Environmental Engineering Reference
In-Depth Information
Unit rotation of circular ring beam:
w
R0
¼
m
R0
p
2
R
=ð
E
c
I
R
Þ¼þ
1
2
25
2
:
=ð
30
1
:
065
Þ¼þ
0
:
1585
%
Deformation modulus for force method:
d
10
¼þ
1
w
R0
h
R
=
2
¼þ
0
:
1585
2
:
20
=
2
¼þ
0
:
1743 mm
d
20
¼
1
w
R0
¼
0
:
1585
%
Statically indeterminate effects (X
1
,X
2
, see Figure 4.47):
Unit bending moments in circular ring beam:
M
R1
¼þ
X
1
h
R
=
2
¼þ
1
½
MN
=
m
2
:
20
=
2
¼þ
1
:
10 MNm
=
m
M
R2
¼
X
2
¼
1
:
00 MNm
=
m
Deformation modulus for force method (cylindrical shell, see [64]):
d
11
¼
M
R1
r
R
h
R
=
2
=ð
E
c
I
R
Þþ
2
X
1
r
u
k
u
=ð
E
c
t
u
Þ
¼þ
1
:
10
2
:
25
2
:
20
=
2
=ð
30
1
:
065
Þþ
2
1
:
00
2
:
675
3
:
601
=ð
30
0
:
35
Þ
¼þ
0
:
0852
þ
1
:
8350
¼þ
1
:
920 mm
u
d
12
¼
M
R2
r
R
h
R
=
2
=ð
E
c
I
R
Þþ
2
X
2
k
=ð
E
c
t
u
Þ
601
2
¼
1
:
00
2
:
25
2
:
20
=
2
=ð
30
1
:
065
Þþ
2
1
:
00
3
:
=ð
30
0
:
35
Þ
393 mm
d
21
¼
M
R1
r
R
=ð
E
c
I
R
Þþ
2
X
1
k
¼
0
:
0775
þ
2
:
4705
¼þ
2
:
u
=ð
E
c
t
u
Þ
601
2
¼
1
:
10
2
:
25
=ð
30
1
:
065
Þþ
2
1
:
00
3
:
=ð
30
0
:
35
Þ
¼
0
:
0775
þ
2
:
4705
¼þ
2
:
393
%
d
22
¼
M
R2
r
R
=ð
E
c
I
R
Þþ
4
X
2
k
u
=
r
u
=ð
E
c
t
u
Þ
601
3
¼þ
1
:
00
2
:
25
=ð
30
1
:
065
Þþ
4
1
:
00
3
:
=
2
:
675
=ð
30
0
:
35
Þ
¼þ
0
:
0704
þ
6
:
6523
¼þ
6
:
723
%
Statically indeterminate analysis:
Determinants for deformation modulus:
393
2
813
10
3
mm
D
¼ d
11
d
22
d
21
d
12
¼
1
:
920
6
:
723
2
:
¼
7
:
Statically indeterminate internal forces:
X
1
¼
1
D
¼ð
0
:
1743
6
:
723
0
:
1585
2
:
393
Þ=
7
:
183
¼
1
:
00
½
MN
=
m
ðd
10
d
22
þ d
20
d
12
Þ=
:
5512
=
7
:
183
¼
0
:
2160 MN
=
m
X
2
¼
1
:
00
½
MNm
=
m
ðd
11
d
20
þ d
21
d
10
Þ=
D
¼ðþ
1
:
920
0
:
1585
þ
2
:
393
0
:
1743
Þ=
7
:
183
¼þ
0
:
7215
=
7
:
183
¼þ
0
:
1005 MNm
=
m
