Civil Engineering Reference
In-Depth Information
U
(
e
)
v
(
e
)
u
(
e
)
u
(
e
)
j
j
j
U
(
e
)
After loading
U
(
e
)
Original
v
(
e
)
u
(
e
)
u
(
e
)
i
i
U
(
e
)
i
(a)
(b)
(c)
Figure 3.4
(a) Bar element at orientation . (b) General displacements of a bar element. (c) Bar element
global displacements.
specification of one or more displacement relations; hence, the displacement for-
mulation of the finite element method includes such situations.
To illustrate the transformation to displacements, Figure 3.4a depicts a bar
element connected at nodes
i
and
j
in a general position in a two-dimensional
(2-D) truss structure. As a result of external loading on the truss, we assume that
nodes
i
and
j
undergo 2-D displacement, as shown in Figure 3.4b. Since the ele-
ment must remain connected at the structural joints, the connected element nodes
must undergo the same 2-D displacements. This means that the element is sub-
jected not only to axial motion but rotation as well. To account for the rotation,
we added displacements
v
1
and
v
2
at element nodes 1 and 2, respectively, in the
direction perpendicular to the element
x
axis. Owing to the assumption of smooth
pin joint connections, the perpendicular displacements are not associated with
element stiffness; nevertheless, these displacements must exist so that the ele-
ment remains connected to the structural joint so that the element displacements
are compatible with (i.e., the same as) joint displacements. Although the element
undergoes a rotation in general, for computation purposes, orientation angle
is
assumed to be the same as in the undeformed structure. This is a result of the
assumption of small, elastic deformations and is used throughout the text.
To now relate element nodal displacements referred to the element coordi-
nates to element displacements in global coordinates, Figure 3.4c shows element
nodal displacements in the global system using the notation
U
(
e
1
=
element node 1 displacement in the global
X
direction
U
(
e
2
=
element node 1 displacement in the global
Y
direction
U
(
e
3
=
element node 2 displacement in the global
X
direction
U
(
e
4
=
element node 2 displacement in the global
Y
direction
