Page 51 - Aircraft Stuctures for Engineering Student

Two-d i mens i ona I pro b I ems

in elasticity

Theoretically we are now in a position to solve any three-dimensional problem
in elasticity having derived three equilibrium conditions, Eqs (1.5), six strain-
displacement equations, Eqs (1.18) and (1.20), and six stress-strain relationships,
Eqs (1.42) and (1.46). These equations are sufficient, when supplemented by
appropriate boundary conditions, to obtain unique solutions for the six stress,
six strain and three displacement functions. It is found, however, that exact
solutions are obtainable only for some simple problems. For bodies of arbitrary
shape and loading, approximate solutions may be found by numerical methods
(e.g. finite differences) or by the Rayleigh-Ritz method based on energy principles
(Chapter 5).
Two approaches are possible in the solution of elasticity problems. We may
solve initially either for the three unknown displacements or for the six unknown
stresses. In the former method the equilibrium equations are written in terms
of strain by expressing the six stresses as functions of strain (see Problem
P. 1.7). The strain-displacement relationships are then used to form three equa-
tions involving the three displacements u, v and w. The boundary conditions
for this method of solution must be specified as displacements. Determination
of u, v and w enables the six strains to be computed from Eqs (1.18) and
(1.20); the six unknown stresses follow from the equations expressing stress as
functions of strain. It should be noted here that no use has been made of the
compatibility equations. The fact that u, and UT are determined directly ensures
that they are single-valued functions, thereby satisfying the requirement of
compatibility.
In most structural problems the object is usually to find the distribution of
stress in an elastic body produced by an external loading system. It is therefore
more convenient in this case to determine the six stresses before calculating any
required strains or displacements. This is accomplished by using Eqs (1.42) and
(1.46) to rewrite the six equations of compatibility in terms of stress. The resulting
equations, in turn, are simplified by making use of the stress relationships
developed in the equations of equilibrium. The solution of these equations auto-
matically satisfies the conditions of compatibility and equilibrium throughout the
body.