Civil Engineering Reference
In-Depth Information
ˆ
⎡
2
⎤
J
(
)
⎡
xx
ˆ
ˆ
⎤
ρ
−
M
xx
⎢
⎥
ww
n
m
NN
⎢
⎥
1
⎢
ˆ
⎥
(
)
2
∑∑
(
)
(
)
J
=
ψψ
x
ˆ
⋅
x
ˆ
⋅
ρ
ρ
x
ˆ
−
x
ˆ
⎢
⎥
MM
n
m
ww
n
m
⎢
⎥
2
yy
N
⎢
⎥
nm
==
11
(
)
⎢
⎥
xx
ˆ
ˆ
ˆ
−
2
⎢
⎥
J
⎣
uu
n
m
⎦
⎢
⎥
⎣
M
zz
⎦
and it is just a matter of choosing
N
sufficiently large. Let us assume that the position of the bridge
is at
z
50 m
and that the relevant length scales of the
u
and
w
components are given by (see
=
f
Eq 3.36):
0.3
x
f
(
)
y
x
y
x
f
f
f
f
L
=
100
z
/10
=
162 m
,
LL
=
/ 354 m
=
and
LL
=
/16
=
10 m
,
u
f
u
u
w
u
such that
y
f
(
)
(
)
x
ˆ
exp
cx
ˆ
cLL
ρ
Δ=
−⋅ Δ
where
=
uu
u
u
u
y
f
(
)
(
)
x
ˆ
exp
c
x
ˆ
cLL
ρ
Δ=
−⋅ Δ where
=
ww
w
w
w
x
f
L
L
N
5
Let us for simplicity set
=
and
=
(which in general will be far too small, as shown in
u
c
= .
The position vector and the influence function are given by
c
3
16
Fig. 7.7). Thus,
=
,
0.9
T
[
]
ˆ
x
=
0.1
0.3
0.5
0.7
0.1
T
()
[
]
ˆ
ψ
x
=
0.1
0.3
0.5
0.3
(
)
(
)
ˆ
ˆ
The influence function multiplications
ψψ
x
⋅
x
are then given by
n
m
()
x
ˆ
n
ψ
0.1
0.3
0.5
0.3
0.1
0.1
0.01
0.03
0.05
0.03
0.01
0.3
0.03
0.09
0.15
0.09
0.03
(
)
0.5
0.05
0.15
0.25
0.15
0.05
x
ˆ
m
ψ
0.3
0.03
0.09
0.15
0.09
0.03
0.1
0.01
0.03
0.05
0.03
0.01
while the covariance coefficients associated with the
u
and
w
components are given by:
(
)
x
ˆ
x
ˆ
n
ρ
Δ
:
uu
0.1
0.3
0.5
0.7
0.9
0.1
1
0.549
0.301
0.165
0.091
ˆ
m
0.3
0.549
1
0.549
0.301
0.165
0.5
0.301
0.549
1
0.549
0.301
x
0.7
0.165
0.301
0.549
1
0.549
0.9
0.091
0.165
0.301
0.549
1
