Civil Engineering Reference
In-Depth Information
U
6
U
4
U
5
F
3
Y
2
U
6
U
3
2
F
3
X
3
U
5
2
2
1
1
Y
U
2
1
X
1
U
1
(a)
(b)
Figure 3.2
(a) A two-element truss with node and element numbers. (b) Global displacement notation.
Conversion of element equations from element coordinates to global coordi-
nates and assembly of the global equilibrium equations are described first in the
two-dimensional case with reference to Figure 3.2a. The figure depicts a simple
two-dimensional truss composed of two structural members joined by pin con-
nections and subjected to applied external forces. The pin connections are taken
as the nodes of two bar elements as shown; node and element numbers, as well
as the selected global coordinate system are also shown. The corresponding
global displacements are shown in Figure 3.2b. The convention used here for
global displacements is that
U
2
i
−
1
is displacement in the global
X
direction of
node
i
and
U
2
i
is displacement of node
i
in the global
Y
direction. The convention
is by no means restrictive; the convention is selected such that displacements in
the direction of the global
X
axis are odd numbered and displacements in the
direction of the global
Y
axis are even numbered. (In using FEM software, the
reader will find that displacements are denoted in various fashions, UX, UY, UZ,
etc.) Orientation angle
for each element is measured as positive from the global
X
axis to the element
x
axis, as shown. Node numbers are circled while element
numbers are in boxes. Element numbers are superscripted in the notation.
To obtain the equilibrium conditions, free-body diagrams of the three con-
necting nodes and the two elements are drawn in Figure 3.3. Note that the exter-
nal forces are numbered via the same convention as the global displacements.
For node 1, (Figure 3.3a), we have the following equilibrium equations in the
global
X
and
Y
directions, respectively:
f
(1
1
cos
F
1
−
1
=
0
(3.1a)
f
(1
1
sin
F
2
−
1
=
0
(3.1b)
