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PRACTICAL ANALYSIS METHODS FOR
CONTINUOUS GIRDER AND CABLE STAYED BRIDGES COMPOSED OF BEAMS WITH
CORRUGATED STEEL WEBS
Hisato KATO1 and Nobuo NISHIMURA2
1Regular member of JSCE, Master Eng., Registered Consulting Engineer, Bridge Engineering and Design Dept., JFE Engineering Corporation
(2-1, Suehiro-cho, Tsurumi-ku, Yokohama 230-8611, Japan) E-mail: [email protected]
2Fellow of JSCE, Dr. Eng., Professor Dept. of Civil Eng., Graduate School of Eng., Osaka University
(2-1, Yamadaoka, Suita, Osaka 565-0879, Japan) E-mail: [email protected]
A developed form of the PC box girders using corrugated steel web are flourishingly under construction. The form has been used for the continuous girders and the usage to be developed even for the cable stayed bridges. Since the corrugated web does not possess rigidity along the span axis, the influence due to shear deformation strongly appears. The authors have developed an elastic equation extending beam-bending theory considering shear deformation. The dual three-moment method and the CWB (Corrugated Web Beam) matrix displacements-method are formularized. The obtained knowledge is so important that the practical bridge engineers should take it into their consideration.
Key Words: composite structure, corrugated steel web, continuous girder, extradosed bridge, cable stayed bridge
1. INTRODUCTION
In order to span deep valleys, prestressed con-crete box girder bridges have been constructed by the cantilever method because of the ease of con-struction and cost-effectiveness. An improved varia-tion of the bridge type, a prestressed concrete box girder bridge composed of beams with corrugated steel webs, has recently been built in increasing numbers1)~3) to meet the severe demands for reduc-tion of public investment, address social needs and enhance performance. The type is now being adopt-ed for bridges with longer spans like extradosed bridges and evolving into cable stayed bridges to capitalize on its strong points4).
This type of bridge offers the following benefits. 1) Use of steel plates for webs that account for ap-
proximately a quarter of the weight of a conven-tional prestressed concrete box girder bridge could reduce the self weight of the superstructure and the size of substructure and increase seismic resistance.
2) Elimination of the need for concrete placement for webs reduces field work and saves labor.
3) Corrugated steel webs resist vertical forces but have extremely low longitudinal rigidity owing to the so called accordion effect. Thus, the upper and lower concrete slabs can be prestressed effi-ciently. Bridges composed of beams with corrugated
steel webs provide the above advantages, but their mechanical properties have not yet been fully iden-tified. Combault et al. (1993)5) presented examples of this type of bridge in France and described the unique bending behavior of beams with corrugated webs. Uehira et al. (1998)6) and Yamaguchi et al. (1997)7) conducted bending and shear tests for beams with corrugated webs and showed that an existing simple calculation method could accurately produce shear strength. They also proposed an e-quation of calculating the strength that considered buckling in the coupled buckling mode and verified the validity of the equation based on a comparison between calculation and test results. Taniguchi and Yoda (1997)8) and Shiromizu et al. (2000)9) made
Structural Eng./Earthquake Eng., JSCE, Vol.21, No.2, 207s-222s, 2004 October
208s
structural analyses by the matrix method while di-viding a beam into multiple elements. The Express-way Technology Center (2001)10) carried out a shear test for a short span and traced the deflection in the elastic range and the ultimate deflection, and strain distribution. These studies traced the ultimate be-havior, and illustrated through testing and finite element analysis (FEM analysis) that the bridges composed of beams with corrugated steel webs ex-hibited complicated behavior under vertical loading, that the webs had low rigidity along the span and that the shear deformation of the webs had a great influence on the bridge.
The authors, based on the existing works, pre-sented a simple structural analysis method to iden-tify the above properties of simple beams under the theme, "expanded beam theory" (Kato et al. 2002)11). (The beam with the above properties is referred to as the corrugated-web beam (CWB) below.) Unlike the "elementary beam theory", which ignores shear deformation and considers the bending of the whole cross section of the beam, the "expanded beam the-ory" handles the accordion effect of the web and the accompanying bending of upper and lower slabs separate from the bending of the whole cross sec-tion. In this study, the expanded beam theory is reinforced, and the double three-moment formula is developed and the CWB matrix displacement meth-od is formulated. Thus, the expanded beam theory is made applicable to continuous beams consisting of CWB elements. Then, analysis methods are pres-ented for continuous girder and cable stayed bridges with CWBs. Examples of analysis are also offered to meet engineers' needs for rational design. The key points that should be taken into consideration in the design procedure of a bridge with CWBs are de-scribed as well. 2. DERIVATION OF AN EXPANDED
BEAM THEORY EQUATION It is assumed that the concrete slabs carry axial
forces N1 and N2, shear forces Q1 and Q2 and mo-ments M1 and M2 and that the web carries shear force Qw only (Fig. 1).
nfg MMNNMMM +=+−+= 221121 )( ηη (2.1)
wnwg QQQQQQ +=++= 21 (2.2)
Vertical displacement v and angle of rotation around the horizontal axis of the web θ are consid-ered. They are related to shear strain of the web γ as
θγ −′= v (' represents d/dx) (2.3)
The relationship of the angle of rotation φ of the line connecting the centers of upper and lower slabs to the angle of rotation θ of the web in the filed of displacement shown in Fig. 2 is expressed by
θφ hv)ee(hd +′+= 21 (2.4)
The axial forces, bending moments and shear forces acting on the upper and lower slabs are represented by
vEIQvEIMEAN ′′′−=′′−=′= 1111111 ,,φη
vEIQvEIMEAN ′′′−=′′−=′−= 2222222 ,,φη
(2.5-6) where, EAi and EIi are axial stiffness and flexural stiffness of slabs that are calculated based on the as-sumption that the slabs are made of steel. The shear of the web is expressed by the following equation using GAw as the shear stiffness.
γww GAQ = (2.7)
Substituting equations (2.5) through (2.7) in equa-tions (2.1) and (2.2) yields
nfnf
f
g
MMEIvEI
AAEvEI
NNMMM
+=′−′′−=
′+−′′−=
−−+=
φ
φηη
ηη
)(
)(2
222
11
221121
(2.8)
γwfwg GAvEIQQQQ +′′′−=++= 21 (2.9)
where, 2
222
1121 , ηη AAIIII nf +=+= (2.10)
nfg III += (2.11)
Transforming equation (2.9) and substituting it in equation (2.4) yield
wgdwfd GAQhhvGAEIhhv /)/()/)(/( −′′′−′=φ (2.12)
If the balance between the stress resultant of the whole cross section and the external force is taken into consideration, the following equation is ob-tained.
qM g −=″ (2.13)
Then, first-order differentiation is applied to equation (2.12), which is substituted in equation (2.8), and the bending moment for the whole cross section Mg is expressed by a function of deflection v.
