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The boundary conditions are
V
a
=
V
d
=
0. These constraints reduce the stiffness (equilib-
rium) relations to
K
bb
V
b
V
c
P
b
P
c
K
bc
=
(4)
K
cb
K
cc
where
P
Xb
P
Zb
P
Xc
P
Zc
P
b
=
P
c
=
α
Substitution of the appropriate values of
in the element stiffness equations gives
1
+
√
2
√
2
/
4
/
4
EA
L
K
bb
=
√
2
+
√
2
/
41
/
4
00
0
EA
L
=
=
K
bc
K
cb
−
1
1
√
2
√
2
+
/
4
−
/
4
EA
L
K
cc
=
√
2
√
2
(5)
−
/
41
+
/
4
These are the same results obtained in Chapter 3, Example 3.5.
5.3.11
Frames
A
frame
or rigid frame is composed of beam elements in which both transverse (bending)
and axial (extension and torsion) effects occur. This differs from the truss of Section 5.3.10,
where each bar element could only extend or compress.
Element Coordinate Transformations
Two-dimensional frames loaded in their planes are to be considered in this section. The
frame is assumed to lie in the
XZ
plane. Local and global force and displacement compo-
nents in the local
xz
and global
XZ
coordinates systems are shown in Fig. 5.20.
The transformation relations to change from local to global components can be established
using the diagrams of Fig. 5.20. In global coordinates, the forces and displacements at the
a
end of the
i
th element of a plane frame are denoted as (Fig. 5.20c)
i
i
F
X
F
Z
M
u
X
u
Z
θ
p
i
a
=
v
i
a
=
(5.88)
a
a
where
M
a
=
i
Ya
. Although according to Eq. (5.17) lower case letters should
be employed to designate element forces referred to global coordinates, it is customary to
use upper case letters for element forces of a framework. The corresponding forces referred
to the local coordinates are designated in the usual fashion, e.g.,
N
xa
. The transformations
from global to local coordinates for the forces at the end of a beam element are readily
established. From Fig. 5.20d, ignoring the superscript
i
,
M
Ya
,
and
i
a
θ
=
θ
N
xa
=
N
a
=
F
Xa
cos
α
−
F
Za
sin
α
(5.89a)
and
V
za
=
V
a
=
α
+
α
F
Xa
sin
F
Za
cos
(5.89b)
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