Civil Engineering Reference
In-Depth Information
we may express Eq. (17.25) for nodes
i
and
j
in matrix form as
.
/
!
.
/
!
F
x
,
i
F
y
,
i
F
z
,
i
F
x
,
j
F
y
,
j
F
z
,
j
F
x
,
i
F
y
,
i
F
z
,
i
F
x
,
j
F
y
,
j
F
z
,
j
λ
x
µ
x
ν
x
000
λ
y
µ
y
ν
y
000
λ
z
µ
z
ν
z
000
000
λ
x
=
(17.27)
0
"
µ
x
ν
x
0
"
000
λ
y
µ
y
ν
y
000
λ
µ
ν
z
¯
¯
z
¯
z
or in abbreviated form
{
F
}=
[
T
]
{
F
}
The derivation of [
K
ij
] for a member of a space frame proceeds on identical lines to
that for the plane frame member. Thus, as before
[
T
]
T
[
K
ij
][
T
]
[
K
ij
]
=
Substituting for [
T
] and [
K
ij
] from Eqs (17.27) and (17.24) gives
λ
2
¯
λ
2
¯
λ
x
µ
x
λ
x
ν
x
−
−
λ
x
µ
x
−
λ
x
ν
x
x
x
µ
2
¯
µ
2
¯
λ
x
µ
x
µ
x
ν
x
−
λ
x
µ
x
−
−
µ
x
ν
x
x
x
ν
2
¯
ν
2
¯
AE
L
λ
x
ν
x
µ
x
ν
x
−
λ
x
ν
x
−
µ
x
ν
x
−
x
x
[
K
ij
]
=
(17.28)
λ
2
¯
λ
2
¯
−
−
λ
x
µ
x
−
λ
x
ν
x
λ
x
µ
x
λ
x
ν
x
x
x
µ
2
¯
µ
2
¯
−
λ
x
µ
x
−
−
µ
x
ν
x
λ
x
µ
x
µ
x
ν
x
x
x
ν
2
¯
ν
2
¯
−
−
−
λ
x
ν
x
µ
x
ν
x
λ
x
ν
x
µ
x
ν
x
x
x
All the suffixes in Eq. (17.28) are
¯
x
so that we may rewrite the equation in simpler
form, namely
.
λ
2
SYM
λµ µ
2
.
λν µν ν
2
.
....................................
AE
L
[
K
ij
]
=
(17.29)
.
λ
2
2
−
−
λµ
−
λν
.
.λµµ
2
µ
2
−
λµ
−
−
µν
.
.λνµν ν
2
ν
2
−
λν
−
µν
−
where
λ
,
µ
and
v
are the direction cosines between the
x
,
y
,
z
and
¯
x
axes, respectively.
The complete stiffnessmatrix for a space frame is assembled from themember stiffness
matrices in a similar manner to that for the plane frame and the solution completed
as before.