'1'' gwd
nw
f
dngg Q
GAhhEIv
GAEI
hhEIvEIM ++−= Ⅳ
(2.14) Based on the relation for the shear throughout the cross section, Qg = Mg',
''1''' gwd
nw
f
dngg Q
GAhhEIv
GAEI
hhEIvEIQ ++−= Ⅴ
(2.15)
209s
Second-order differentiation is applied to the both sides of equation (2.14), which is substituted in equation (2.13). Then, the following six-order differential equation is obtained. Relationships, Qg' = -q and Qg" = -q', are used.
''11 qEI
qEIGA
hh
EIv
EIGA
hh
II
vff
wd
nf
wd
n
g +−=− ⅣⅥ
(2.16)
If f
wd
n
g
EIGA
hh
II
=2α is applied, the solution of the dif-
ferential equation is obtained as
4065
34
2321
)sinh()cosh( xCxCxC
xCxCxCCv
+++
+++=
αα (2.17)
where, C0x4 is a particular solution where a full distributed load is applied, and C1 through C6 are the constants determined by the boundary condi-tions.
The constants for a simple beam under various loading conditions are listed in Table 1. Obtained by consecutively differentiating equation (2.17) are
( ) ( )( ) ( ) 3
065
2432
4coshsinh
32
xCxCxC
xCxCCv
+++
++=′
αααα
(2.18)
( ) ( )( ) ( ) 2
02
62
5
43
12sinhcosh
62
xCxCxC
xCCv
+++
+=′′
αααα
(2.19)
( )( ) ( ) xCxCxC
Cv
03
63
5
4
24coshsinh
6
+++
=′′′
αααα
(2.20) Without any loading on the beam, the deforma-
tion of the element is expressed by the above equa-tions with C0 = 0. Then, Qg' in equation (2.14) is 0. Substituting equations (2.19) and (2.20) yields
( ) ( )xEICEICM ggg 62 43 −+−= (2.21)
Since Qg = Mg',
( )gg EICQ 64 −= (2.22)
Substituting equations (2.18), (2.20) and (2.22) in equation (2.12) yields
( ))cosh()sinh(
632
65
2432
xCxCxCxCC
αηαη
βφ
+++++= (2.23)
where, 2αβ
f
g
II
= and n
f
II
αη −= .
Performing first-order differentiation for Equation (2.23) yields
( ) ( ))sinh()cosh(
62
65
43
xCxCxCC
ααηαηα
φ
+++=′
(2.24) Inserting Equation (2.19) into the first term of equa-tion (2.8) yields
( ) ( )( ) ( ))sinh()cosh(
622
62
5
43
xEICxEIC
xEICEICM
ff
fff
αααα −+−+
−+−=
(2.25) Substituting equation (2.24) in the second term of Equation (2.8) yields
Q1
Qw
Q2 M2
M1N1
N2
η1
η2
εn εf εe
e1
x y
Fig. 1 Internal forces and strain distribution
v’
θ v’
γ
θ
v’
v’ e1
e2
h
η1
η2
θ
φ
hd
AA’⊥A’C’’BB’⊥B’B’’CC’⊥C’C’’
Fig. 2 Displacement field
Before deformation
After deformation
a) Displacement field in the web Point C’’
Point C
Point C’ Point B’’
Point B’ Point A’ Point B
Point A
b) Displacement field throughout the beam
210s
( ) ( )( )( ) ( )( )xEICxEIC
xEICEICM
nn
nnn
ααηαηα sinhcosh62
65
43
−+−+−+−=
(2.26) 3. DOUBLE THREE-MOMENT
FORMULA Out of the constants in Table 1 based on the ex-
panded beam theory, the constants for the case of concentrated moments at beam ends are employed to induce a three-moment formula based on the ex-panded beam theory as a three-moment formula is induced based on the elementary beam theory (Komatsu 1982)12). Inserting rigidity and load into a given three-moment formula easily produces solu-tions to statically indeterminate problems such as continuous girders by manual calculation or using a spreadsheet. The three-moment formula is also use-ful for verification during the development of com-puter programs as described later. The beam ele-ments based on the elementary and expanded beam theories are referred to as general and CWB ele-ments below, respectively. In the elementary beam theory, a three-moment formula is induced so that the angles of rotations φ i on both sides of a node be-
tween two members are equal to each other. In the expanded beam theory, the angles of deflection v'i are also made equal to each other as well as the an-gles of rotations φ i. The continuities of φ i and v'i are maintained, so the continuities of moment Mni of entire beam induced by the axial force of the slab and of moment Mfi induced solely for the slab are guaranteed accordingly. Suffix i indicates the num-ber of the node to be considered. (The three-mo-ment formula based on the expanded beam theory is referred to as the double three-moment formula be-low.)
Equations are obtained separately from the bal-ance between the shear forces at the ends of left and right members on both sides of node i and the exter-nal force acting at the node in the expanded beam theory. The process is the same as in the elementary beam theory.
If the constants that determine the shape of a beam in the case where a moment acting at the right end are represented by Cm1 through Cm6,
)cosh()sinh(32
65
2432
xCxCxCxCCv
mm
mmm
αααα ++++=′ (3.1)
( ))cosh()sinh(
632
65
2432
xCxCxCxCC
mm
mmm
αηαη
βφ
+++++= (3.2)
P
L
q
L L M
中央集中荷重 支点集中モーメント 分布荷重
C0 0 0 gEI
q24
C1 0 0 fg
n
IEIIq
4α−
C2 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+ 2
2
82 αf
n
g IIL
EIP
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
f
gf
g III
LEIML ρ
α 21
61
gfg
n
EIqL
IEIIqL
242
3
2 +α
C3 0 0 fg
n
IEIIq
22α−
C4 gEI
P12
− gLEI
M6
− gEI
qL12
−
C5 0 0 fg
n
IEIIq
4α
C6 ( )2cosh2 3 LIEI
PI
fg
n
αα−
f
gf
g III
LEIM ρ
αα
−
)sinh(2 ( )( )2cosh
2sinh4 L
LIEI
Iq
fg
n
α
α
α−
表 2 弾性方程式の係数
ρ は作用モーメント M のうち床版で負担されるモーメント M f の分担率を表し、M f=ρ・M なる関係を有する
C0
C1
C2
C3
C4
C5
C6
Concentrated loading at midpoint Moment concentrated at right support Distributed loading
Table 1 Constants of integration under typical loading conditions
ρ indicates the percentage of Mf carried by slabs in the total acting moment M. Mf = ρM.
