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5.3
Displacement Method
The most commonly used method today for the analysis of large structural systems is
the displacement method. Although the displacement method can be employed without
reference to its roots as a variational method (in such a case it is sometimes called the
direct
stiffness method
), it is perhaps best understood when it is considered to be a variationally
based approach. The basis of the displacement method is the principle of virtual work.
Since, as shown in Chapter 2, the principle of virtual work is equivalent to the global form
of the equations of equilibrium, it is understandable that the displacement method is also
referred to as the
equilibrium method
.
5.3.1 Nodal Displacement Equations Based on the Principle of Virtual Work
The principle of virtual work relations, which are designated as equations
C
in Chapter 2,
are expressed in terms of displacements for an elastic solid by [Chapter 2, Eq. (2.58c)]
S
p
δ
u
T
k
D
u
dV
u
T
p
V
u
T
p
dS
V
δ
−
V
δ
dV
−
=
0
(5.29)
where
u
D
T
ED
u
(5.30)
A different form of the principle of virtual work is useful in establishing the fundamen-
tals of the displacement method. The structural system is to be modeled in terms of elements
for which the responses and applied loading are represented by forces and displacements at
the nodes. This modeling amounts to spatial discretization of the structure. From Chapter 4,
Eq. (4.62), the virtual work for element
i
is
k
D
=
i
v
iT
k
i
v
i
p
i
0
−
(δ
W
i
+
δ
W
e
)
=
δ
(
−
)
(5.31)
The element nodal
di
splacements are
v
i
and the nodal forces representing the effects of
applied loading are
p
i
0
. If the structural system is modeled as
M
elements, the principle of
virtual work for all elements becomes
M
M
i
v
iT
k
i
v
i
p
i
0
−
(δ
W
i
+
δ
W
e
)
=−
1
(δ
W
i
+
δ
W
e
)
=
1
δ
(
−
)
=
0
(5.32)
i
=
i
=
It should be understood that Eq. (5.32) represents the summation of internal and external
virtual work done by the element forces at the nodes, i.e., this equation contains the total
virtual work of all elements of the system.
A useful form of the principle of virtual work is obtained by expressing the element
displacements
v
i
in terms of the unknown global nodal displacements
V
. This can be
accomplished by enforcing the nodal compatibility conditions. The nodal displacements
of the various elements joined at a particular node must match the values of the system
displacements of the node. This requirement should not be surprising, since the principle
of virtual work (Chapter 2) corresponds to the equilibrium equations provided that the
displacements are kinematically admissible. In order to implement these compatibility
conditions, the local (element) end displacements must be transformed to the directions
of the global coordinate system. Assume
v
i
has been transformed to
v
i
. The compatibility
conditions for node
k
of Fig. 5.11, where four elements meet, are
v
k
=
v
k
=
v
k
=
v
k
=
V
k
(5.33)
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