Page 53 - Aircraft Stuctures for Engineering Student

38 Two-dimensional problems in elasticity

so that

and

Also

Substituting as before in Eq. (1.21) and simplifying by use of the equations of
equilibrium we have the compatibility equation for plane strain
(g+&)(n.x+oy)
ax
-+- dY
1
=
--
I-u(ax a,>
The two equations of equilibrium together with the boundary conditions, from Eqs
(1.7), and one of the compatibility equations (2.4) or (2.5) are generally sufficient for
the determination of the stress distribution in a two-dimensional problem.

The solution of problems in elasticity presents difficulties but the procedure may be
simplified by the introduction of a stress function. For a particular two-dimensional
case the stresses are related to a single function of x and y such that substitution
for the stresses in terms of this function automatically satisfies the equations of
equilibrium no matter what form the function may take. However, a large proportion
of the infinite number of functions which fulfil this condition are eliminated by the
requirement that the form of the stress function must also satisfy the two-dimensional
equations of compatibility, (2.4) and (2.5), plus the appropriate boundary conditions.
For simplicity let us consider the two-dimensional case for which the body forces
are zero. The problem is now to determine a stress-stress function relationship
which satisfies the equilibrium conditions of

and a form for the stress function giving stresses which satisfy the compatibility
equation