Civil Engineering Reference
In-Depth Information
i
j
k
l
j
i
i
i
j
j
k
(b) Element stiffness matrix
j
e
1
e
2
i
k
l
(c) Default rigid connection
e
1
j k
e
2
i
l
(a) Global stiffness matrix
(d) Connected with joint
Figure 3.7
(a-d) Assembling global stiffness matrix and processing displacement
relationship.
All displacements at the connection of adjacent elements are continuous
by default, or the connection is rigid from element to element as shown in
Figure 3.7c. It is obvious that the global stiffness of node
j
will be the sum
of submatrices of both elements
e
an
( )
. These results are due to one-
to-one mapping of element stiffness and global stiffness during assembling
matrices. However, the relationship between element stiffness and global
stiffness does not have to be one to one. When this happens, a matrix-pro-
cessing technique,
the displacement relationship
, will be used. Taking the
simulation of commonly used joints as example, the principle of displace-
ment relationship is discussed briefly next in this section.
As shown in Figure 3.7d, two beam elements are connected with a joint.
Four nodes,
i
e
1
2
, , ,and
, in the global matrix will be needed to have enough
degrees of freedom to represent the extra rotation at the joint. If each node
is assumed to have six (6) degrees of freedom, node
j
j k
l
and
will be sharing
five (5) of them and each node has one rotation independent of one another.
The relationships of displacements between nodes
j
k
and
will be that the
five (5) shared displacements of node
k
are mapped to those of node
j
,
and their rotation is separated. When assembling
e
1
, it is a usual summing
process. When assembling
e
2
, matrix elements corresponding to shared dis-
placements at node
k
will be added to node
j
instead, rather than to node
k
as is normally done. Only the rotation matrix elements will be added to its
own position, node
k
in global. This type of relationship is often called the
master-slave relationship.
k
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