Page 57 - Aircraft Stuctures for Engineering Student

42 Two-dimensional problems in elasticity

method to the torsion of solid sections (Chapter 3) and to the problem of a beam
supporting shear loads (Section 2.6).
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In the examples of Section 2.3 we have seen that a particular stress function form may
be applicable to a variety of problems. Different problems are deduced from a given
stress function by specifying, in the first instance, the shape of the body and then
assigning a variety of values to the coefficients. The resulting stress functions give
stresses which satisfy the equations of equilibrium and compatibility at all points
within and on the boundary of the body. It follows that the applied loads must be
distributed around the boundary of the body in the same manner as the internal
stresses at the boundary. Thus, in the case of pure bending (Fig. 2.2(a)) the applied
bending moment must be produced by tensile and compressive forces on the ends
of the plate, their magnitudes being dependent on their distance from the neutral
axis. If this condition is invalidated by the application of loads in an arbitrary fashion
or by preventing the free distortion of any section of the body then the solution of the
problem is no longer exact. As this is the case in practically every structural problem it
would appear that the usefulness of the theory is strictly limited. To surmount this
obstacle we turn to the important principle of St. Venant which may be summarized
as stating:

that while statically equivalent systems of forces acting on a body produce substan-
tially diferent local efects the stresses at sections distant from the surface of loading
are essentially the same.
Thus, at a section AA close to the end of a beam supporting two point loads P the
stress distribution varies as shown in Fig. 2.3, whilst at the section BB, a distance
usually taken to be greater than the dimension of the surface to which the load is
applied, the stress distribution is uniform.
We may therefore apply the theory to sections of bodies away from points of
applied loading or constraint. The determination of stresses in these regions requires,
for some problems, separate calculation (see Chapter 11).

Fig. 2.3 Stress distributions illustrating St. Venant‘s principle.