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σ are the unknown parameters, i.e., with generalized stresses, A T is the matrix
of direction cosines of the normals to the boundaries [Eq. (1.57)], and N σ contains the
polynomials expressing the distribution of stresses. For the boundary displacements, choose
a trial solution such as
where
N B v (6.152)
where v are the nodal displacements, and N B defines the assumed boundary displacements.
Note that when the structure is discretized into elements, the vector p in Eq. (6.150) con-
tains the reaction forces on the boundaries of the elements. Hence, the second relationship
of Eq. (6.151) should be applied to all elements. Also, u is the boundary displacement which
can occur on all elements. Thus, Eq. (6.150) can be expressed as
u
=
V δ σ T E 1 σ dV
dS
M
W =
u T p
p T u
δ
S pi
+ δ
)
i
(6.153)
N 1
N 2
u T p dS
p T u dS
+
S pj δ
S uk δ
=
0
j
=
1
k
=
1
where M is the total number of elements, N 1 is the number of elements where boundary
tractions are applied, N 2 is the number of elements where boundary displacements are
prescribed, S pi is the boundary of the i th element, and S pj and S uk are the boundaries of the
j th and k th elements among the N 1 and N 2 elements. With the hybrid method it is common
to organize the element matrix, with the help of condensation, to obtain an element stiffness
matrix.
6.9
Generalized Finite Element Methods
The hybrid method of Section 6.8 is one of the most important of the generalized finite
element methods. The generalized or mixed variational forms AB, AD, CB, and CD of
Chapter 2, Table 2.4, can be used as the basis of further development of generalized or
mixed finite element methods [Wunderlich, 1972; Wunderlich, 1983].
6.9.1
Discretization of Principles
Discretization should begin with the selection of appropriate trial functions, e.g., use
] T
u
=
N u
u
where
u
=
[
u 1
u 2 ···
(6.154)
for displacements and
N s
σ
=
σ p
+
N σ
σ
(
or
s
=
s p
+
s
)
(6.155)
for stresses (or stress resultants), where
σ contain the unknown parameters, N u and
N σ contain the polynomials, and σ p (or s p ) is the particular solution portion resulting from
the prescribed stresses. These can be gathered together using the state vector z , giving
u and
u
σ
or
u
s
N z
z
=
z p
+
z
with
z
=
z
=
(6.156)
N u
0
σ p
0
N z =
z p =
0
N
σ
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