Civil Engineering Reference
In-Depth Information
Equations 3.9 through 3.14 are equivalent to the matrix form
k
(1)
c
2
1
k
(1)
s
1
c
1
0
0
−
k
(1)
c
2
1
−
k
(1)
s
1
c
1
k
(1)
s
1
c
1
k
(1)
s
2
1
0
0
−
k
(1)
s
1
c
1
−
k
(1)
s
2
1
U
1
U
2
U
3
U
4
U
5
U
6
F
1
F
2
F
3
F
4
F
5
F
6
0
0
k
(2)
c
2
2
k
(2)
s
2
c
2
−
k
(2)
c
2
2
−
k
(2)
s
2
c
2
0
0
k
(2)
s
2
c
2
k
(2)
s
2
2
−
k
(2)
s
2
c
2
−
k
(2)
s
2
2
(3.15)
=
k
(1)
c
2
k
(1)
s
1
c
1
+
k
(2)
s
2
c
2
1
+
k
(2)
c
2
−
k
(1)
c
2
12
−
k
1
s
1
c
1
−
k
(2)
c
2
2
−
k
(2)
s
2
c
2
2
k
(1)
s
1
c
1
+
k
(1)
s
2
1
+
k
(2)
s
2
−
k
(1)
s
2
−
k
(2)
s
2
c
2
−
k
(2)
s
2
−
k
1
s
1
c
1
1
2
k
(2)
s
2
c
2
2
The six algebraic equations represented by matrix Equation 3.15 express the
complete set of equilibrium conditions for the two-element truss. Equation 3.15
is of the form
(3.16)
where
[
K
]
is the global stiffness matrix,
{
U
}
is the vector of nodal displace-
ments, and
{
F
}
is the vector of applied nodal forces. We observe that the global
stiffness matrix is a 6
[
K
]
{
U
}={
F
}
6 symmetric matrix corresponding to six possible global
displacements. Application of boundary conditions and solution of the equations
are deferred at this time, pending further discussion.
×
3.3 ELEMENT TRANSFORMATION
Formulation of global finite element equations by direct application of equilib-
rium conditions, as in the previous section, proves to be quite cumbersome ex-
cept for the very simplest of models. By writing the nodal equilibrium equations
in the global coordinate system and introducing the displacement formulation,
the procedure of the previous section implicitly transformed the individual ele-
ment characteristics (the stiffness matrix) to the global system. A direct method
for transforming the stiffness characteristics on an element-by-element basis
is now developed in preparation for use in the direct assembly procedure of the
following section.
Recalling the bar element equations expressed in the element frame as
u
(
e
1
u
(
e
2
u
(
e
1
u
(
e
2
f
(
e
)
1
f
(
e
)
2
1
k
e
AE
L
−
1
−
k
e
=
=
(3.17)
−
11
−
k
e
k
e
the present objective is to transform these equilibrium equations into the global
coordinate system in the form
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
F
(
e
1
F
(
e
2
F
(
e
3
F
(
e
4
K
(
e
)
=
(3.18)
