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If the displacement
u
0 and the solution corresponds
to the Coulomb solution of Example 1.6 for the torsion of a shaft of circular cross-section.
A uniqueness study of
=
0 at the bar end
(
x
=
0
)
,
then
c
=
(e.g., see Little (1973)) shows that the warping function may be
determined only up to a constant. The shear stresses obtained from
ω
ω
involve derivatives
of
and, hence, are found uniquely. However, because of the constant, care must be taken
in the direct use of
ω
ω
.
1.9.6 Force Form of the Governing Differential Equations
To obtain a force form of the governing equations, a stress function is introduced such that
the conditions of equilibrium are satisfied. Then, the compatibility equations are expressed
in terms of the stress function. For the torsion problem, Prandtl
25
introduced a stress function
ψ(
y, z
)
defined as
=
∂ψ
∂
=−
∂ψ
∂
τ
z
,
τ
(1.155)
xy
xz
y
This is usually called the
Prandtl stress function
. Note that the condition of equilibrium
of Eq. (1.144a) is identically satisfied. The solution to the torsion problem requires the
calculation of the stress function
.
Based on the strains of Eq. (1.142), the meaningful compatibility equations of Eq. (1.28)
are
ψ
1
2
∂
1
2
∂
1
2
∂
1
2
∂
y
z
−
γ
+
∂
γ
=
0
,
∂
γ
−
γ
=
0
(1.156)
xz
y
z
xy
z
y
zx
xy
Since
γ
=
∂
z
ψ/
γ
=−
∂
ψ/
G,
G
xy
xz
y
Eq. (1.156) can be written
1
2
∂
y
∂
z
ψ
=
2
∂
z
∂
z
ψ
=
1
2
y
2
2
y
2
+
∂
0
,
−
+
∂
0
(1.157)
y
z
In order for both of the equations in Eq. (1.157) to hold,
must be a constant.
To find this constant, introduce stress-displacement relations obtained by substituting Eq.
(1.142) into Eq. (1.143). Then differentiate
(∂
+
∂
)ψ
τ
xy
with respect to
z
and
τ
xz
with respect to
y
.
Thus
∂ψ
∂
φ
1
2
∂τ
xy
∂
=
∂
∂
+
∂
ω
2
z
=
∂
ψ
=−
G
z
z
z
∂
z
∂
y
(1.158)
φ
∂τ
xz
y
=
∂
−
∂ψ
∂
+
∂
2
ω
2
y
=−
∂
ψ
=−
G
−
1
∂
∂
∂
∂
y
y
y
z
so that
2
2
∂
z
2
+
∂
ψ
ψ
φ
2
φ
y
2
=−
2
G
or
∇
ψ
=−
2
G
(1.159)
∂
∂
25
Ludwig Prandtl (1875-1953) was a German engineer who is best known for his pioneering work in aerodynamics.
In 1900, he received his doctorate under August F
o
ppl in Munich. He established engineering mechanics at the
University of G
o
ttingen, where he guided many students who have since played leading roles in mechanics.
He initiated the development of several areas of mechanics with fundamental studies. He is the founder of the
boundary layer theory of fluid mechanics. He cooperated with Theodore von Karman in significant research on
airfoil theory, drag, and turbulent flows. One of his first papers points out that by using a soap film, information
on the distribution of torsional stresses on a cross-section can be obtained. This is the
membrane analogy
.
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