L
P
L
M
q
L
211s
( )gmg EICQ 64 −= (3.3)
However, Cm1 = Cm3 = Cm5 = 0 according to Table 1. A condition shown in Fig. 3 a) is assumed where
counter-clockwise rotational moment M acts at the right end of the element (hereinafter referred to as Condition r). Then, v' and φ at the right and left ends are obtained. The first suffix r to v' and φ represents Condition r, and the second suffix l or r indicates that the variate is at the left or right end. At the left end: at x=0
α62 mmrl CCv +=′
( ) ηβφ 642 6 mmmrl CCC ++= (3.4-5) At the right end: at x=L
( ) )cosh(3 62
42 LCLCCv mmmrr αα++=′
( ) ηβφ 642 6 mmmrl CCC ++= (3.6-7) A condition subjected to clockwise rotational
moment M at the left end is marked by suffix l (hereinafter referred to as Condition l). v' and φ are obtained at the left and right ends under the condi-tion.
rllrrrll vvvv ′−=′′−=′ , (3.8-9)
rllrrrll φφφφ −=−= , (3.10-11)
v' is calculated using
( )nff
fnnf
f
gf
MMIMIMI
III
+
−=
−ρ (3.12)
α62 mmrl CCv +=′
( )
( ) ng
ff
n
g
MLL
LEI
MLLI
ILEI
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
ααα
ααα
sinh11
61
sinh11
61
2
2 (3.13)
( ) )cosh(3 62
42 LCLCCv mmmrr αα++=′
( )
( ) ng
ff
n
g
ML
LL
LEI
ML
LLI
ILEI
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−=
αα
α
α
αα
α
α
sinh)cosh(1
31
sinh)cosh(1
31
2
2 (3.14)
If fn II≡γ
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−≡≡
LLEID
EILD
gg ααα sinh111
6 216
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−≡−≡
LL
LEID
EILD
gc
g αα
α
α sinh)cosh(11
3 23
(3.15-17) are assumed,
( ) ( ) nfrl MDDMDDv 1616 −++=′ γ (3.18)
( ) ( ) ncfcrr MDDMDDv −++=′ 33 γ (3.19) v' is considered at node i for members j and j+1
on both sides of node i. v' is identified by the third suffix, j or j+1 (Fig. 4). Considering that v' is continuous at node i yields
11,1,1,,,, ++++ +′+′+′=+′+′+′ jjoljrljlljjorjrrjlr vvvvvv ξξ (3.20) Substituting Equations (3.18) through (3.19) in Equation (3.20) yields
( ) ( )( )( )( ) ( )
011,,
1,1,11,61,1,111,6
,1,1,3,,3
,1,11,3,,3
1,,1,61,,1,6
=−+′−′+
−−+−
−+−+
++++
−−+−
++
+++++++
++
+++
−−
jjjoljor
injjifjjj
injcjjcj
ifjcjjjcjj
injjifjjj
vv
MDDMDD
MDDDD
MDDDD
MDDMDD
ξξ
γ
γγ
γ
(3.21) where, v'or, j and v'ol, j+1 are v's at the right end of
member j and at the left end of member j+1, respec-tively under the loading on the members, and ξ j and ξ j +1 are the angles of rotation of members j and j+1, respectively due to sinkage at nodes i-1, i and i+1.
They have the relationship,
111
111111
+++
−+ −⎟⎟⎠
⎞⎜⎜⎝
⎛++−=− i
ji
jji
jjj v
Lv
LLv
Lξξ
(3.22)
P i
Q ll,i+1 or Q rl,i+1 Q lr,i or Q rr,i
M f,i-1 or M n,i-1
M f i or M n i M f,i+1 or M
L j L j;1
Fig. 4 Double three-moment formula
Node i-1 Member j
Node i Member j+1
Node i+1 a) Numbers of members and nodes
b) Acting moment
c) Shear force at node i
ξj+1
ξj
M v’rlor φrl M
v’rr or φrr
(＋)
(－)
v’ll or φll (＋)
v’lr or φlr
(－)
Fig. 3 v' and φ when a moment acts at the end of a member
a) Condition r b) Condition l
212s
Assuming
1,,,0
1,11,6,61,111,6,5
1,1,3,,3,4
1,11,3,,3,3
,1,6,2,1,6,1
,
,
+
+++++
++
+++
′+′−≡
+−≡−−≡
−+−≡
+++≡
+−≡−−≡
joljorj
jjjjjjj
jcjjcjj
jcjjjcjjj
jjjjjjj
vvU
DDUDDU
DDDDU
DDDDU
DDUDDU
γ
γγ
γ
1,31,2,1 1,11,1 ++ −≡+≡−≡ jjjjjjj LTLLTLT (3.23-28)
yields
jijinjifj
ijinjifj
ijinjifj
UvTMUMU
vTMUMU
vTMUMU
,01,31,,61,,5
,2,,4,,3
1,11,,21,,1
=+++
+++
++
+++
−−−
(3.29) Similar calculations are made for φ as shown below.
( ) ηβφ 642 6 mmmrl CCC ++=
( )
( ) nn
f
f
g
g
fn
f
f
n
f
g
g
MLI
ILLI
ILEI
MLI
ILI
ILI
ILEI
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛++−+=
αααα
αααα
sinh11
61
sinh11
61
22
22
(3.30)
( ) )cosh(63 62
42 LCLCC mmmrr αηβφ +++=
( )
( ) nn
f
f
g
g
fn
f
f
n
f
g
g
ML
LII
LLIIL
EI
ML
LII
LII
LIIL
EI
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛++−−+=
αα
α
αα
αα
α
αα
sinh)cosh(1
31
sinh)cosh(1
31
22
22
(3.31) Assuming
( )⎟⎟⎠
⎞⎜⎜⎝
⎛+≡⎟
⎟⎠
⎞⎜⎜⎝
⎛−≡
LII
LEIE
LIIL
EIE
n
f
gf
g
g αααα sinh111,
61
2126
( )( )⎟
⎟⎠
⎞⎜⎜⎝
⎛+≡⎟
⎟⎠
⎞⎜⎜⎝
⎛−−≡
LL
II
LEIE
LIIL
EIE
n
f
gc
f
g
g αα
α
αα sinhcosh11,
31
223
(3.32-33) yields
( ) ( ) nfrl MEEMEE 1616 −++= γφ (3.34)
( ) ( ) ncfcrr MEEMEE −++= 33 γφ (3.35)
Considering that φ continues at node i from the right end of member j through the left end of mem-ber j+1 yields
11,1,1,,,, ++++ +++=+++ jjoljrljlljjorjrrjlr ξφφφξφφφ (3.36)
Substituting Equations (3.34) and (3.35) in Equation (3.36) yields
( ) ( )( )( )( ) ( )
011,,
1,1,11,61,1,111,6
,1,1,3,,3
,1,11,3,,3
1,,1,61,,1,6
=−+−+
−−+−
−+−+
++++
−−+−
++
+++++++
++
+++
−−
jjjoljor
injjifjjj
injcjjcj
ifjcjjjcjj
injjifjjj
MEEMEE
MEEEE
MEEEE
MEEMEE
ξξφφ
γ
γγ
γ
(3.37) Assuming
1,,,0
1,11,6,61,111,6,5
1,1,3,,3,4
1,11,3,,3,3
,1,6,2,1,6,1
,
,
+
+++++
++
+++
+−≡
+−≡−−≡
−+−≡
+++≡
+−≡−−≡
joljorj
jjjjjjj
jcjjcjj
jcjjjcjjj
jjjjjjj
V
EEVEEV
EEEEV
EEEEV
EEVEEV
φφ
γ
γγ
γ
(3.38-42) yields
jijinjifj
ijinjifj
ijinjifj
VvTMVMV
vTMVMV
vTMVMV
,01,31,,61,,5
,2,,4,,3
1,11,,21,,1
=+++
+++
++
+++
−−− (3.43)
The shear Qg acting on the cross section is repre-sented by Q below.
