Information Technology Reference
In-Depth Information
It can also be shown that
a
∗
=
b
T
, where
a
∗
is defined by
V
a
∗
v
. Although
a
∗
a
=
=
I
, the
product
aa
∗
is not the identity matrix.
This relationship between the equilibrium matrix
b
∗
and the kinematic matrix
a
T
for
discrete systems is akin to the relationship between the kinematic operator
D
and the
equilibrium operator
D
T
for continuous systems (Chapter 1). With the help of these new
definitions, the equilibrium equations in the displac
e
ment form of Eq. (5.44) are readily
obtained. From the nodal equilibrium relations
p
=
b P
and the identities of Eqs. (5.52) and
(5.56), we find
b
∗
p
a
T
p
=
P
=
(5.57)
The unassembled set of stiffness equations would be
p
=
kv
. Introduce this into Eq. (5.57),
giving
a
T
kv
=
P
(5.58)
From the nodal connectivity equation (Eq. 5.37) it follows that
a
T
kaV
=
P
or
KV
=
P
(5.59)
where
K
is again given by
a
T
ka
. Thus, without direct reference to the principle of virtual
work, it is evident that the equations of equilibrium provide the basis for the displacement
relations of Eq. (5.44).
With the derivation of the congruent transformation representation of
K
, it can be
observed that the system matrix
K
contains all the basic equations:
a
T
ka
=
K
(5.60)
Conditions of
Compatibility
Equilibrium
Material
Law
5.3.3
Transformation of Coordinates
a
T
ka
,
it was assumed that all
of the element forces, displacements, and stiffness matrices were available in the form
referred to the global coordinates. The transformation
a
T
ka
can be generalized to include
a transformation from local to global coordinates for the element stiffness matrix. From
Eq. (5.28), the stiffness matrix
k
i
In the previous development of the transformation
K
=
in local coordinates is transformed to global coordinates
using
k
i
T
i
k
i
T
iT
=
(5.61)
Then
K
=
a
T
ka
becomes
k
i
T
i
]
a
k
a
T
diag [
T
iT
=
a
T
K
=
a
(5.62)
with
T
i
T
i
T
i
=
a
(5.63)
T
i
where
T
i
has now replaced
I
in Eq. (5.37).
Search WWH ::
Custom Search