8/17/2019 Analysis of Suspension Bridge
1/27
2. Theory of Elastic Catenary
- 1 -
2 .
2 . 1 -
&
i (0,0)
j (l x , l y)
P ( x,y,p)
x
y
F ix
F iy
F jx
F jy
l x
l y
l o , E, Ao , w
:
:
:
:
:
: C
: -
:
: -
w
dp
dx
dy
dθ
( 2 . 1 )
( ∵ sin ≈
, cos ≈
, sin
cos )
8/17/2019 Analysis of Suspension Bridge
2/27
2. Theory of Elastic Catenary
- 2 -
P
F ix
F iy
T
w, s
x
y
x(s)
y(s)
< F >
( 2 . 2 )
( 2 . 2 )
( )
∴
( 2 . 3 )
, ,
, ,
( )
①
E . ( 2 . 2 ) E . ( 2 . 1 ) ,
( 2 . 4 )
②
F E . ( 2 . 2 ) E . ( 2 . 3 ) ,
∴
8/17/2019 Analysis of Suspension Bridge
3/27
2. Theory of Elastic Catenary
- 3 -
E . ( 2 . 4 ) ,
( 2 . 5 )
E . ( 2 . 5 )
ln
A ,
ln
,
(2.6)
③
F E . ( 2 . 2 ) E . ( 2 . 3 ) ,
∴
E ( 2 . 4 ) ,
( 2 . 7 )
E . ( 2 . 7 )
A ,
,
(2.8)
8/17/2019 Analysis of Suspension Bridge
4/27
2. Theory of Elastic Catenary
- 4 -
④
E . ( 2 . 6 ) ,
(2.9)
E ( 2 . 8 ) ,
(2.10)
, G :
:
E . ( 2 . 4 ) ,
:
( 2 . 1 1 )
( 2 . 1 2 )
, , ( 2 . 1 3 )
, ( 2 . 1 4 )
, -
.
( E . ( 2 . 6 ) ) , ( E , ( 2 . 8 ) ) ( E . ( 2 . 4 ) ) .
8/17/2019 Analysis of Suspension Bridge
5/27
2. Theory of Elastic Catenary
- 5 -
⑤ ( )
F E ( 2 . 3 ) ,
∴
E ( 2 . 4 ) ,
( 2 . 1 5 )
E . ( 2 . 1 5 )
A ,
,
(2.16)
.
ln
( 2 . 1 7 )
, ,
,
8/17/2019 Analysis of Suspension Bridge
6/27
2. Theory of Elastic Catenary
- 6 -
⑥ - -
F E . ( 2 . 7 ) E . ( 2 . 5 )
( 2 . 1 8 )
- G - :
tan ( ) ( 2 . 1 9 )
- G - :
tan ( ) ( 2 . 1 9 )
8/17/2019 Analysis of Suspension Bridge
7/27
2. Theory of Elastic Catenary
- 7 -
2 - 1
;
i (0,0)
j (l x , l y)
P ( x,y,p)
x
y
F ix
F iy
F jx
F jy
l x
l y
l o , E, Ao , w
- : × ,
, ,
- G : ,
D - - .
)
- - -
,
( )
,
,
,
8/17/2019 Analysis of Suspension Bridge
8/27
2. Theory of Elastic Catenary
- 8 -
( ) ( )
0 - 1 0 . 0 0 ( . G . ) - 1 . 0 0 ( . G . )
1 - 1 2 . 0 9 3 6 8 2 . 6 8
2 - 2 3 . 8 5 1 6 6 4 . 2 6
3 - 4 6 . 6 5 1 6 6 4 . 1 3
4 - 8 9 . 8 8 1 6 6 3 . 7 8
5 - 1 6 9 . 1 8 1 6 6 2 . 5 5
6 - 3 0 7 . 7 4 1 6 5 8 . 4 9
7 - 5 3 4 . 6 5 1 6 4 6 . 4 7
8 - 8 7 8 . 8 6 1 6 1 5 . 3 4
9 - 1 3 6 3 . 0 1 1 5 4 6 . 6 9
1 0 - 2 0 0 5 . 3 9 1 4 1 8 . 0 8
1 1 - 2 8 2 8 . 8 9 1 2 0 8 . 7 2
1 2 - 3 8 6 3 . 4 8 9 0 2 . 8 1
1 3 - 5 1 3 0 . 8 8 4 9 3 . 0 4
1 4 - 6 6 0 3 . 6 2 - 7 . 7 5
1 5 - 8 1 3 1 . 4 1 - 5 4 1 . 8 3
1 6 - 9 3 7 8 . 1 8 - 9 8 4 . 1 9
1 7 - 9 9 9 2 . 3 4 - 1 2 0 3 . 8 0
1 8 - 1 0 1 0 2 . 0 8 - 1 2 4 3 . 2 0
1 9 - 1 0 1 0 4 . 9 7 - 1 2 4 4 . 2 4
2 0 - 1 0 1 0 4 . 9 7 - 1 2 4 4 . 2 4
* . G . :
B ,
E . ( 2 . 6 ) E ( 2 . 8 ) . A ,
E . ( 2 . 4 ) .
