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5.4.1
Nodal Force Equations Based on the Principle of Complementary Virtual Work
In Chapter 2 the principle of complementary virtual work relations for an elastic solid,
which are designated as Eq. (D), take the form [Chapter 2, Eq. (2.78)]
V
δ
σ
T
E
−
1
σ
dV
p
T
−
+
S
u
δ
u
dS
=
0
(5.116)
For a system of beam (bar) elements, this becomes
x
δ
p
T
E
−
1
p
dx
p
T
p
T
−
x
δ
+
u
dx
−
[
δ
u
]
k
=
0
|
|
|
|
|
|
Element
Applied
Applied
Contributions
displacements displacements
(5.117)
along the
at the
elements
nodes
|
||
|
Internal Work
External work
As with the displacement method, for the force method the structure is modeled by
M
elements. Forces
p
i
and displacements
v
i
at the ends of the elements compose the responses.
The principle of complementary virtual work,
W
i
−
δ
W
e
=
−
δ
0, for this discretized repre-
sentation would take the form
M
M
p
i
R
f
i
p
i
R
−
v
i
=
p
iT
v
i
v
i
1
δ
(
−
)
=
1
δ
0
(5.118)
i
=
i
=
which is analogous to Eq. (5.32) for the displacement method. The notation of Chapter 4
is utilized here, e.g., the element flexibility matrix
f
i
f
i
p
i
R
in which the
reduced force and displacement vectors are introduced. The terms in parentheses in Eq.
(5.118) give the compatibility contributions for the individual elements. As expected by
analogy with the displacement method, the internal complementary virtual work provides
a relationship for the element flexibility matrix. As indicated in Chapter 4, the fl
ex
ibility
matrix is formed only for restrained systems. The applied displacement vector
v
i
is the
result of either influences, like thermal loading, distributed along the member or imposed
displacements at the nodes. If the summations of Eq. (5.118) were to be expressed as system
matrices, the following equation would be expected:
is defined by
v
i
R
=
P
T
δ
(
V
−
V
)
=
0
(5.119)
The summation of the nodal compatibility is expressed in terms of the unknown global
nodal forces
P
The requirement that has to be satisfied is that the nodal forces of the var-
ious elements joined at each node must be in equilibrium with the applied nodal forces.
Imposition of the equilibrium conditions is an expected requirement, since the principle
of complementary virtual work corresponds to the kinematic conditions provided that the
forces are in equilibrium. Of course, the local (element) nodal forces may need transforma-
tion to the global coordinate system to ease the establishment of equilibrium conditions.
.
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