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VM
]
T
In matrix form, with
z
=
[
wθ
and upon integration by parts applied to some of the
terms, this appears as
−
00
∂
0
w
V
M
p
z
0
0
0
x
L
0
δ
00
1
∂
x
z
T
dx
+
[Boundary terms]
=
0
(7)
1
k
s
GA
∂
x
1
−
0
1
EI
0
∂
0
−
x
z
Euler's equations of (6) are
Equilibrium conditions
∂
x
M
=
V
∂
x
V
=−
p
z
(8)
Kinematical equations including the material law
=
∂
θ
=
κ
=
(θ
+
∂
w)
=
γ
M
EI
EI
V
k
s
GA
k
s
GA
(9)
x
x
Boundary conditions at
x
=
0 and
x
=
L
M
=
0or
δθ
=
0
V
=
0or
δw
=
0
(10)
In summary, by utilizing the kinematic conditions initially, the remaining governing beam
equations have been derived by using a variational theorem in which both displacements
and forces are varied independently.
EXAMPLE 2.12 Torsion, A Field Theory Example
The global form of the governing equations for the torsion problem will be derived using
a generalized variational principle formed as a combination of the principles of virtual
work and complementary virtual work [Zeller,
197
9]. To do so, consider a bar of uniform
cross-section and length
L
subjected to a torque
M
t
.
The cross-sectional shape is arbitrary.
Suppose the left end
(
x
=
a
=
0
)
is fixed and the right end
(
x
=
b
=
L
)
,
where the torque
is applied, is free.
Begin with principle
C
,
the principle of virtual work, as given by Eq. (2) of Example 2.8.
This includes the term
M
t
δφ
for the applied torque at the boundary
x
=
L,
as well as a
term
M
t
δφ
that “extends” the principle for the displacement boundary condition
(δφ
=
0
)
at
x
This condition is usually satisfied for kinematically admissible displacements.
Then
C
becomes
V
(τ
xy
δγ
xy
+
τ
xz
δγ
xz
)
=
0
.
dV
−
[
M
t
δφ
]
L
+
[
M
t
δφ
]
0
=
0
(C) or (1)
The underline is used to indicate the term that is “extending” the principle.
Integration by parts will connect
C
to
A
. We choose to repeat some of the formulation
presented in Example 2.8. First substitute the strain-displacement relations of Chapter 1,
Eq. (1.142) into (1), giving
xy
−
δφ
∂ω
∂
z
xz
−
δφ
∂ω
∂
y
dV
−
φ
δ
∂ω
∂
−
φ
δ
∂ω
∂
τ
y
+
+
τ
z
−
y
z
V
−
[
M
t
δφ
]
L
+
[
M
t
δφ
]
0
=
0
(2)
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