( ) ( )LMEI
LEIMEICQ g
ggmrl =−−=−= 6
664
( )LMEICQ gmrr =−= 64
rllrrrll QQQQ −=−= , (3.44-46) Representing the vertical load on node i by Pi
and considering the balance between shear forces from members j and j+1 on both sides of node i yields
01,1,1,,,, =++++−−− +++ ijoljrljlljorjrrjlr PQQQQQQ (3.47)
( )
0
11
1,,1
1,1,
,,1
1,1,
=++−+
+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+−
+
++
++
+
−−
ijoljorj
inif
inifjjj
inif
PQQL
MM
MMLLL
MM (3.48)
Assuming
ijoljorj
jjjj
jjjjj
jjjj
PQQW
TLWW
TLLWW
TLWW
−−≡
−=≡=
−=−−≡=
−=≡=
+
+
+
1,,,0
,31,6,5
,21,4,3
,1,2,1
1
11
1
(3.49-52) yields
213s
jinjifj
injifjinjifj
WMWMW
MWMWMWMW
,01,,61,,5
,,4,,31,,21,,1
=++
+++
++
−− (3.53)
Equations (3.29) and (3.43) are related to the moments { }1,,1, ;; +− ififif MMM and { }1,,1, ;; +− ininin MMM on three continuous nodes. Incorporating Equation (3.53) concerning the balance of shear forces at node i yields simultaneous equations with such vari-ables as moments at node i, Mf and Mn, and deflec-tion vi. 4. EXAMPLE OF A DOUBLE THREE-
MOMENT FORMULA In this chapter, a double three-moment formula
related to CWB elements is used to calculate mo-ments Mf and Mn acting on the intermediate sup-ports of a three-span continuous bridge composed of beams with corrugated webs throughout the bridge length under a full distributed load. It is then verified whether the ratio Mf /Mg = Mf /(Mf + Mn) agrees with the solution obtained by the CWB ma-trix method described later. A simple parametric analysis is also made using a spreadsheet to verify whether the influence of CWB decreases and whether the ratio Mf /Mg becomes lower over a longer span as described by Kato et al. (2002)11). The analysis is made while varying the span length from 51 to 64, 80, 100, 125, 156 and 195 m and as-suming rigidities If = 0.00217 m4 and In = 0.76612 m4 based on the cross section in Fig. 5 b). The equation obtained in Chapter 3 is expressed as a matrix for the case with four nodes.
PAM = (4.1) where,
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
0000000000
00
00
0000000000
3333
626262
525252
22223131
424242616161
323232515151
12122121
222222414141
121212313131
1111
212121
111111
TTWVUWVU
TTTTWVUWVUWVUWVU
TTTTWVUWVUWVUWVU
TTWVUWVU
TA (4.2)
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
=444333
222111
;;;;;
;;;;;;
vMMvMM
vMMvMM
nfnf
nfnfTM (4.3)
{ }020202010101 ;;;;; WVUWVU=TP (4.4)
In this example, nodes at the ends, i = 1 and 4, are hinged nodes, so Mfi = Mni = vi = 0 (i = 1 and 4). Considering vi = 0 (i = 2 and 3) and handling Mfi and Mni (i = 2 and 3) as unknowns yield
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
02
02
01
01
3
3
2
2
42322212
42322212
61514131
61514131
VUVU
MMMM
VVVVUUUUVVVVUUUU
n
f
n
f
(4.5).
The above simultaneous equations are solved for each span to obtain Mf /Mg and the result is plotted in a graph (Fig. 6). Mf /Mg = 0.0277 at L = 80 m, equivalent to Mf /Mg shown in Fig. 10 b) based on the matrix displacement method. It was verified that Mf /Mg decreased and CWB had smaller influence over a longer span. Although Mf /Mg is small, the normal stress caused by Mf which is on the surface
a) Analytical model of three-span continuous bridge
図－6中間支点の Mf / Mg
M f/M g
0.000
0.010
0.020
0.030
0.040
0.050
50 100 150 200 250
中央支間長 L m
Mf/Mg
0.0277
80.0
Fig. 6 Mf /Mg at intermediate points
Length of center span Lm
a) Analytical model of three-span continuous bridge
b) Cross section
10,000
3,00
030
025
0
6,000
Web t = 16
L m 0.8L m 0.8L m
L =51～195m
i=1 i=2 i=3 i=4j=1 j=2 j=3
If =21,650cm4
In =768,300cm4
Full distributed load
Fig. 5 Example of double three-moment formula
214s
of the concrete cross section, is nearly level with the stress throughout the cross section induced by Mn acting on the whole cross section of the CWB. Great care should therefore be taken during design. 5. CWB MATRIX DISPLACEMENT
METHOD Textbooks on structural mechanics start with the
description of the three-moment method, or so called stress method, based on the elementary beam theory and finally discuss the matrix analysis of framed structures13) that considers the continuity of stress resultants at respective nodes (Weaver 1980). In the expanded beam theory, analysis by the matrix displacement method is also possible. In this chap-ter, the in-plane matrix displacement method based on the expanded beam theory (referred to as the CWB matrix displacement method below) is formu-lated.
The matrix displacement method based on the elementary beam theory involves the solution of si-multaneous equations treating displacements u and v and angle of rotation φ as unknowns. In the ex-panded beam theory, the first-order differentiation v' of displacement v needs to be considered in addi-tion to the above displacements. The relationship between the displacement and acting forces at the ends of a CWB element is derived. The displace-ment is expressed by Equation (5.1) where the both ends of the element are represented by i and j.
{ }jjjjiiii vvuvvu φφ ;;;;;;; ′′ (5.1)
The forces at the ends of the element is expressed by
{ }jnjfjyjxinifiyix MMPPMMPP ,,,,,,,, ;;;;;;; (5.2)
The displacement in the direction of member axis u and the axial force Px are independent of other vari-ables, so they are excluded and three degrees of freedom are considered at each node. First, the rela-tionship of displacements at nodes to constants C1 through C6 are derived. Substituting x = 0 and L in Equations (2.17), (2.18) and (2.23) yields
at x = 0 51 CCvi += (5.3)
α62 CCvi +=′ (5.4)
ηβφ 642 6 CCCi ++= (5.5)
at x = L
)sinh()cosh( 65
34
2321
LCLCLCLCLCCvjαα ++
+++= (5.6)
( ) ( )( ) ( )LCLC
LCLCCvj
αααα coshsinh32
65
2432
++
++=′ (5.7)
( ))cosh()sinh(
632
65
2432
LCLC
LCLCCj
αηαη
βφ
++
+++= (5.8)
The relationship is expressed by a matrix.
