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In order to express discrete forms of the principles of Chapter 2, Table 2.4, we will need
several expressions based on Eqs. (6.154, 6.155, and 6.156) and p
A T σ of Eq. (1.57)
=
σ T N T
σ
u T
= δ
u T N u ,
σ T
δ
z T
= δ
z T N z )
δ
δ
= δ
,
or
δ
z
=
N z
z
,
σ T N T
σ
p T
A T σ
T
δ
= δ(
)
= δ
A
Then, the four principles of Table 2.4 can be discretized. Each of the generalized variational
theorems will involve a summation over all M elements. We obtain, if D u =
D
σ =
D , for
the two symmetric forms,
From CB:
+
,
.
1
. N u D T N
N u
D T σ p +
z T
N u
0
p V
M
0
σ
3
δ
···
·
···
z
·········
dV
-
.
N T
E 1 σ p + 0
V i
σ (
)
i
=
1
N T
N T
σ
E 1 N σ
σ (
DN u
)
.
1
.
"
z T
N u
z T
N u P
N u p p
······
P T u
p
0
0
3
dS
+ δ
···
0
+ δ
···
·
···
z
+
dS
#
.
S pi
S ui
P T N u
0
=
0
(6.157)
Form AD:
+
,
.
0
1
3
.
z T
N u (
D T N σ )
N u (
0
p V )
·········
D σ p
+
M
δ
···
·
···
z
dV
-
N T
σ
D N u
.
V i
N T
E 1 σ p
0
σ (
+
)
=
i
1
N T
σ
E 1 N
σ
.
0
1
3
.
"
z T
z T
N u P
N u (
0
p p
p
)
0
···
P T u
dS
+ δ
+ δ
···
·
···
z
+
······
0
dS
#
.
S pi
S ui
P T N u
0
=
0
(6.158)
A T σ p .
The discretized forms of generalized variational principles represent the same basic equa-
tions as the continuous forms. Depending on the variational principle, some of the funda-
mental equations for solids, i.e., kinematics, material law, or equilibrium, are satisfied and
others are the resulting best possible approximations. The same applies to the static and
geometric boundary conditions. If the trial functions satisfy one of the fundamental con-
ditions, then the corresponding terms will fall out. For example, the boundary terms are
dropped when the boundary conditions are satisfied by the trial solutions.
The system of equations for the unknowns
A T N
where P
=
and p p =
σ
z are found using the fundamental lemma of
the calculus of variations applied to Eq. (6.158), since
δ
z are arbitrary variations.
The forms of Eq. (6.157) lead to the element matrix for CB
0
i
T
c i
N T
=
with
=
σ (
DN u )
dV
f
V i
(6.159)
N T
σ
E 1 N
f
=
dV
σ
V i
For linear trial functions for all state variables, these expressions are particularly easy to
integrate.
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