Civil Engineering Reference
In-Depth Information
3.8 THREE-DIMENSIONAL TRUSSES
Three-dimensional (3-D) trusses can also be modeled using the bar element,
provided the connections between elements are such that only axial load is trans-
mitted. Strictly, this requires that all connections be ball-and-socket joints. Even
when the connection restriction is not precisely satisfied, analysis of a 3-D truss
using bar elements is often of value in obtaining preliminary estimates of mem-
ber stresses, which in context of design, is valuable in determining required
structural properties. Referring to Figure 3.7 which depicts a one-dimensional
bar element connected to nodes
i
and
j
in a 3-D global reference frame, the unit
vector along the element axis (i.e., the element reference frame) expressed in the
global system is
1
L
[(
X
j
−
(
e
)
=
X
i
)
I
+
(
Y
j
−
Y
i
)
J
+
(
Z
j
−
Z
i
)
K
]
(3.53)
or
(
e
)
z
K
(3.54)
Thus, the element displacements are expressed in components in the 3-D global
system as
=
cos
x
I
+
cos
y
J
+
cos
u
(
e
1
U
(
e
)
1
U
(
e
)
2
U
(
e
)
3
=
cos
x
+
cos
y
+
cos
z
(3.55)
u
(
e
2
U
(
e
)
4
U
(
e
)
5
U
(
e
)
6
=
cos
x
+
cos
y
+
cos
z
(3.56)
Here, we use the notation that element displacements 1 and 4 are in the global
X
direction, displacements 2 and 5 are in the global
Y
direction, and element
displacements 3 and 6 are in the global
Z
direction.
U
3
j
1
(
e
)
U
3
j
2
U
3
j
j
U
3
i
1
Y
X
i
Y
U
3
i
2
Z
U
3
i
X
Z
Figure 3.7
Bar element in a 3-D global coordinate
system.
