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the bar is nearly the same. That is,
(1.140)
This assumption is attributed to Saint Venant and hence this formulation leads to what is
known as
Saint-Venant's theory of torsion
. It is convenient to introduce
u
=
f
(
y, z
)
φ
in the expression
)
=−
φ
ω(
for
u
.
Thus, letting
f
(
y, z
y, z
)
=−
φ
ω(
u
y, z
)
(1.141)
where
is referred to as the
warping
function and the minus sign has been inserted for later
convenience.
In summary, the displacements for bars of arbitrary cross-sectional shape are
ω
=−
φ
ω(
v
=−
φ
xz,
w
=
φ
xy
)
u
y, z
,
The strain-displacement relations of Eqs. (1.18) become
x
=
∂
u
x
=−
φ
ω(
y, z
)
=
0
∂
y
=
∂v
∂
y
=
0
z
=
∂w
∂
z
=
0
(1.142)
y
=−
φ
z
=
∂v
∂
x
+
∂
u
+
∂ω
∂
γ
xy
∂
y
z
=−
φ
=
∂w
∂
x
+
∂
+
∂ω
∂
u
γ
−
y
xz
∂
z
=
∂w
∂
y
+
∂v
z
=
φ
x
−
φ
x
γ
=
0
yz
∂
These kinematic relations will be used to develop governing equations for a bar that
has a constant cross-section with constant torque. However, once the relationship between
the applied torque and induced stresses is established, this theory can be applied to bars
with fewer restrictions, e.g., with nonconstant cross-sections and torques (Wempner, 1973;
Pilkey, 2002).
1.9.2 Material Laws
The constitutive relations of Eqs. (1.34a) reduce for the strains of Eq. (1.142) to
σ
=
σ
=
σ
=
τ
=
0
x
y
z
yz
τ
=
G
γ
xy
,
τ
=
G
γ
(1.143)
xy
xz
xz
1.9.3
Equations of Equilibrium
Since
σ
x
=
σ
y
=
σ
z
=
τ
yz
=
0
,
the equations of equilibrium [Eqs. (1.51)] are given by
∂τ
y
+
∂τ
xy
xz
=
0
(1.144a)
∂
∂
z
∂τ
yx
∂
=
∂τ
zx
∂
x
=
0
(1.144b)
x
where the applied loadings have been set equal to zero. Equations (1.144b) imply that the
shear stresses
τ
τ
yx
and
zx
are independent of
x
.
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