ACx = (5.9) where
{ }jjjiii vvvv φφ ;;;;; ′′=x (5.10)
{ }654321 ;;;;; CCCCCC=TC (5.11)
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+
=
hh
hh
hh
CSLLCSLL
SCLLL
ηηβ
αα
ηβ
α
632103210
10601000010
010001
2
2
32A (5.12)
where ( ) ( )LSLC hh αα sinh,cosh == Then, the relationship of the member-end forces
to constants C1 through C6 is derived. Substituting x = 0 and L in Equations (2.22), (2.25) and (2.26) yields the equations below. These equa-tions are, however, induced for the normal direction in the ordinary beam theory. In a right-handed sys-tem of a generalized frame element, the signs of Py, i, Mf, j, Mn, j need to be reversed (Fig. 7). Mf and Mn, and v' and φ are in the same rotational direction, re-
Px, j Py, j
Qg
Mg
v’
Mg
Qg
v
Px, i
Py, iu, j
v j
u, i
v, i
Mf, i or Mn, i Mf, j or Mn, j
v’ j or φj v’ i or φi
Fig. 7 Positive directions of sectional forces and displacements
a) Beam theory
b) Frame analysis
q M
v’cl or φcl
M
v’ql or φql
Fig. 8 Calculation of uniform distributed load
a) Condition c b) Condition q
215s
spectively.
( )ggiy EICQP 64, =−= (5.13)
( ) ( )253, 2 αfffif EICEICMM −+−== (5.14)
( ) ( )ηαnnnin EICEICMM −+−== 53, 2 (5.15)
( )ggjy EICQP 64, −== (5.16)
( ) ( )( ) ( ))sinh()cosh(
622
62
5
43
,
LEICLEIC
LEICEIC
MM
ff
ff
fjf
αααα ++
+=
−= (5.17)
( ) ( )( )( ) ( )( )LEICLEIC
LEICEIC
MM
nn
nn
njn
ααηαηα sinhcosh62
65
43
,
+++=
−= (5.18)
They are expressed by a matrix as
BCf = (5.19) where, { }jnjfjyinifiy MMPMMP ,,,,,, ;;;;;=f
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−
=
hnhnnn
hfhfff
g
nn
ff
g
SEICEILEIEISEICEILEIEI
EIEIEIEIEI
EI
ηαηα
αα
ηα
α
62006200
0060000020000200006000
22
2
B
(5.20) (1) Stiffness matrix
The vector C of the integration constant is ex-pressed by C = A-1x from Equation (5.9). If C is substituted in Equation (5.19), stiffness matrix K is expressed by
KxxBAf 1 == − (5.21) If the elements of matrix A-1 is represented by
i,ja , the following equations are obtained.
( ) ( )( )( ) ( ) 6356,16255,1
6154,16353,1
6252,16151,1
/,//,/
/,/
FSFFIaFSFFIaFSFFaFSFFICa
FSFFICaFSFFISa
hnhn
hhnh
hnhhgh
+=+−=
−=+−=
+=+= α
( )
( )gg
ggn
gfg
IFFaIFFaIFFaIFFIa
IFFIaIFFa
/,//,/
/,/
346,2245,2
144,2343,2
242,2141,2
==
−=+=
+==
( )
( )( ) ( ) LIFLIaLIFLIa
LIFLaLIFLIa
LIFLIaLIFLa
gngf
ggn
gfg
2/3,2/3
2/3,2/3
2/3,2/3
32
6,322
5,3
12
4,332
3,3
22
2,312
1,3
−=−=
=+−=
+−=−=
gg
gg
gg
IFaIFa
IFaIFa
IFaIFa
/,/
/,/
/,/
36,425,4
14,433,4
22,411,4
==
−==
==
( )
( )( ) ( ) 6356,56255,5
6154,56353,5
6252,56151,5
/,//,/
/,/
FSFFIaFSFFIaFSFFaFSFFICa
FSFFICaFSFFa
hnhn
hhnh
hnhh
+−=−=
=−=
+−=−=
( )( )
6346,66245,6
6144,66343,6
6242,66141,6
/,//,/
/,/
FFFaFFFaFFFaFFFIa
FFFIaFFFa
n
n
−=−=
=+−=
−=−=
where,
( )( ) ( )( )( ){ }
( )( ) ggnh
gnnhm
fhmghm
gh
hmfnhm
IFIICF
IIFFISFFISICF
FISF
LCLIISF
αβ
β
α
α
≡−−≡
−≡−≡
−≡
−≡
++−≡
65
403
02
01
2220
,/61
/6,//2
/2
2/12
(5.22-28) (2) Load
In frame analysis, the concentrated loads or mo-ments at nodes are considered as external loads dur-ing calculation. Distributed loads acting throughout the beam therefore need to be replaced with equiva-lent nodal forces at both ends of the beam. De-scribed below is a method of calculation where a full distributed load is applied to a CWB element. For a general element, the shear forces caused in the beam by a full distributed load are given as the con-centrated nodal loads at both ends of the element. Moments are also given at both ends so that the an-gle of deflection at a beam end becomes identical to the value under a full distributed load. The same procedure is adopted for the CWB element. Mf and Mn need to be determined so that the angle of deflection v' and the angle of rotation φ become identical to those under a full distributed load.
The element is symmetrical about its center un-der a full distributed load. It is easily assumed that such a shape can be created by applying equivalent moments at both ends of the element that are of the same scale and in opposite directions. The condition where conditions r and l used for deriving the three-moment formula occur simultaneously, or the con-dition where counter-clockwise and clockwise rota-tional moments act at the right and left ends simul-
216s
taneously (condition c), is indicated by first suffix c. Then, v' and φ at the left end are obtained.
( ) ( ))cosh(13 62
4 LCLCvvv mmllrlcl αα −+−=′+′=′ (5.29)
( ) ( ))cosh(13 62
4 LCLC mmllrlcl αηφφφ −+−=+= (5.30)
The condition under distributed load q (condition q) is indicated by first suffix q, and v' and φ at the left end are obtained. The above conditions are shown in Fig. 8 a) and b). If the constants under the full distributed load in Table 1 are represented by Cq0 through Cq6,
3065
2432
4)cosh()sinh(
32
xCxCxC
xCxCCv
qqq
qqq
+++
++=′
αααα
(5.31)
( ) ( )( ) ( ) ( ) 0
365
2432
424coshsinh
632
qqq
qqq
CxxxCxC
xCxCC
++++
+++=
βαηαη
βφ
(5.32) At the left end, x = 0.
