal torques in shafts or shaft segments,
• angles of twist in shafts or shaft segments, and
• rotation angles of specific points or rigid components.
Internal Torque Sign convention
Moments in general, and internal torques specifically, are conveniently represented by a
double-headed vector arrow. This convention is based on the right-hand rule:
• Curl the fingers of your right hand in the direction that the moment tends to rotate.
The direction that your right thumb points indicates the direction of the doubleheaded
vector arrow.
• Conversely, point your right-hand thumb in the direction of the double-headed
vector arrow, and the fingers of your right hand curl in the direction that the
moment tends to rotate.