Civil Engineering Reference
In-Depth Information
this equation by
cos
,
we obtain the equilibrium equation for the node in the
X
direction of the global coordinate system. Similarly, multiplying by
sin
, the
Y
direction global equilibrium equation is obtained. Exactly the same procedure
with the second equation expresses equilibrium of element node 2 in the global
coordinate system. The same desired operations described are obtained if we
premultiply both sides of Equation 3.24 by
[
R
]
T
, the transpose of the transfor-
mation matrix; that is,
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
f
(
e
)
1
cos
cos
0
f
(
e
)
1
f
(
e
)
2
[
R
]
T
k
e
[
R
]
f
(
e
1
sin
−
k
e
sin
0
=
=
−
k
e
k
e
0
cos
f
(
e
2
cos
0
sin
f
(
e
)
2
sin
(3.25)
Clearly, the right-hand side of Equation 3.25 represents the components of the
element forces in the global coordinate system, so we now have
U
(
e
1
U
(
e
2
U
(
e
3
U
(
e
4
F
(
e
1
F
(
e
2
F
(
e
3
F
(
e
4
[
R
]
T
k
e
[
R
]
−
k
e
(3.26)
=
−
k
e
k
e
Matrix Equation 3.26 represents the equilibrium equations for element nodes 1
and 2, expressed in the global coordinate system. Comparing this result with
Equation 3.18, the element stiffness matrix in the global coordinate frame is seen
to be given by
[
R
]
T
k
e
[
R
]
K
(
e
)
=
−
k
e
(3.27)
−
k
e
k
e
Introducing the notation
c
=
cos
,
s
=
sin
and performing the matrix multi-
plications on the right-hand side of Equation 3.27 results in
c
2
c
2
sc
−
−
sc
K
(
e
)
=
s
2
s
2
sc
−
sc
−
(3.28)
k
e
c
2
c
2
−
−
sc
sc
s
2
s
2
−
sc
−
sc
where
k
e
=
AE
/
L
is the characteristic axial stiffness of the element.
Examination of Equation 3.28 shows that the symmetry of the element stiff-
ness matrix is preserved in the transformation to global coordinates. In addition,
although not obvious by inspection, it can be shown that the determinant is zero,
indicating that, after transformation, the stiffness matrix remains singular. This is
to be expected, since as previously discussed, rigid body motion of the element
is possible in the absence of specified constraints.