α62 qqlq CCv +=′ (5.33) ( ) ηβφ 642 6 qqqql CCC ++= (5.34)
The M-q relationship is obtained when the loads un-der the c and q conditions are applied simultane-ously and v' and φ at the left end are 0.
From 0=′+′ qlcl vv
( ) ( )0
)cosh(13
62
62
4
=++−+−
α
αα
mm
CCLCLC (5.35)
From 0=+ qlcl φφ
( ) ( )( ) 06
)cosh(13
642
62
4
=++−+−
ηβ
αη
qqq
mm
CCCLCLC (5.36)
The equations are solved for M and ρ by substituting Cm1 through Cm6, and Cq0 through Cq6 listed in Table 1 in Equations (5.35) and (5.36).
12
2qLM −= (5.37)
( ) ( ) ( )( )( )⎟
⎟⎠
⎞⎜⎜⎝
⎛
−−−−
−=)cosh(1
6sinh1 64
2
LCC
MLEI
II qqg
g
f
αηα
ηαβααρ
(5.38) Based on the above, the nodal forces on points i and j of a CWB element that are equivalent to a full dis-tributed load on the CWB element are expressed by
( ) ( )( )12/1,12/,2/ 2,
2,, qLMqLMqLP inifiy ρρ −===
( ) ( )( )12/1,12/,2/ 2,
2,, qLMqLMqLP jnjfjy ρρ −−=−==
(5.39-40)
(3) Condensation Prestressed concrete box girder bridges com-
posed of beams with corrugated webs have corru-gated steel plates for webs in general sections and usually have concrete webs at supports. In frame analysis, the model of a bridge adopts general frame elements for the concrete web and CWB elements for the corrugated web. The former have three de-grees of freedom at each node and the latter four de-grees of freedom. The continuity of moment through the connection should be maintained. The unbalance in number of degrees of freedom needs to be resolved for modeling. The length of the general element at the support is normally 3 to 5 m, short for the beam height, so the shear deformation in the element should be taken into consideration during analysis. A model is made while ensuring the continuity between the angle of deflection v' of the general element at the connection with the CWB element and v' on the CWB element side φ on the CWB element side is made identical to v'. In matrix calculation, unnecessary degrees of freedom are re-moved by condensation and simultaneous equations are solved.
6. EXAMPLE OF A CONTINUOUS GIRDER BRIDGE In this chapter, the behavior of the three-span
continuous girder bridge shown in Fig. 9 is ana-lyzed by the matrix method derived above. A full distributed load is applied throughout the bridge length. A uniform cross section is assumed for sim-plicity (Fig. 5). A model of a continuous girder bridge of cross section with the same rigidity as that of the uniform cross section but without any corru-gated webs, or a bridge with general elements throughout the bridge length, is defined as the model in the "general element case". The model is compared with the models in two other cases. The general element is based on the elementary beam theory that ignores the flexural stiffness of the web, namely uses the second moment of inertia about the centroid of entire cross section only for slabs and ignores the shear deformation.
As described earlier, the corrugated steel plates immediately above supports are replaced by con-crete webs on ordinary bridges composed of corru-gated webs. The case with general elements over a length of 2 m each on the end supports and over a length of 4 m each on intermediate supports is de-fined as the "case with CWB in part of the bridge". The case where CWB elements are assumed over
217s
the total length of the bridge is defined as the "case with CWB throughout the bridge".
Fig. 10 a) shows deflections. Fig. 10 b) shows the ratios of the moment on the top flange concrete slab to the total moment (ρ =Mf /Mg). Fig. 10 c) shows the bending stresses acting on the top surface of the slab (σ = Mf / If・e1 + Mn / In・η1, for e1 and η1, refer to Fig. 2). The deflection is expressed as a non-dimensional deflection using the deflection at the midpoint between intermediate supports in the general element case. The bending stress is ex-pressed as a non-dimensional stress using the stress at intermediate supports in the general element case.
According to the figure showing deflections, the deflection of the bridge composed of beams with corrugated webs exceeds that of the bridge with general elements by approximately 30% at midspan. The deflection at midspan was greater on a bridge with CWB elements because the angle of deflection fluctuated more greatly near the supports. No significant variance was found between the cases with CWB in part of the bridge and with CWB throughout the bridge.
The ratio of the moment on the slab to total mo-ment was at a uniform level and ρ = Mf /Mg = If /Ig for the case with general elements. ρ was great near the supports for the case with CWB elements.
80m64m 64m
2m 4m 4m 2m60m 76m 60m
Fig. 9 Analytical model of continuous girder bridge
Full distributed load
Corrugated web Corrugated web Corrugated web
図－11 斜張橋の解析モデル
4m
[email protected]+15.5=9090m 18090m
20m
9@1=
9
L C 上層より①～⑨
q=760N/cm (1 面ケーブル当たFig. 11 Analytical model of cable stayed bridge
1st through 9th stay cables from the top
(per cable plane)
c) 曲げ応力比-1.00
-0.50
0.00
0.50
1.00
1.50
2.00
2.50
0 20 40 60 80 100
比
LCb) Mf/Mg の比
-0.16
-0.12
-0.08
-0.04
0.00
0.04
0.08
0 20 40 60 80 100
比
0.0277
a) たわみ
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 20 40 60 80 100
比
一般要素
部分CWP
全CWP
LC
Fig. 10 Analytical results for continuous girder bridge
General element Partial CWB Total CWB
a) Deflection
b) Mf /Mg ratio
c) Bending stress
RatioRatio
Ratio
218s
b) 3本タイプ斜張橋発生応力
-1.60
-1.40
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
00 20 40 60 80m
σ/σ
o
CWB軸応力
CWB下FLG下曲げ応力
CWB下FLG上曲げ応力
σo=9本タイプ塔基 部 下フランジ下面 Mnによる曲げ応力
CL
CL
a) 9本タイプ斜張橋発生応力
-1.60
-1.40
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
00 20 40 60 80 m
σ/σ
o
CWB軸応力
CWB下FLG下曲げ応力
CWB下FLG上曲げ応力
一般要素軸応力
一般要素下FLG下曲げ応力
CL
σo=9本タイプ塔基 部 下フランジ下面 Mnによる曲げ応力
注 1
Fig. 13 Stresses on the cable stayed bridge
σo: Bending stress due to Mn on the bottom surface of the lower flange at the base of the pylon in the nine-stay-cable type
σo: Bending stress due to Mn on the bottom surface of the lower flange at the base of the pylon in the nine-stay-cable type*1
CWB element axial stressBending stress on the bottom surface of the lower flange of CWB elementBending stress on the top surface of the lower flange of CWB elementGeneral element axial stressBending stress on the bottom surface of the lower flange of the general element
a) Stresses on a cable stayed bridge with nine stays
CWB element axial stressBending stress on the bottom surface of the lower flange of CWB elementBending stress on the top surface of the lower flange of CWB element
σo for nine-stay-cable type is used for comparison with Fig a).
