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The Membrane Analogy
It was noted by Prandtl that the governing equations for the Saint-Venant torsion problem
closely resemble the equilibrium equations for a flat membrane lying in the
yz
plane sub-
jected to a lateral pressure. If
N
is the uniform tension per unit length in the membrane,
p
is the lateral pressure, and
u
is the lateral displacement of the membrane, the differential
equation of equilibrium is
∂
2
u
y
2
+
∂
2
u
p
N
z
2
=−
(1.165)
∂
∂
Comparison of this with Eq. (1.159) shows that the relations are the same if
u
replaces
ψ
and
φ
. Since the volume between a deformed membrane and the
yz
plane is
p
/
N
replaces 2
G
A
udydz
, it follows from Eq. (1.163) that the twisting moment is equal to twice the volume
of the membrane. It can be shown that the contour lines showing constant
u
correspond to
lines of constrant stress function and that the slope of the membrane is equal to the value
of the shear stress. The membrane analogy opens the door to a wide range of experimental
studies of torsional problems. The membrane is stretched over a hole with a boundary the
same as the boundary of a cross-section of the bar subject to torsion.
1.9.7 Mixed Form of the Governing Differential Equations
The equations as originally derived in Sections 1.9.1, 1.9.2, and 1.9.3 are almost in mixed
form already. Normally, a mixed form is established by first writing the constitutive rela-
tions in terms of displacements rather than strains. Thus, if the equations of Eq. (1.142) are
substituted into the relations of Eq. (1.143),
φ
z
,
φ
+
∂ω
∂
+
∂ω
∂
τ
xy
=−
G
τ
xz
=−
G
−
y
(1.166)
y
z
The combination of this and the differential equilibrium conditions forms the desired mixed
relations. To gather these relations together, as was done in Eq. (1.75), consider the state
vector of displacement and stress variables
ω
τ
(1.167)
xz
τ
xy
Then Eqs. (1.166) and (1.144a) take the form
φ
0
∂
∂
ω
G
0
−
0
0
0
z
y
+
φ
=
∂
z
τ
10
G
y
(1.168)
xz
∂
τ
01
z
y
xy
These exhibit the usual characteristics of mixed formulations of having unknowns with
no higher than first order derivatives and of not involving derivatives of the material
parameters.
Equation (1.168) along with the boundary conditions of Eq. (1.145) and the resultant
moment condition of Eq. (1.148) form a complete set of equations for the mixed formulation
of the Saint-Venant torsion problem. Equations (1.168) provide the stresses
τ
xz
,
τ
xy
, and the
φ
. By setting
M
t
of Eq. (1.148) equal to the applied torque
wa
rping function as a function of
φ
can be computed.
M
t
,
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