Civil Engineering Reference
In-Depth Information
The contribution from element
m
to concentrated loads in node
p
is then given by
(see Fig. 7.9.a)
Q
⎡⎤
y
2
L
⎛ ⎞
V
BL
⎢⎥
ρ
⎛ ⎞
(
)
ˆ
()
(
)
m
ˆ
Q
t
Q
q
xt
,
B ψ v
=
=
⋅
=
⋅
⋅
⋅
⋅
(7.44)
⎜ ⎟
⎢⎥
⎜ ⎟
p
z
m
q
m
p
⎜ ⎟
⎝ ⎠
m
2
2
2
⎝ ⎠
⎢
⎣⎦
m
Q
θ
p
p
m
where
L
is the element length, and where it has been assumed that the nodal
discretisation is such that all wind velocity properties with sufficient accuracy may be
allotted to node
p
, and that they are constants along the span of the element. (This is a
simplification that is not mandatory, but otherwise, an element integration scheme has to
be adopted.) Comparing the definition of nodal loads shown in Fig. 7.3 to the element
load components shown in Fig. 7.9, it is seen that the contribution from element
m
to
the load vector in node
p
is defined by (see Fig. 7.9.b and c)
R
R
⎡⎤
⎢⎥
⎧
⎡
1
T
0
QQ Q
0
0
⎤
for a horizontal element
⎢⎥
2
−
⎣
⎦
⎪
y
z
θ
⎢⎥
⎪
R
R
3
R
=
=
(7.45)
⎢⎥
⎨
⎢⎥
⎪
⎡
p
m
4
T
⎢⎥
⎪
⎣
−
QQ
0
0
0
−
Q
⎤
for a vertical element
R
R
⎦
⎩
z
y
θ
5
⎢
⎢
⎣⎦
6
p
m
and thus,
⎛ ⎞
⎛ ⎞
2
V
BL
ρ
(
)
ˆ
()
⎡
⎤
ˆ
R
t
=⋅
θ Q
=
⋅
θ B ψ v
⋅
⋅
⋅
(7.46)
⎜ ⎟
⎜ ⎟
p
m
p
⎜ ⎟
m
⎣
q
m
p
⎦
m
m
2
2
m
⎝ ⎠
⎝ ⎠
m
p
where indices
m
and
p
indicate quantities associated with element
m
at node
p
, and
where
T
⎧
⎡
010 0 00
001 0 00 for a ho izontal element
000 100
⎤
⎪
⎢
⎥
⎪
⎢
⎥
⎪
⎢
⎥
−
⎪⎣
⎦
θ
=
(7.47)
⎨
⎪ ⎡
m
T
010000
1
⎤
⎪
⎢
⎥
0
0
0
0
0
for a vertical element
−
⎪ ⎢
⎥
⎪
⎢
0 0000 1
⎥
−
⎩
⎣
⎦