b) Stresses on a cable stayed bridge with three stays
The bending stress was also great near the sup-ports in the case of CWB elements. This is ascrib-able to the increase in bending solely of the slab due to the increase in ρ near the supports as described
earlier. Care should be taken accordingly in design. The behavior of the beam in the area except those near the support is nearly the same as that identified by analysis using the general element. The cross
Upper slab
Fig. 12 Cross section of main girder at the cable anchorage
Steel diaphragm Cable anchorage
Corrugated web
Lower slab
Table 2 Structural specification of cable stayed bridge (per cable plane) Girder
(regarded as steel) E(N/ mm2) If (cm4) In(cm4) A(cm2) Aw(cm2)
2.058x105 21,650 768,300 3,214 480 Pylon E(N/ mm2) I (cm4) A(cm2)
3.742x104 1,067x106 80,000
Cable E(N/ mm2)Top layer: 1st through 3rd stay cables
Middle layer: 4th through 6th stay cables
Bottom layer: 7th through 9th stay cables
1.960x105 N- PWS109 N- PWS85 N- PWS73
219s
section in the area can therefore be designed using the result of analysis based on the elementary beam theory. The above discussions apply only to the be-havior under a full distributed load. A separate study is required for the case of a concentrated load at midspan or for a cable anchorage of a cable stayed bridge. 7. EXAMPLE OF A CABLE STAYED
BRIDGE In this chapter, a cable stayed bridge with three
spans of 90 m, 180 m and 90 m of the same cross section as that of the sample continuous girder bridge and of the profile shown in Fig. 11 is as-sumed for analysis. Nine stay cables are connected to each pylon (nine-stay-cable type). For compari-son, a bridge with three stay cables (three-stay-cable type) is also examined. The main girder is rigidly connected to the bases of the pylons. The full cross section of the girder is made of concrete over a length of 4 m, the width of the pylon. Where a corrugated steel plate web is adopted for the beam of the cable stayed bridge, the cable anchorage is lo-cated at level with the upper slab and at the outer-most edge of the slab as shown in Fig. 12. A steel diaphragm is installed inside the girder at each cable anchorage. The corrugated web is longitudinally continuous even at the cable anchorage.
The CWB element has four degrees of freedom at each node while the stay cable has two degrees of freedom. For formulating the total stiffness matrix, the stiffness of a cable is incorporated for the first and second degrees of freedom, or for the horizontal and vertical degrees of freedom. At the cable anchorages at the girder and the pylon, the initial prestressing force for the cable is regarded as the axial force and its horizontal and vertical compo-nents are applied as external forces. At the same time, the moment is applied to the girder that is ob-tained by multiplying the distance between the cen-troid of the full girder cross section and the anchor-age by the horizontal force.
In the nine-stay-cable type, stay cables are grouped in three sets of three cables and are repre-sented from the left by PWS-73, -85 and -109. A prestress equivalent to approximately 90% of the to-tal dead load is provided. In the three-stay-cable type, each stay cable is regarded to be equivalent to a group of three stay cables of the nine-stay-cable type. A comparison is made between the "general element case" with general elements and the "CWB case" with CWB elements throughout the girder. Bridge erection using cranes and bents is assumed.
Full dead and live loads (760 N/cm) are applied. Only the left side of the symmetrical structure is analyzed. The structural specifications are given in Table 2.
Fig. 13 a) and b) show the stress distribution in the concrete lower slab in the left half of the center span. (The bending stress was calculated by σ =±Mf / If・e2 + Mn / In・η2. + or - in the first term indi-cates the stress on the bottom or top surface of the lower slab. For e2 and η2, refer to Fig. 2.) Non-dimensional stress is obtained using σ0, the bending stress due to Mn at the base of the pylon in the nine-stay-cable type with CWB elements.
For either the nine- or three-stay-cable type, the bending stress in the general element case is nearly identical to the stress due to Mn at the base of the pylon in the CWB case. In the analysis of the nine-stay-cable type, the stress on the lower slab due to Mn at the rigid connection to the pylon is in com-pression. In the case of the CWB element, the stress due to Mf is additionally applied, and the stress on the bottom surface of the lower slab becomes larger than that in the case of the general element by 40%. On the top surface on the other hand, the stress due to Mf offsets the stress due to Mn and becomes 60% smaller. The stress due to the axial force is the same either in the general or CWB element case. The stress of the slab fluctuates according to Mf at the cable anchorages in the CWB element case. At the midpoint of the center span, the relationship between the stresses on the top and bottom surfaces in terms of magnitude was reversed longitudinally in front of and at the back of the cable anchorage.
The properties at the base of the pylon identified by the analysis of the three-stay-cable type are the same as those obtained by the analysis of the nine-stay-cable type. The fluctuation of stress at the ca-ble anchorage is much greater than for the nine-stay-cable case.
The above discussions have found that analyzing a cable stayed bridge composed of beams with gen-eral elements cannot trace the stress characteristics at the base of the pylon, so stress is likely to be cal-culated on the dangerous side. For a cable stayed bridge with a small number of stays, drastic fluctua-tion of stress near the cable anchorage cannot be captured by general element analysis. Using CWB for analysis enables precise identification of the ba-sic stress characteristics, so the method is found effective for verifying the result of FEM analysis. The findings should be fully taken into considera-tion especially during the design of longitudinal slab cross section or for estimating prestress.
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8. FEM ANALYSIS OF A CABLE STAYED BRIDGE In this chapter, three-dimensional FEM analysis
is made of a model of the same structure as the sam-ple cable stayed bridge discussed in Chapter 7. The analysis result is compared with that obtained using CWB elements. The objective is to verify the valid-ity of the CWB matrix method. Fig. 14 shows the analysis model. Fig. 15 shows the details of the girder. Only the left half of the structure is analyzed. The stay cables, pylon and the girder on the left side of the pylon are modeled as general frame elements. The right side is modeled using finite elements. The cross section of the girder is assumed to be symmet-rical. The stay cables are anchored to the concrete upper slab. The slab is represented by eight-node solid elements. The corrugated web is represented by three-node shell elements. The functions of the diaphragm at the cable anchorage are expressed using a rigid bar with its ends pinned to the upper and lower slabs. A horizontal rigid bar is installed in the upper slab to avoid stress concentration at the anchorage. A combination of dead and live loads is distributed uniformly throughout the length of the girder as in Chapter 7.
