Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 52

FIGURE 1.18
Shear stresses and notation used to find the resultant moment on a cross-section.

which can be considered as a relationship between the angle of twist and the applied torque.
These relations are referred to as the Coulomb solution for the torsion of a bar of circular
cross section. The shear stresses of (3) are solutions to the theory of elasticity problem in
that the kinematic, constitutive, and equilibrium conditions are satisfied. The shear stresses
τ xy and

τ xz of (3) probably do not accurately reflect the actual shear stress on the ends of
the bar where the torsional loads are applied. In reality, the shear stress components of (3)
are usually accurate at a distance of several bar diameters from the ends. This concept that
there is a redistribution of the stress values at a distance from the ends is referred to as
Saint-Venant's principle 24 .
The stress distribution represented by (3) is independent of the axial coordinate x , so that
the stress distribution remains the same for any cross section along the shaft. From (3) and (8)

M t z
J

M t y
J

τ =− τ = (9)

xy xz

The resultant shear stress lies in the plane of the cross-section and is perpendicular to the
radius r from the origin. The magnitude of this resultant stress

τ

is

J z 2

M t M t r

J

τ = τ

xy

+ τ

xz

= + y 2 = (10)

which is a relationship between the torque M t and the shear stress for cylindrical bars of
hollow or solid circular cross-sections.

was zero. For cross-sections of arbitrary
shape, there is experimental evidence that the axial deformation of each cross-section along

In the case of bars of circular cross-sections, u ( y, z )

24 In 1855, Barre de Saint-Venant enunciated a useful principle that now bears his name. In essence, this principle
can be stated as the redistribution of loading, resulting from a set of forces acting on a small region of the surface
of an elastic body being replaced by a statically equivalent set of forces, causing significant changes in the stress
distribution only in the neighborhood of the loading, while stresses remain essentially the same in those portions
of the body located at large distances from the applied loading. By “large distances” are meant distances great
in comparison with the dimensions of the surface on which the loading is applied. “Statically equivalent” sets of
forces mean that the two distributions of loadings have the same resultant force and moment.