Civil Engineering Reference
In-Depth Information
(
)
x
ˆ
x
ˆ
n
ρ
Δ
:
ww
0.1
0.3
0.5
0.7
0.9
0.1
1
0.0408
0.0017
0.0001
≈ 0
ˆ
m
0.3
0.0408
1
0.0408
0.0017
0.0001
x
0.5
0.0017
0.0408
1
0.0408
0.0017
0.7
0.0001
0.0017
0.0408
1
0.0408
0.9
≈
0
0.0001
0.0017
0.0408
1
The inner products
() ( )
(
)
() ( )
(
)
x
ˆ
x
ˆ
x
ˆ
and
x
ˆ
x
ˆ
x
ˆ
are then:
ψψ ρ
⋅
⋅
Δ
ψψ ρ
⋅
⋅
Δ
n
m
uu
n
m
uu
n
1
2
3
4
5
m
2
() ( )
(
)
10
x
ˆ
x
ˆ
x
ˆ
⋅
ψψ ρ
⋅
⋅
Δ
n
m
uu
1
1
1.647
1.505
0.495
0.091
2
1.647
9
8.235
2.709
0.495
3
1.505
8.235
25
8.235
1.505
4
0.495
2.709
8.235
9
1.647
5
0.091
0.495
1.505
1.647
1
n
1
2
3
4
5
m
2
() ( )
(
)
10
x
ˆ
x
ˆ
x
ˆ
⋅
ψψ ρ
⋅
⋅
Δ
n
m
ww
1
1
0.1225
0.0085
0.0003
≈ 0
2
0.1225
9
0.612
0.0153
0.0003
3
0.0085
0.612
25
0.612
0.0085
4
0.0003
0.0153
0.612
9
0.1224
5
0.0003
0.0085
0.1224
1
≈ 0
The normalised joint acceptance functions are given by
1
⎡
55
⎤
1
2
ˆ
∑∑
() ( )
(
)
J
x
ˆ
x
ˆ
x
ˆ
0.2
=⋅
ψψ ρ
==
⋅
⋅
Δ ≈
⎢
⎥
MM
n
m
uu
zz
2
5
⎣
⎦
nm
11
1
⎡
55
⎤
1
2
ˆ
ˆ
∑∑
(
)
(
)
(
)
J
J
x
ˆ
x
ˆ
x
ˆ
0.14
=
=
⋅
ψψ ρ
==
⋅
⋅
Δ
≈
⎢
⎥
MM
MM
n
m
ww
xx
yy
2
5
⎣
⎦
nm
11
and thus
1
⎛
⎞
22
0.14
VBLC I
′
⎠
σ
≈
⋅
ρ
⎜
⎟
MM
M w
xx
2
⎝
1
⎛
⎞
22
0.07
VBLCI
′
⎠
σ
≈
⋅
⎜
ρ
⎟
MM
Lw
yy
2
⎝
1
⎛
⎞
22
0.2
VDLCI
⎠
σ
≈⋅
⎜
ρ
⎟
MM
Du
zz
2
⎝
