Page 22 - Aircraft Stuctures for Engineering Student

1.3 Equations of equilibrium 7

direction then positive shear stresses are in directions opposite to the positive direc-
tions of the appropriate axes.
Two types of external force may act on a body to produce the internal stress system
we have already discussed. Of these, surface forces such as PI, P2 . . . , or hydrostatic
pressure, are distributed over the surface area of the body. The surface force per unit
area may be resolved into components parallel to our orthogonal system of axes and
these are generally given the symbols X, Y and Z. The second force system derives
from gravitational and inertia effects and the forces are known as body forces.
These are distributed over the volume of the body and the components of body
force per unit volume are designated X, Y and 2.

Generally, except in cases of uniform stress, the direct and shear stresses on opposite
faces of an element are not equal as indicated in Fig. 1.5 but differ by small amounts.
Thus if, say, the direct stress acting on the z plane is a, then the direct stress acting on
the z+ Sz plane is, from the first two terms of a Taylor’s series expansion,
a, + (aaJaz)Sz.
We now investigate the equilibrium of an element at some internal point in an
elastic body where the stress system is obtained by the method just described.
In Fig. 1.6 the element is in equilibrium under forces corresponding to the stresses
shown and the components of body forces (not shown). Surface forces acting on the
boundary of the body, although contributing to the production of the internal stress
system, do not directly feature in the equilibrium equations.

Fig. 1.6 Stresses on the faces of an element at a point in an elastic body.