Page 54 - Aircraft Stuctures for Engineering Student

2.3 Inverse and semi-inverse methods 39

The English mathematician Airy proposed a stress function 4 defined by the
equations

Clearly, substitution of Eqs (2.8) into Eqs (2.6) verifies that the equations of
equilibrium are satisfied by this particular stress-stress function relationship. Further
substitution into Eq. (2.7) restricts the possible forms of the stress function to those
satisfying the biharmonic equation

a44 a44 a4q5
-+2- +-=O
ax4 ax2ay2 ay4
The final form of the stress function is then determined by the boundary conditions
relating to the actual problem. Thus, a two-dimensional problem in elasticity with
zero body forces reduces to the determination of a function 4 of x and y, which
satisfies Eq. (2.9) at all points in the body and Eqs (1.7) reduced to two-dimensions
at all points on the boundary of the body.

The task of finding a stress function satisfying the above conditions is extremely
difficult in the majority of elasticity problems although some important classical
solutions have been obtained in this way. An alternative approach, known as the
inverse method, is to specify a form of the function 4 satisfying Eq. (2.9), assume
an arbitrary boundary and then determine the loading conditions which fit the
assumed stress function and chosen boundary. Obvious solutions arise in which q5
is expressed as a polynomial. Timoshenko and Goodier' consider a variety of
polynomials for 4 and determine the associated loading conditions for a variety of
rectangular sheets. Some of these cases are quoted here.

Example 2.1
Consider the stress function
4 = AX^ + BX~ + cy2
where A, B and C are constants. Equation (2.9) is identically satisfied since each term
becomes zero on substituting for 4. The stresses follow from
@q5
ux= -=2c
aY2

To produce these stresses at any point in a rectangular sheet we require loading
conditions providing the boundary stresses shown in Fig. 2.1.