CONTINUUM MECHANICS for ENGINEERS

where ψ(x 1,x 2) is called the warping function. Note that the warping is independent
of x 3 and therefore the same for all cross sections. Also, we assume
that x 3 is a centroidal axis although this condition is not absolutely necessary.
As in the analysis of the circular shaft we may again use Eq 4.7-5 along
with Eq 6.2-2 to calculate the stress components from Eq 6.8-6. Thus
σ = σ = σ = σ = 0 ; σ = Gθ ψ −x
; σ Gθ ψ x
11 22 33 12 13 , 1 2 23 , 2 1
(6.8-7)
It is clear from these stress components that there are no normal stresses
between the longitudinal elements of the shaft. The first two of the equilibrium
equations, Eq 6.4-1, are satisfied identically by Eq 6.8-7 in the absence
of body forces, and substitution into the third equilibrium equation yields
which indicates that ψ must be harmonic
(6.8-8)
(6.8-9)
on the cross section of the shaft.
Boundary conditions on the surfaces of the shaft must also be satisfied.
On the lateral surface which is stress free, the following conditions based
upon Eq 3.4-8, must prevail
(6.8-10)
noting that here n 3 = 0. The first two of these equations are satisfied identically
while the third requires
which reduces immediately to
( ) = ( − )
σ , + σ , + σ , = Gθ(
ψ, + ψ,
)=
13 1 23 2 33 3 11 22 0
2
ψ = 0
σ n + σ n = 0; σ n + σ n = 0; σ n + σ n = 0
11 1 12 2 21 1 22 2 31 1 32 2
( )
( ) + ( + ) =
Gθψ − x n Gθψ x n
, 1 2 1 , 2 1 2 0
dψ
ψ, n + ψ,
n = = xn −xn
dn
1 1 2 2 2 1 1 2
(6.8-11a)
(6.8-11b)
Therefore, ψ x must be harmonic in the cross section of the shaft shown
1, x2
in Figure 6.8, and its derivative with respect to the normal of the lateral
surface must satisfy Eq 6.8-11b on the perimeter C of the cross section.
We note further that in order for all cross sections to be force free, that is,
in simple shear over those cross sections
∫∫ ∫∫ ∫∫
σ dx dx = σ dx dx = σ dx dx =
13 1 2 23 1 2 33 1 2 0
(6.8-12)