Civil Engineering Reference
In-Depth Information
*
⎡⎤
P
(
)
⎡
2/
DBC
⎤
⎡ ⎤
.7
12.5
3.75
1
2
⎢⎥
D
⎡⎤
*
3
*
3
H
C
′
V
⎛
⎞ ⎡ ⎤
⎡
V
31.25
9.375
⎢
⎥
⎢ ⎥
⎤
L
M
*
1
*
1
H
C
′
⇒
⎢⎥
=−
=−
and
⎢⎥
⎜ ⎟
⎢ ⎥ ⎢
=
=
⎢
⎥
⎢ ⎥
⎥
L
B
B
C
′
ω
ω
⎢⎥
⎢⎥
A
⎝ ⎠
⎣ ⎦
⎣
⎦
i
⎢
⎥
⎢ ⎥
⎣⎦
i
C
′
A
⎣ ⎦
⎢⎥
⎣
⎦
M
⎣⎦
Thus,
(
)
*
2
2
*
∫
BH
B A
dx
φφ
+
φ
z
3
3
θ
θ
2
2
*
2
2
*
aaBH
aB A
ρ
B
ρ
B
+
L
exp
z
θ
3
θ
3
0.136
κ
=
⋅
=
⋅
=
ae
i
(
)
2
m
2
m
2
2
2
2
2
2
aaa
∫
φφφ
dx
++
++
i
i
y
z
y
z
θ
θ
L
(
)
2*
2 *
*
∫
PH
Adx
φ
+
φ
+
φ φ
y
1
z
1
z
1
θ
2*
2 *
*
2
2
aP
aH
aaBA
B
B
+
+
ρ
ρ
L
exp
y
1
z
1
z
1
θ
0.0069
ζ
=
⋅
=
⋅
= −
(
)
ae
i
4
m
4
m
2
2
2
2
2
2
aaa
∫
φφφ
++
dx
++
i
i
y
z
y
z
θ
θ
L
x
0
The following is then obtained at
=
r
⎡
3
⎤
⎛⎞
π
a
EI
⎢
⎥
⎜
⎝⎠
y
z
L
⎢
⎥
1
⎡
σ
⎤
V
yy
⎢
⎥
⎡
⎤
2
⎢
⎥
2
3
3
⎛ ⎞
ρ
BV
J
πω
π
⎛⎞
⎢
⎥
ˆ
⎢
⎥
i
a
EI
⎢
σ
⎥
=
⋅
⋅
⋅
⋅
⎜ ⎟
⎜⎟
(
) (
)
VV
ii
⎢
z
y
⎥
zz
⎢
⎥
4
mB
ω
L
⎢
⎥
1
⎝⎠
−
κ
⋅
ζ
−
ζ
⎝ ⎠
i
i
ae
i
ae
⎢
⎥
⎣
⎦
i
i
σ
⎢
⎥
⎣
M
xx
⎦
2
⎢
⎛
2
⎞
⎥
EI
⎛⎞
π
π
w
a
GI
⎢
⎜
+
⎟
⎥
⎜⎟
⎜
t
⎟
θ
2
L
L
⎢
⎝⎠
⎥
⎝
⎠
⎣
⎦
(
)
x
x
x
0
⎡
σ
=
⎤
154 kN
⎡
⎤
VV
r
yy
⎢
⎥
⎢
⎥
(
)
0
3.8 kN
61 kNm
⇒
⎢
σ
σ
=
⎥
=
⎢
⎥
VV
r
zz
MM
⎢
⎥
⎢
⎥
(
)
0
=
⎣
⎦
⎢
⎥
⎣
r
⎦
xx
The following is obtained at
r
xL
=
/ 2
:
2
⎡
⎤
⎛⎞
π
1
a
EI
⎢
⎥
⎡
⎤
⎜⎟
2
2
⎡
σ
⎤
z
y
3
⎛ ⎞
L
ρ
BV
J
mB
πω
M
yy
⎢
⎝⎠
⎥
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
⎝⎠
⎣
⎦
(
)
⎡
xL
xL
/2
⎤
σ
=
612 kNm
24480 kNm
⎡
⎤
MM
r
yy
⎢
⎥
⇒
=
⎢
⎥
(
)
⎢
/2
⎥
σ
=
⎣
⎦
⎣
MM
r
⎦
zz
