Elasticity_ Barber

7.3 Solution for the stress function
7.3 Solution for the stress function 97
Once the body force potential V has been determined, the next step is to find
a suitable function φ, which satisfies the compatibility condition (7.8) and
which defines stresses through equations (7.1–7.3) satisfying the boundary
conditions of the problem. There are broadly speaking two ways of doing this.
One is to choose some suitable form (such as a polynomial) without regard to
equation (7.8) and then satisfy the constraint conditions resulting from (7.8)
in the same step as those arising from the boundary conditions. The other
is to seek a general solution of the inhomogeneous equation (7.8) and then
determine the resulting arbitrary constants from the boundary conditions.
7.3.1 The rotating rectangular beam
As an illustration of the first method, we consider the problem of the rectangular
beam −a < x < a, −b < y < b, rotating about the origin at constant
angular velocity Ω, all the boundaries being traction-free (see Figure 7.1).
Figure 7.1: The rotating rectangular bar.
as
The body force potential for this problem is obtained from equation (7.22)
V = − 1 2 ρΩ2 (x 2 + y 2 ) , (7.23)
and hence the stress function must satisfy the equation
∇ 4 φ = 2ρ(1 − ν)Ω 2 , (7.24)
from (7.8, 7.23).
The geometry suggests a formulation in Cartesian coördinates and equation
(7.24) leads us to expect a polynomial of degree 4 in x, y. We also note
that V is even in both x and y and that the boundary conditions are homogeneous,
so we propose the candidate stress function
φ = A 1 x 4 + A 2 x 2 y 2 + A 3 y 4 + A 4 x 2 + A 5 y 2 , (7.25)
which contains all the terms of degree 4 and below with the required symmetry.