Page 287 - Introduction to Continuum Mechanics

The Elastic Solid 271

Now that we have a possible stress distribution, let us consider the nature of the boundary
tractions. As is the case with simple extension, the lateral surface is obviously traction-free.
On the end face xj = /, we have a surface traction

which gives a resultant force system

where A is the cross-sectional area, /22, ^33, and /23 are the moments and product of inertia of
the cross-sectional area. On the face*i = 0, the resultant force system is equal and opposite
to that given above.

we will set a - 0 to make RI = 0 so that there is no axial forces acting at the end faces.
We now assume, without any loss in generality, that we have chosen the *2 and x^ axis to
coincide with the principal axes of the cross-sectional area (e.g., along lines of symmetry) so
that /23 = 0. In this case, from Eqs. (ix) and (x), we have ft = -M-$/Iy$ and y - M2//22 so that
the stress distribution for the cylindrical bar is given by

and all other TJJ = 0.
To investigate the nature of the deformation that is induced by bending moments, for
simplicity we let M 3 = 0. The corresponding strains are

These equations can be integrated (we are assured that this is possible since the strains are
compatible) to give the following displacement field: