Page 18 - Aircraft Stuctures for Engineering Student

1

Basic elasticity

We shall consider, in this chapter, the basic ideas and relationships of the theory of
elasticity that are necessary for the development of the analytical work in the
remainder of the book. The treatment is divided into three broad sections: stress,
strain and stress-strain relationships. The third section is deferred until the end of
the chapter to emphasize the fact that the analysis of stress and strain, for example
the equations of equilibrium and compatibility, does not assume a particular
stress-strain law. In other words, the relationships derived in Sections 1.1 to 1.14
inclusive are applicable to non-linear as well as linearly elastic bodies.

Consider the arbitrarily shaped, three-dimensional body shown in Fig. 1.1. The body
is in equilibrium under the action of externally applied forces Pi, P2.. . and is
assumed to comprise a continuous and deformable material so that the forces are
transmitted throughout its volume. Thus, at any internal point 0 there is a resultant
force 6P. The particle of material at 0 subjected to the force SP is in equilibrium so
that there must be an equal but opposite force 6P (shown dotted in Fig. 1.1) acting on
the particle at the same time. If we now divide the body by any plane nn containing 0
then these two forces SP may be considered as being uniformly distributed over a
small area 6A of each face of the plane at the corresponding points 0 as in Fig. 1.2.
The stress at 0 is then defined by the equation

The directions of the forces 6P in Fig. 1.2 are such as to produce tensile stresses on
the faces of the plane nn. It must be realized here that while the direction of SP is
absolute the choice of plane is arbitrary, so that although the direction of the stress
at 0 will always be in the direction of 6P its magnitude depends upon the actual
plane chosen since a different plane will have a different inclination and therefore a
different value for the area 6A. This may be more easily understood by reference to
the bar in simple tension in Fig. 1.3. On the cross-sectional plane mm the uniform
stress is given by PIA, while on the inclined plane m'm' the stress is of magnitude
PIA'. In both cases the stresses are parallel to the direction of P.