Civil Engineering Reference
In-Depth Information
In Equation 3.18,
[
K
(
e
)
]
represents the element stiffness matrix in the global co-
ordinate system, the vector
{
F
(
e
)
on the right-hand side contains the element
nodal force components in the global frame, displacements
U
(
e
1
and
U
(
e
3
are
parallel to the global
X
axis, while
U
(
e
2
and
U
(
e
4
are parallel to the global
Y
axis.
The relation between the element axial displacements in the element coordinate
system and the element displacements in global coordinates (Equation 3.4) is
u
(
e
1
}
U
(
e
1
cos
U
(
e
2
sin
(3.19)
=
+
u
(
e
2
U
(
e
3
cos
U
(
e
4
sin
(3.20)
=
+
which can be written in matrix form as
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
u
(
e
1
u
(
e
2
cos
sin
0
0
(3.21)
=
=
[
R
]
0
0
cos
sin
where
cos
sin
0
0
(3.22)
[
R
]
=
0
0
cos
sin
is the transformation matrix of element
axial
displacements to global displace-
ments. (Again note that the element nodal displacements in the direction perpen-
dicular to the element axis,
v
1
and
v
2
, are not considered in the stiffness matrix
development; these displacements come into play in dynamic analyses in
Chapter 10.) Substituting Equation 3.22 into Equation 3.17 yields
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
f
(
e
)
1
f
(
e
)
2
k
e
cos
−
k
e
sin
0
0
(3.23)
=
−
k
e
k
e
0
0
cos
sin
or
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
f
(
e
)
1
f
(
e
)
2
k
e
[
R
]
−
k
e
=
(3.24)
−
k
e
k
e
While we have transformed the equilibrium equations from element displace-
ments to global displacements as the unknowns, the equations are still expressed
in the element coordinate system. The first of Equation 3.23 is the equilibrium
condition for element node 1 in the element coordinate system. If we multiply
