Elasticity_ Barber

33.7 Approximations using Castigliano’s second theorem 509
system with respect to x and y. For example, Timoshenko and Goodier 5 give
a solution of the same problem using the two degree of freedom function
ϕ = (x 2 − a 2 )(y 2 − a 2 )[A 1 + A 2 (x 2 + y 2 )] ,
which gives a result for the torque within 0.15% of the exact value.
33.7.2 The in-plane problem
We showed in Chapter 4 that the most general in-plane stress field satisfying
the equations of equilibrium (2.5) in the absence of body force can be expressed
in terms of the scalar Airy stress function φ through the relations
σ xx = ∂2 φ
∂y 2 ;
σ xy = − ∂2 φ
∂x∂y ;
σ yy = ∂2 φ
∂x 2 . (33.37)
We also showed in §4.4.3 that the traction boundary-value problem — i.e. the
problem of a two-dimensional body with prescribed tractions on the boundaries
— has a solution in which the stress components are independent of
Poisson’s ratio. We can therefore simplify the following treatment by considering
the special case of plane stress (σ zx =σ zy =σ zz =0) with ν =0 for which
equation (33.4:i) reduces to
(
σ
2
xx + σyy)
2
U 0 =
= 1
4µ
+ σ2 xy
4µ
[ (∂ 2 ) 2
φ
∂x 2 +
2µ
( ∂ 2 ) 2
φ
∂y 2 + 2
We express the trial stress function φ in the form
( ∂ 2 ) 2
]
φ
. (33.38)
∂x∂y
φ = φ P + φ H , (33.39)
where φ P is any particular function of x, y that satisfies the traction boundary
conditions and φ H is a function satisfying homogeneous (i.e. traction-free)
boundary conditions and that contains one or more degrees of freedom A k .
The stress components (33.37) will define a traction-free boundary if
φ = 0
and
∂φ
∂n = 0 , (33.40)
on the boundary, where n is the local normal. In order to satisfy these conditions,
we note that if f i (x, y) = 0 defines a line segment that is part of
the boundary Γ , and if f i is a continuous function in the vicinity of the line
segment, then we can perform a Taylor expansion about any point on the
5 loc. cit. Art.111.