< > < C
>
8/17/2019 Analysis of Suspension Bridge
9/27
2. Theory of Elastic Catenary
- 9 -
2 - 2
;
i (0,0)
j (l x , l y)
P ( x,y,p)
x
y
F ix
F iy
F jx
F jy
l x
l y
l o , E, Ao , w
- : × ,
, ,
- G : ,
D - - .
)
( )
( )
0 - 2 0 . 0 ( . G . ) 1 . 0 ( . G . )
1 - 2 0 . 2 7 1 5 3 7 9 . 8 0
2 - 4 0 . 0 5 8 0 8 4 . 6 6
3 - 7 9 . 0 8 8 0 8 5 . 5 0
4 - 1 5 4 . 6 8 8 0 8 5 . 5 0
5 - 2 9 8 . 0 2 8 0 8 5 . 5 0
6 - 5 6 0 . 8 2 8 0 8 5 . 5 0
7 - 1 0 1 9 . 5 9 8 0 8 5 . 5 0
8 - 1 7 6 9 . 2 2 8 0 8 5 . 5 0
9 - 2 9 0 0 . 4 3 8 0 8 5 . 5 0
1 0 - 4 4 7 3 . 2 3 8 0 8 5 . 5 0
1 1 - 6 5 0 9 . 8 1 8 0 8 5 . 5 0
1 2 - 8 9 9 5 . 9 2 8 0 8 5 . 5 0
1 3 - 1 1 8 3 5 . 6 8 8 0 8 5 . 5 0
1 4 - 1 4 7 2 9 . 0 7 8 0 8 5 . 5 0
1 5 - 1 7 0 6 2 . 6 3 8 0 8 5 . 5 0
1 6 - 1 8 2 1 2 . 8 7 8 0 8 5 . 5 0
1 7 - 1 8 4 2 2 . 4 5 8 0 8 5 . 5 0
1 8 - 1 8 4 2 8 . 2 5 8 0 8 5 . 5 0
1 9 - 1 8 4 2 8 . 2 5 8 0 8 5 . 5 0
2 0 - 1 8 4 2 8 . 2 5 8 0 8 5 . 5 0
* . G . :
8/17/2019 Analysis of Suspension Bridge
10/27
2. Theory of Elastic Catenary
- 10 -
B ,
E . ( 2 . 6 ) E ( 2 . 8 ) . A ,
E . ( 2 . 4 ) .
< > < C
>
8/17/2019 Analysis of Suspension Bridge
11/27
2. Theory of Elastic Catenary
- 11 -
1
Givens : , , , , and
Unknowns : ,
Newton-Raphson formula
,
( )
,
,
:
8/17/2019 Analysis of Suspension Bridge
12/27
2. Theory of Elastic Catenary
- 12 -
2
Givens : , , , , and
Unknowns : ,
Newton-Raphson formula
where,
,
,
: iteration order
8/17/2019 Analysis of Suspension Bridge
13/27
2. Theory of Elastic Catenary
- 13 -
3
Givens : , , , , and
Unknowns : ,
Newton-Raphson formula
where,
,
,
: iteration order
8/17/2019 Analysis of Suspension Bridge
14/27
2. Theory of Elastic Catenary
- 14 -
2 . 2 -
&
i (0,0,0)
F ix
F iy
F iz
F ix
F iy
F iz
j (l x , l y , l z )
P ( x,y,z,p)
x
y
z
l o , E, Ao , w
l x
l y
l z
:
:
: -
:
: -
:
: C
: -
:
: -
( 2 . 1 5 )
8/17/2019 Analysis of Suspension Bridge
15/27
2. Theory of Elastic Catenary
- 15 -
i (0,0,0)
F ix
F iy
F iz
P ( x,y,z,p)
x
y
z x(s)
y(s)
z(s)
w, s
T
< F >
( 2 . 1 6 )
( 2 . 1 6 )
( 2 . 1 6 )
( )
∴
( 2 . 1 7 )
, , ,
, , ,
( )
①
E . ( 2 . 1 6 ) E . ( 2 . 1 5 ) ,
( 2 . 1 8 )
8/17/2019 Analysis of Suspension Bridge
16/27
2. Theory of Elastic Catenary
- 16 -
②
F E . ( 2 . 1 6 ) E . ( 2 . 1 7 ) ,
∴
E . ( 2 . 1 8 ) ,
( 2 . 1 9 )
E . ( 2 . 1 9 )
ln
A ,
ln
,
(2.20)
③
F E . ( 2 . 1 6 ) E . ( 2 . 1 7 ) ,
∴
E ( 2 . 1 8 ) ,
( 2 . 2 1 )
E . ( 2 . 2 1 )
A ,
8/17/2019 Analysis of Suspension Bridge
17/27
2. Theory of Elastic Catenary
- 17 -
,
(2.22)
④
F E . ( 2 . 1 6 ) E . ( 2 . 1 7 ) ,
∴
E . ( 2 . 1 8 ) ,
( 2 . 2 3 )
E . ( 2 . 2 3 )
ln
A ,
ln
,
(2.24)
8/17/2019 Analysis of Suspension Bridge
18/27
2. Theory of Elastic Catenary
- 18 -
⑤ ,
E . ( 2 . 2 0 ) , E . ( 2 . 2 2 ) E . ( 2 . 2 4 ) , , ,
(2.25)
(2.26)
(2.27)
, G :
: ,
E . ( 2 . 1 8 ) ,
:
( 2 . 2 8 )
( 2 . 2 9 )
( 2 . 3 0 )
, , , ( 2 . 3 1 )
,
( 2 . 3 2 )
8/17/2019 Analysis of Suspension Bridge
19/27
3. Elastic Catenary Element for Finite Element Analysis
- 19 -
3 .
