Civil Engineering Reference
In-Depth Information
which is of the form
{
F
}=
[
K
ij
]
{
δ
}
where [
K
ij
] is the stiffness matrix for the beam.
It is possible to write Eq. (17.31) in an alternative form such that the elements of [
K
ij
]
are pure numbers. Thus
.
/
!
.
/
!
F
y
,
i
M
i
/
L
F
y
,
j
M
j
/
L
−
−
−
12
6
12
6
v
i
θ
i
L
v
j
θ
j
L
EI
L
3
−
6462
=
0
"
−
12
6
12
6
0
"
−
6264
This form of Eq. (17.31) is particularly useful in numerical calculations for an
assemblage of beams in which
EI
/
L
3
is constant.
Equation (17.31) is derived for a beam whose axis i
s a
ligned with the
x
axis so that
the stiffness matrix defined by Eq. (17.31) is actually [
K
ij
] the stiffness matrix referred
to a local coordinate system. If the beam is positioned in the
xy
plane with its axis
arbitrarily inclined to the
x
axis then the
x
and
y
axes form a global coordinate system
and it becomes necessary to transform Eq. (17.31) to allow f
or
this. The procedure
is similar to that for the truss member of Section 17.1 in that [
K
ij
] must be expanded
to allow for the fact that nodal displacements
w
i
and
w
j
, which are irrelevant for the
beam in local coordinates, have components
w
i
,
v
i
and
w
j
,
v
j
in global coordinates.
Thus
w
i
v
i
θ
i
w
j
v
j
θ
j
00 0 00 0
0
/
L
3
6
/
L
2
12
/
L
3
6
/
L
2
−
−
−
0
6
/
L
2
4
/
L
0
/
L
2
2
/
L
00 0 00 0
0
−
0
[
K
ij
]
=
EI
(17.32)
12
/
L
3
6
/
L
2
0
/
L
3
6
/
L
2
−
6
/
L
2
2
/
L
0
/
L
2
0
−
4
/
L
We may deduce the transformation matrix [
T
] from Eq. (17.17) if we remember that
although
w
and
v
transform in exactly the same way as in the case of a truss member
the rotations
θ
remain the same in either local or global coordinates.
Hence
λµ
0 000
−
µλ
0 000
001 000
000
λµ
0
000
[
T
]
=
(17.33)
µλ
0
000 001
−
