Civil Engineering Reference
In-Depth Information
Thus, the load vector in node
p
is obtained by adding up the contributions from all
adjoining elements. i.e.
2
2
⎛ ⎞
V
⎡
BL
⎤
⎛
V
⎞
ρ
ρ
⎛ ⎞
(
)
ˆ
ˆ
()
∑
∑
ˆ
ˆ
t
R
=
R
=
θ B ψ v
⋅
⋅
=
⋅
Q v
⋅
(7.48)
⎜ ⎟
⎜
⎟
⎢
⎥
⎜ ⎟
p
p
⎜ ⎟
q
p
⎜
⎟
m
2
2
2
⎝ ⎠
m
⎣
⎦
⎝ ⎠
⎝
⎠
m
m
m
p
p
where
BL
⎛
⎞
ˆ
ˆ
∑
Q
=
⋅
θ B ψ
(7.49)
⋅
⋅
⎜
⎟
p
q
2
⎝
⎠
m
m
The total system load vector is then given by
T
()
RRRR
t
=
⎣
⎡
""
(7.50)
⎤
⎦
1
p
N
where
N
is the total number of nodes. Since the content of this load vector is
considered quasi-static the relationship
()
()
()
()
Kr R
holds, and because
t
t
ut
and
wt
⋅
=
()
()
are both zero mean variables then
t
r
are also zero mean variables.
Thus, it is seen from to Eqs. 7.6 - 7.9 that the fluctuating background quasi-static part of
the element force vector
t
as well as
R
()
F
m
t
is given by
F
F
F
⎡⎤
⎢⎥
⎢⎥
⎢⎥
1
2
{
}
()
3
()
()
1
()
⎡
−
⎤
t
t
t
t
F
=
=
k
⋅
d
=
k
⋅
⎡
A r
⋅
⎤
=
k
⋅
A
⋅
K
⋅
R
(7.51)
⎢⎥
⎣
⎦
m
m
m
m
m
m
m
⎣
⎦
F
F
F
⎢⎥
⎢⎥
⎢⎥
⎢
⎣⎦
4
5
6
m
The covariance matrix between cross sectional force components
2
1
⎡
⎤
σ
Cov
Cov
Cov
Cov
Cov
F
F F
F F
F F
F F
F F
1 2
1 3
1 4
1 5
1 6
⎢
⎥
⎢
2
⎥
Cov
Cov
Cov
Cov
σ
F
F F
F F
F F
F F
⎢
2
2 3
2 4
2 5
2 6
⎥
⎢
⎥
2
Cov
Cov
Cov
σ
F
F F
F F
F F
⎢
⎥
3
3 4
3 5
3 6
Cov
=
⎢
(7.52)
F
mm
⎥
2
Cov
Cov
σ
⎢
F
F F
F F
⎥
4
4 5
4
6
⎢
⎥
2
5
Sym
.
σ
Cov
⎢
F
F F
⎥
5 6
⎢
⎥
2
6
σ
⎢
⎥
⎣
F
⎦
