Civil Engineering Reference
In-Depth Information
Fig. 7.6
Simply supported beam type of bridge
Introducing the non-dimensional spanwise coordinate
ˆ
x
xL
/
=
and the function
x
ˆ
when
x
ˆ
1 / 2
≤
⎧
()
x
ˆ
ψ
=
⎨
1
x
ˆ
when 1/2<
x
ˆ
1
−
≤
⎩
M
M
M
it is readily seen that the influence functions for
,
and
at midspan are given by
x
y
z
L
Gx
L
()
()
()
()
()
()
M
Gx
ˆ
x
ˆ
ˆ
x
ˆ
Gx
ˆ
x
ˆ
=−
ψ
=⋅
ψ
=−
⋅
ψ
M
y
M
x
2
2
Thus
2
⎡
⎤
⎛
⎞
⎢
⎥
⎜
σ
⎟
M
xx
⎢
⎥
⎜
⎟
1
2
⎢
⎥
⎜
22
⎟
VBLC I
′
ρ
⎜
⎟
⎢
Mw
⎥
⎝
⎠
⎢
⎥
ˆ
⎡
2
⎤
2
J
⎢
⎥
(
)
⎛
⎞
⎡
x
ˆ
⎤
ρ
Δ
MM
⎢
x
x
⎥
⎢
σ
⎥
ww
11
⎜
⎟
⎢
⎥
M
yy
⎢
ˆ
⎥
() ()
(
)
2
∫∫
x
ˆ
x
ˆ
x
ˆ
dx dx
ˆ
ˆ
J
⎢
⎥
=
ψψ ρ
⋅
⋅
Δ
=
⎜
⎟
⎢
⎥
1
1
2
ww
1
2
⎢
M
M
⎥
y
y
⎢
⎥
⎜
22
⎟
VBLCI
′
(
)
ρ
⎢
⎥
⎜
⎟
00
x
ˆ
ρ
Δ
⎢
⎥
Lw
ˆ
⎢
⎥
⎣
⎦
2
4
uu
⎝
⎠
J
⎢
⎥
⎣
M
M
⎦
⎢
⎥
z
z
2
⎢
⎥
⎛
⎞
⎢
⎜
σ
⎟
⎥
M
zz
⎢
⎥
⎜
⎟
1
2
⎢
⎜
22
⎟
⎥
ρ
VDLCI
⎜
⎟
Du
⎢
⎥
⎝
⎠
⎣
⎦
Cov
and since
=−
σ
⋅
σ
it will suffice to calculate the joint acceptance functions
MM
MM
MM
x
y
x
x
y
y
on the right hand side of the equation above. Let us divide the span into
N
segments and calculate
the load effect at the midpoint of each of these segments. Then the volume integral is replaced by a
double summation (and
dx
ˆ
1/
N
=
), i.e.
