Civil Engineering Reference
In-Depth Information
where
(
)
T
(
)
T
x
0
x
0
T β
⋅
=
⋅
β
=
⋅
T
=
i
r
i
r
6
6
4
⎡
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎛
⎞
π
EI
⎛
⎞
π
⎛⎞
π
⎛⎞
π
2
2
(
)
w
⎢
a
EI
a a
EI EI
−
a a
EI
GI
+
00
⎜
⎟
⎜⎟
⎜⎟
⎜⎟
⎜
y
z
y
z
z
y
y
θ
z
t
⎟
2
L
L
L
L
EI
⎢
⎝⎠
⎝⎠
⎝⎠
⎝
⎠
⎢
6
4
⎛
2
⎞
⎢
π
π
π
⎛⎞
2
⎛⎞
(
)
2
w
a
EI
a a
EI
GI
00
−
+
⎜
⎟
⎢
⎜⎟
⎜⎟
⎜
z
y
z
y
t
θ
⎟
L
L
2
L
⎝⎠
⎝⎠
⎝
⎠
⎢
⎢
2
2
⎛
⎞
π
EI
⎛⎞
π
⎢
2
w
a
GI
+
00
⎜
⎟
⎜⎟
⎢
θ
⎜
t
⎟
2
L
L
⎝⎠
⎝
⎠
⎢
⎢
Sym
.
0
0
0
⎢
⎥
⎦
⎣
x
0
V
,
V
and
M
As could be expected, at
=
it is only
that applies, and thus
r
x
3
⎡
⎤
⎛⎞
π
⎢
a
EI
⎥
⎜
⎝⎠
y
z
L
⎢
⎥
1
⎡
σ
⎤
V
yy
⎢
⎥
2
⎡
⎤
2
⎢
⎥
3
3
BV
J
⎛ ⎞
ρ
πω
π
⎛⎞
⎢
⎥
ˆ
⎢
i
⎥
a
EI
⎢
σ
⎥
=
⋅
⋅
⋅
⋅
⎜ ⎟
⎜⎟
VV
ii
(
) (
)
⎢
z
y
⎥
zz
4
mB
⎢
⎥
L
ω
⎢
⎥
1
−
κ
⋅
ζ
−
ζ
⎝⎠
⎝ ⎠
i
i
ae
i
ae
⎢
⎥
⎣
⎦
i
i
σ
⎢
⎥
⎣
M
xx
⎦
2
⎢
2
⎥
⎛
⎞
π
EI
⎛⎞
π
w
a
GI
+
⎢
⎜
⎟
⎥
⎜⎟
⎜
θ
t
⎟
2
L
L
⎢
⎝⎠
⎥
⎝
⎠
⎣
⎦
1
1
r
xL
/ 2
and
ρ
=
,
ρ
=
ρ
= −
. At
=
V
yz
VM
VM
yx
zx
S
πω
⋅
L
L
L
⎛
⎞
i
ˆˆ
⎛
⎞ ⎛
⎞
QQ
T
T
x
ii
x
x
Cov
= =
⋅
T β
⋅
=⋅
β
=⋅
T
⎜
⎟
⎜
⎟ ⎜
⎟
FF
r
(
) (
)
i
r
i
r
R
i
2
2
2
⎝
⎠
41
−
κ
⋅
ζ
−
ζ
⎝
⎠ ⎝
⎠
ae
i
ae
i
i
where
000
0
0
⎡
⎤
⎢
⎥
00
0
0
⎢
⎥
⎢
0
0
0
⎥
⎢
⎥
L
L
⎛
⎞ ⎛
⎞
T
T
4
6
T β
⋅
x
=
⋅
β
x
=
⋅
T
=
⎢
π
π
⎥
⎛⎞
⎛⎞
⎜
⎟ ⎜
⎟
(
)
2
i
r
i
r
2
Sym
.
a
EI
a a
EI EI
2
2
−
⎝
⎠ ⎝
⎠
⎢
⎥
⎜⎟
⎜⎟
z
y
z
y
y
z
L
L
⎝⎠
⎝⎠
⎢
⎥
⎢
⎥
4
⎛⎞
π
2
⎢
(
)
⎥
2
a
EI
⎜⎟
y
z
⎢
L
⎥
⎝⎠
⎣
⎦
As could be expected, at
r
xL
=
/ 2
it is only
M
and
M
that applies, and thus
y
z
⎡
2
⎤
π
1
⎛⎞
a
EI
⎢
⎥
2
⎡
⎤
⎜⎟
σ
2
z
y
⎡
⎤
3
L
BV
J
mB
⎛ ⎞
πω
M
yy
ρ
⎢
⎝⎠
⎥
ˆ
⎢
⎥
i
⎢
⎥
=
⋅
⋅
⋅
⋅
⎜ ⎟
(
) (
)
⎢
⎥
ii
4
ω
⎢
⎥
2
⎢
σ
⎥
1
−
κ
⋅
ζ
−
ζ
⎝ ⎠
⎛⎞
π
⎣
MM
⎦
i
i
⎢
⎥
ae
i
ae
zz
⎣
⎦
i
i
a
EI
⎜⎟
⎢
y
z
⎥
L
⎣
⎝⎠
⎦
