Civil Engineering Reference
In-Depth Information
Fig. 7.12
Four element simply supported beam
Let us consider the simple case of
N
=
4
as illustrated in Fig. 7.12, and adopt the data given in
the table below.
B
(m)
D
(m)
C
′
C
′
I
I
L
L
C
y
f
y
f
L
L
L
L
M
u
w
D
u
w
x
f
x
f
x
f
u
u
u
20
4
0.7
5
1.5
0.2
0.1
1/4
1/3
1/16
⎛
⎞
⎛
⎞
Δ
x
3
Δ
x
⎛
⎞
(
)
exp
⎜
⎟
exp
x
ˆ
exp
⎜
⎟
exp
4
x
ˆ
ρ
=
−
=
−
Δ
ρ
=
−
=
− Δ
⇒
and
⎜
⎟
uu
ww
⎜
y
f
⎟
4
⎜
y
f
⎟
⎝
⎠
L
L
⎝
⎠
⎝
⎠
u
w
x
xx
ˆ
ˆ
and
ˆ
x
xL
/
where
Δ=
−
=
. In this case there are only two separations:
p
k
(
)
(
)
⎧
ρ
ρ
L
/ 4 . 2
=
⎧
ρ
ρ
L
/ 4 . 9
=
1
1
⎪
⎪
uu
ww
uu
ww
x
ˆ
x
ˆ
Δ=
⇒
⎨
and
Δ=
⇒
⎨
(
)
(
)
4
L
/ 4 . 6
2
L
/ 4 . 4
=
=
⎪
⎩
⎪
⎩
ˆ
ˆ
ˆ
⎡
⎤
Cov
Cov
Cov
RR
RR
RR
22
23
24
2
⎢
⎥
2
⎛
VBL
⎞
⎢
ρ
ˆ
ˆ
⎥
Then
Cov
=
Cov
Cov
⎜
⎟
RR
⎜
⎟
⎢
RR
RR
24
33
34
⎥
⋅
⎝
⎠
⎢
⎥
ˆ
Sym
.
Cov
⎢
⎥
⎣
RR
⎦
44
where
0
0
0
0
0
0
⎡
⎤
⎢
⎥
0
0.003136
0
0
0
0
⎢
⎥
⎢
0
0
0.25
1.5
0
0
⎥
−
ˆ
ˆ
ˆ
Cov
=
Cov
=
Cov
=
⎢
⎥
RR
RR
RR
4
22
33
4
0
0
−
1.5
9
0
0
⎢
⎥
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0
0
0
0
0
⎢
⎥
⎣
⎦
0
0
0
0
0
0
⎡
⎤
⎢
⎥
0
0.002572
0
0
0
0
⎢
⎥
⎢
0
0
0.09
0.54
0
0
⎥
−
ˆ
ˆ
Cov
=
Cov
=
⎢
⎥
RR
RR
23
34
0
0
−
0.54
3.24
0
0
⎢
⎥
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0
0
0
0
0
⎢
⎥
⎣
⎦
