Civil Engineering Reference
In-Depth Information
(see Fig. 2.14 and Eq. 2.99, and assuming for simplicity that
x
Δ
is a constant), each
rendering a base moment contribution
()
( )
()
M
t
GxQt
=
⋅
(2.113)
k
M
k
k
such that the total bending moment at the base is
N
()
∑
(2.114)
()
M t
Mt
=
k
k
1
=
Its variance is then given by (see Eq. 2.100)
2
T
T
⎡
N
⎤
1
1
2
2
()
∑
()
lim
∫
M t
dt
lim
∫
M
t
dt
σ
=
⎡
⎤
=
⎢
⎥
⎣
⎦
M
k
T
T
T
T
→∞
→∞
⎣
⎦
k
=
1
0
0
T
1
[
] [
]
∫
=
lim
M
++ ++
.....
M
.....
M
⋅
M
++ ++
.....
M
.....
M
dt
1
k
N
1
k
N
T
T
→∞
0
T
NN
1
2
∑∑
∫
()
()
lim
M
tM tdt
⇒
σ
=
⎡
⋅
⎤
(2.115)
⎣
⎦
M
n
m
T
T
nm
→∞
==
11
0
As can be seen, the transition from a single summation to a double summation is
necessary to capture all the cross products. Introducing Eqs. 2.112 and 2.113, the
following is obtained:
T
NN
1
L
L
⎡
⎤
2
∑∑
∫
()( )
(
)
(
)
lim
Gxqxt
,
Gxqxt
,
t
σ
=
⋅
⎢
⎥
M
M
n
y
n
M
m
y
m
T
N
N
T
nm
⎣
⎦
→∞
==
11
0
2
⎧
T
⎫
NN
1
L
⎪
⎪
⎛
⎞
∑∑
()
( )
(
)
(
)
GxGx
lim
∫
qxt
,
qxt
,
t
=
⋅
⋅
⎡
⋅
⎤
⋅
⎨
⎬
⎜⎟
⎣
⎦
Mn
Mm
yn
ym
T
N
T
⎝⎠
→∞
⎪
⎪
⎩
⎭
nm
11
==
0
2
σ
⋅
L
⎛
⎞
⎧
NN
⎫
⎪
⎪
q
y
2
∑∑
()
( )
()
GxGx
x
⇒
σ
=
⎜
⎟
⋅
⋅
⋅
ρ
Δ
(2.116)
⎨
⎬
M
M
n
M
m
q
y
⎜
N
⎟
⎪
⎪
⎩
⎭
⎝
⎠
nm
11
==
(
)
x
where
ρ
Δ
is the covariance coefficient to the distributed load, and where
q
y
x
xx
Δ=
−
. The expression in Eq. 2.116 is equivalent to that which was obtained in
mn
Eq. 2.103.