In order to compare the results of FEM analysis with the analysis using CWB, three-stay-cable type is selected because it emphasizes stress fluctuation further and helps identify the difference between the two types of analysis more easily than the nine-stay-cable type. Fig. 16 shows stresses on the upper and lower slabs of the girder on the right side of the py-lon. Table 3 shows the displacements at the top of the pylon and at the midpoint of the center span. In the FEM analysis, the sum of axial and flexural
components of stress is calculated, so the sum of these components is shown in Fig. 16. (1) Properties at the cable anchorage
The stress distribution on the lower slab was al-most the same either in the FEM analysis or the analysis using CWB (Fig. 16 a)). One of the small differences is that the local minimum of stress on the bottom surface of the lower slab was slightly smaller in the FEM analysis than in the analysis us-ing CWB. In the FEM analysis, stresses in the ele-ments in the vicinity of the anchorage are plotted while in the analysis using CWB, the stress was calculated from the stress resultant at the anchorage. This means that the local minimum in the analysis using CWB should have been below the result of the FEM analysis. In the FEM analysis using CWB, no shear lag along the width of the slab was consid-ered, so the result of the FEM analysis was expected to be smaller than in the analysis using CWB. In a separate analysis in which the slab was regarded as steel and its width was narrowed to reduce the influ-ence of shear lag in the slab, the local minimum was nearly the same as the one in the analysis using CWB. The stress distribution on the upper slab was almost the same either in the FEM analysis or the analysis using CWB (Fig. 16 b)). In the FEM analy-sis, stress concentration due to cable anchorage was found at a point near the cable anchorage on the side of the pylon despite the attachment of a horizontal rigid bar. Thus, the compressive stress was larger than in the analysis using CWB. The difference in result was outstanding at the anchor-age because different methods were adopted for plotting the stress as described earlier. The local
Fig. 15 Details of FEM analysis model
Stay cableBar element
Slab represented by eight-node solid elements
Corrugated web represented by three-node shell elements
Horizontal rigid bar
Vertical rigid bar
Fig. 14 FEM analysis model for a cable stayed bridge
Table 3 Displacements at the top of the pylon and midpoint of center span (cm)
Horizontal displacement
at the top of the pylon
Deflection at the midpoint of center span
CWB FEM
1.1 0.9
-13.6 -14.4
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maximum in the analysis using CWB exceeded that in the FEM analysis. (2) Properties at the rigid connection to the pylon
As for the outstanding increase in stress at the rigid connection with the pylon described in Chap-ter 7, the results of FEM analysis and CWB analy-sis were in good agreement with each other and the absolute maximum and minimum were accurately identified on the sides of the slabs opposite to the web, or on the bottom surface of the lower slab and on the top surface of the upper slab. On the web sides of the upper and lower slabs, on the other hand, no significant variance was found in the FEM analysis as compared with the CWB analysis.
The result of CWB analysis was in good agree-ment with the result of FEM analysis with respect to the displacement of the top of the pylon and the de-flection of the girder.
As discussed above, the analysis using CWB cannot take into consideration the effects of the lo-cal stress concentration at the cable anchorage on the upper slab or of the shear lag along the width of the slab. A similar variance is found between the frame analysis and FEM analysis of an ordinary structure. Except for the local factors, therefore, it was verified that the analysis using CWB could accurately trace the overall behavior of a cable stayed bridge composed of beams with corrugated webs and the basic stress distribution characteristics.
-25
-20
-15
-10
-5
0
5
10
0 20 40 60 80
CL
a) 下フランジ発生応力
-10
-5
0
5
0 20 40 60 80
CL
N/mm2
b) 上フランジ発生応力
m
Local minimum in the analysis using CWB
Local minimum in FEM analysis
On the top surface in the FEM analysis
On the bottom surface in the analysis using CWB On the top surface in the analysis using CWB On the bottom surface in the FEM analysis
Absolute minima in the FEM analysis and the analysis using CWB
Fig. 16 Comparison of the results of FEM analysis and the analysis using CWB for cable stayed bridge
a) Stresses in the lower flange b) Stresses in the upper flange
Absolute maximum in the analysis using CWB Absolute maximum in the FEM analysis
Local maximum in the analysis using CWB
Local maximum in the FEM analysis
On the top surface in the FEM analysis
On the bottom surface in the analysis using CWBOn the top surface in the analysis using CWBOn the bottom surface in the FEM analysis
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9. CONCLUSIONS The elementary beam theory handles a moment
per cross section perpendicular to the axis of a member. The expanded beam theory handles the bending moment Mf carried by the slab and moment Mn consisting of the axial forces of upper and lower slabs. The three-moment formula and the displace-ment method that are developed based on the elementary beam theory can provide for the frame analysis of a general beam element. Likewise, the double three-moment formula and the matrix dis-placement method for four degrees of freedom per node that were developed based on the expanded beam theory enable in-plane frame analysis of a beam element with corrugated webs. The findings of this study are described below.
1) The expanded beam theory was evolved and the double three-moment formula was developed that ensured the continuities at each node of the first-order differentiation v' of deflection and of the angle of rotation φ of the line between the centers of upper and lower slabs. The double three-moment formula enables the analysis of a continuous beam with a corrugated steel plate web.
2) The CWB matrix displacement method that considers four degrees of freedom at a node, horizontal and vertical movements, v' and φ, enables the in-plane frame analysis of beam elements with corrugated steel plate web.
3) An analysis of a continuous girder with a corrugated steel plate web by the CWB matrix method revealed that the moment carried solely by the slab near the intermediate support increased substantially and that remarkably large normal stress occurred.
4) As a result of an analysis of a cable stayed bridge composed of beams with corrugated steel plate webs by the CWB matrix displacement method, it was found that the stress carried solely by the slab increased at the rigid connection be-tween the main girder and the pylon. Where a small number of stay cables were used, stress fluctuated greatly near the cable anchorage.
5) The result of analysis of a cable stayed bridge composed of beams with corrugated steel plate webs was compared with the result of a three- dimensional FEM analysis. It was verified even though the analysis using CWB could not take into consideration the stress concentration at the cable anchorage or shear lag in the slab, it could accu-rately trace the overall behavior of the structure and the basic characteristics of stress distribution.
6) The characteristics described in 3) and 4) were identified using the CWB element and comparing the CWB analysis result with the result of general element analysis or FEM analysis. The findings should be fully considered in actual design. REFERENCES 1) Tagawa, K., Okamoto, H. and Nakada, K. : A study on
corrugated web girders, NKK Technical Report, Vol. 71, pp. 25-33, October 1976. (in Japanese)
2) Combault, J. : The Maupre Viaduct Near Charolles, France, Proc.of AISC Engineering Conference, Miami Beach, 12-1~22, 1988.6.
3) Mizuguchi, K., Ashizuka, K., Furuta, K., Ohura, T., Taki, K. and Kato, T. : Design and construction of the Mototani Bridge, Bridge and Foundation Engineering, Vol. 32, No. 9, pp. 2-10, 1998. (in Japanese)
4) Japan Prestressed Concrete Engineering Association: New development of prestressed concrete-21st century, Proceed-ings of the 30th prestressed concrete technology conference, February 2002. (in Japanese)
5) Combault, J., Lebon, J.D., and Pei, G. : