ABSTRACT
In this chapter we formulate and solve the boundary problem for computing the displacement and stress fields of prismatic bodies having cross sections of arbitrary geometry (solid or with holes)(see Fig. 6.1) made from isotropic, linearly elastic materials. The dimensions of the cross sections of these bodies are small as compared to
1their length. We call such bodies prismatic bodies. We choose the x axis to be any axis parallel to their axis. The bodies are initially in a reference, stress-free, strain-free state
oof mechanical and thermal equilibrium at a uniform temperature T . Subsequently, they reach a second state of mechanical equilibrium due to the application on each of their end
1 1surfaces (x = 0 and x = L) of a distribution of external tractions which is statically 1equivalent to a given concentrated torsional moment of magnitude M . We assume that
the distribution of the external tractions is applied in such a way that it does not restrain the end surfaces of the bodies from warping. In practice we often do not know the distribution of traction acting on the end surfaces of prismatic bodies. However, we do know the magnitude of their resultant force and moment. On the basis of the principle of Saint Venant (see Section 5.3) all distributions of traction, acting on the end surfaces of the prismatic bodies under consideration, which are statically equivalent, produce essentially the same distribution of components of stress on parts of the bodies which are sufficiently removed from their end surfaces. Thus, we can only ensure that the stress field of such bodies yields a distribution of traction on each one of their end surfaces
1which is statically equivalent to a given torsional moment of magnitude M . On the
(6.1)
(6.2a)
(6.2b)
(6.2c)
(6.2d)
(6.2e)
(6.2f)
basis of the above presentation, referring to relations (5.14), the boundary conditions for the prismatic bodies under consideration are
On their lateral surfaces
1 1On their end surfaces (x = 0 and x = L)
2 3 2 3where e and e are the x and x coordinates, respectively, of the shear center of the cross 2 3sections of the prismatic bodies. Notice that since F and F vanish the moment of the
1components of tractions and about any axis parallel to the x axis must be equal to
1M . These boundary conditions do not restrain a body from moving as a rigid-body. Thus, we anticipate that the components of displacement of a body obtained from the solution of the boundary value problem under consideration will include an unspecified rigid-body motion of this body. We will eliminate this rigid-body motion by assuming that it is possible to restrain the body from moving as a rigid-body without inhibiting the warping of its end surfaces.
