Civil Engineering Reference
In-Depth Information
individual component is now labeled as associated with a specific row-column
position of the global stiffness matrix and can be added directly to that location.
For example, Equation 3.38 shows that the
k
(2)
24
component of element 2 is to be
added to global stiffness component
K
46
(and via symmetry
K
64
). Thus, we can
take each element in turn and add the individual components of the element stiff-
ness matrix to the proper locations in the global stiffness matrix.
The form of Equations 3.37 and 3.38 is convenient for illustrative purposes
only. For actual computations, inclusion of the global displacement numbers
within the element stiffness matrix is unwieldy. A streamlined technique suitable
for computer application is described next. For a 2-D truss modeled by spar
elements, the following conventions are adopted:
1.
The global nodes at which each element is connected are denoted by
i
and
j
.
2.
The origin of the element coordinate system is located at node
i
and the
element
x
axis has a positive sense in the direction from node
i
to node
j
.
3.
The global displacements at element nodes are
U
2
i
−
1
,
U
2
i
,
U
2
j
−
1
, and
U
2
j
as noted in Section 3.2.
Using these conventions, all the information required to define element con-
nectivity and assemble the global stiffness matrix is embodied in an
element-
node connectivity
table, which lists element numbers in sequence and shows the
global node numbers
i
and
j
to which each element is connected. For the two-
element truss of Figure 3.2, the required data are as shown in Table 3.2.
Using the nodal data of Table 3.2, we define, for each element, a
1
×
4
ele-
ment displacement location vector
as
L
(
e
)
=
12
j
]
(3.39)
where each value is the global displacement number corresponding to element
stiffness matrix rows and columns 1, 2, 3, 4 respectively. For the truss of Fig-
ure 3.2, the element displacement location vectors are
L
(1)
=
[2
i
−
1
i
2
j
−
[1256]
(3.40)
L
(2)
=
[3456]
(3.41)
Before proceeding, let us note the quantity of information that can be
obtained from simple-looking Table 3.2. With the geometry of the structure
defined, the (
X
,
Y
) global coordinates of each node are specified. Using these
data, the length of each element and the direction cosines of element orientation
Table 3.2
Element-Node Connectivity Table
for Figure 3.2
Node
Element
i
j
1
1
3
2
2
3
