Basic Equations: Differential Form - Mechanics of Structures: Variational and Computational Methods - page 49

or lower surfaces, τ = 0at z = t / 2 , and we find for the shear stress τ

xz at z xz

t 2
4 −

z 2

t / 2

V
I

V
2 I

τ xz = zdz = (1.138)

z

This indicates that the shear stress varies parabolically with z . This stress is zero at the top
and bottom of the cross-section

( z =± t / 2 ) and has its maximum value at the centroidal

axis ( z = 0).

1.9 Torsion, An Example of Field Theory Equations

Torsional stresses occurring on the cross-section of a bar present an interesting example of
structural behavior. They are of particular concern here because governing equations are
similar to those occurring in a variety of mechanics problems, which are often referred to
as field theory problems. Included in this class are several fluid mechanics problems.

1.9.1 Kinematical Relationships
Coulomb, 23 while studying the resistance of thin wires to torsion, formulated a solution that
was based on the assumption that a cross-section of a bar remains plane after deformation
and the cross-sections merely rotate. Navier erroneously applied this assumtion to the
general case of torsion of bars with noncircular cross-sections. Proper formulation of the
twisting bar, however, must account for deformation of cross-sections in the axial direction.
Saint-Venant provided the correct solution for a bar in torsion. He used the semi-inverse
method in which assumptions are made at the outset as to the deformations. These assumed
deformations, which include an unknown function, are then shown to satisfy the equilib-
rium conditions and the surface boundary conditions. That is, the unknown function is
chosen to satisfy these relations.
It then follows from the uniqueness principle for the solution of the elasticity equations
that the initial displacement assumptions were correct and the solution is exact. Saint-
Venant was guided by the solution for a shaft with a circular cross-section for which the
assumed deformation is formed of “rigid” rotations of the plane cross-sections. In addition,
deformation of the cross-section in the axial direction was incorporated. This is referred to
as the warping of the cross-section. However, it is assumed that there are no restraints placed
on the warping along the bar.
Consider a section of a shaft having a constant cross-section and a constant torque along
the bar axis. Turn first to the case where the cross-sections remain plane and simply rotate as
rigid surfaces so that the displacement in the x direction is zero. Suppose displacements are
small relative to the cross-sectional dimensions. Also, assume the profile of the cross-section,
i.e., the shape of the section, is not distorted due to twisting. As indicated in Fig. 1.17a, a
cross-section rotates about an axis of twist through an angle φ. This is the angle of twist of

the bar. One face of a shaft segment of length dx will rotate d φ relative to the other face.

23 Charles Augustin Coulomb (1736-1806) probably furthered the mechanics of elastic bodies more than any
other 18th century scientist. He was educated as a military engineer in France and served for thirty years in the
military. His theories of friction, strength of materials, and torsion are still used. Of his accomplishments in applied
mechanics, only his friction theory was accepted prior to the 19th century. As a result of his work in electricity
and magnetism, the unit of quantity of electricity, the coulomb, was named for him. Near the end of his life, he
devoted his efforts to hospital reform and the improvement of education in his native land, France.