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EXAMPLE 2.8 Torsion, An Example of Field Theory Equations
The governing equations for the Saint Venant torsion problem were considered in Chapter
1, Section 1.9, as an example of the theory of elasticity. The principle of virtual work will
now be used to provide equilibrium relations and static boundary conditions by assuming
kinematically admissible displacements at the outset.
As in Chapter 1, Section 1.9, an element of shaft from x
b with a constant
torque M t along the bar axis will be examined. Suppose this torque is caused by an applied
torque M t at the x
=
a to x
=
=
a end. As shown in Chapter 1, Eq. (1.142), the kinematic conditions
are
x =
0 ,
y =
0 ,
z =
0 ,
γ yz =
0
=− φ z
,
=− φ
+ ∂ω
+ ∂ω
γ
γ
y
(1)
xy
xz
y
z
From Eq. (2.54),
a
δ
W
=− δ(
W i
+
W e
) =
V
δγ
+ τ
δγ
)
dV
M t
δφ |
=
0
(2)
xy
xy
xz
xz
Substitution of the kinematic relations (1) into (2) yields
τ xy z
+ τ xz y
dA d x
+ ∂ω
δφ + φ δ ∂ω
∂ω
δφ φ δ ∂ω
b
a
M t δφ |
y
y
z
z
x
A
dA
xz ∂ω
xy ∂ω
δφ
=
A
xz y
τ
xy z
)
dA
τ
z + τ
dx
y
x
A
A φ
dA d x
δ ∂ω
δ ∂ω
b
a
τ
y + τ
M t
δφ |
=
0
(3)
xy
xz
z
x
where x is the coordinate along the axis of the bar, and A is the cross-sectional area. We will
apply Green's theorem to the second and third surface integrals.
Green's theorem of Appendix II, Eq. (II.6), applied to a typical term in (3) appears as
xy ∂ω
dA
A ∂τ
dz
ds ds
xy
A τ
=
τ
ω
y ω
dA
(4)
xy
y
Then, with Chapter 1, Eq. (1.148),
∂τ xz
(
M t )δφ b
z + ∂τ xy
δ(ωφ )
M t
a +
dA d x
y
A
L
dy
ds + τ xy
dz
ds
δ(ωφ )
τ xz
ds
=
0
(5)
S p
where L is the length of the bar segment. Since the variations are arbitrary, it can be con-
cluded that
τ xz,z + τ xy, y =
0on A
(equation of equilibrium)
(6a)
dy
ds + τ
dz
ds =
τ
0or
τ
xz a z
+ τ
xy a y
=
0on S p
(6b)
xz
xy
(static boundary condition)
M t
=
A
xz y
τ
xy z
)
dA
=
M t
on
a, b
(6c)
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