,
.
E , ( 2 . 2 5 ) E . ( 2 . 2 7 ) ( , )
;
( 3 . 1 )
( 3 . 1 )
( 3 . 1 )
,
;
( 3 . 2 )
( 3 . 2 )
( 3 . 2 )
E . ( 3 . 2 ) ;
( 3 . 3 )
,
8/17/2019 Analysis of Suspension Bridge
20/27
3. Elastic Catenary Element for Finite Element Analysis
- 20 -
, ,
,
E E . ( 3 . 3 ) , ,
.
( 3 . 4 )
A , ,
, E . ( 3 . 4 ) ;
F ix + ΔF ix
F iy+ ΔF iy
w, l o
x
y F jx+ ΔF jx
F jy+ ΔF jy
From
(∵ )
∴
( 3 . 4 )
, - ,
;
, , ( 3 . 5 )
8/17/2019 Analysis of Suspension Bridge
21/27
3. Elastic Catenary Element for Finite Element Analysis
- 21 -
E . ( 3 . 5 ) E . ( 3 . 4 ) , .
Δ
Δ
Δ
Δ
Δ
Δ
ΔΔ
ΔΔ Δ Δ
( 3 . 6 )
,
8/17/2019 Analysis of Suspension Bridge
22/27
4. Initial Shape Analysis of Suspension Bridge
- 22 -
4 .
C D .
E 2 - 1 E 2 - 2 , ( )
. , ( )
. C D
,
, .
4 . 1 ( )
,
.
E , ( 2 . 1 1 ) E . ( 2 . 1 2 ) ( , )
;
( 4 . 1 )
( 4 . 1 )
,
;
( 4 . 2 )
( 4 . 2 )
E . ( 4 . 2 ) ;
( 4 . 3 )
,
8/17/2019 Analysis of Suspension Bridge
23/27
4. Initial Shape Analysis of Suspension Bridge
- 23 -
,
,
E E . ( 4 . 3 ) , ,
.
( 4 . 4 )
A , , (
E . ( 2 . 1 3 ) ) , E . ( 4 . 4 ) ;
( 4 . 4 )
- ,
;
, , ( 4 . 5 )
E . ( 4 . 5 ) E . ( 4 . 4 ) ,
.
( 4 . 6 )
,
E . ( 4 . 6 ) ;
8/17/2019 Analysis of Suspension Bridge
24/27
4. Initial Shape Analysis of Suspension Bridge
- 24 -
( 4 . 7 )
,
:
:
:
:
:
:
4 . 2
E . ( 4 . 7 )
,
.
( F E )
E . ( 4 . 7 ) .
( 4 . 8 )
, :
:
,
. F , E . ( 4 . 8 )
:
( 4 . 9 )
, :
:
:
:
( - )
8/17/2019 Analysis of Suspension Bridge
25/27
4. Initial Shape Analysis of Suspension Bridge
- 25 -
E . ( 4 . 9 )
(
) . ,
F E
E . ( 4 . 9 ) .
, - ( ) .
- ( , ∼ )
( ) ,
E . ( 4 . 9 ) ,
F E .
S
xi< 수 정 로 >
< ( ) C ( ) >
, E . ( 4 . 9 )
:
( 4 . 1 0 )
, :
:
:
:
D , E . ( 4 . 9 )
8/17/2019 Analysis of Suspension Bridge
26/27
4. Initial Shape Analysis of Suspension Bridge
- 26 -
( 4 . 1 1 )
, :
E . ( 4 . 1 1 )
.
E . ( 4 . 1 1 ) ,
.
( 4 . 1 2 )
( 4 . 1 2 )
( )
E . ( 2 . 1 1 ) E . ( 2 . 1 4 )
.
( 4 . 1 3 )
( 4 . 1 3 )
E . ( 4 . 1 3 ) ,
- E 2 . 1 2 . 2 .
( 4 . 1 4 ) , :
E . ( 4 . 1 4 ) E . ( 2 . 1 3 ) ,
- . ,
.
( - ) .
.
8/17/2019 Analysis of Suspension Bridge
27/27
4. Initial Shape Analysis of Suspension Bridge
E . ( 2 . 1 1 )
E . ( 2 . 1 2 )
.
A ,
.
C D ( ) ,
. F
, , E . ( 4 . 1 1 ) .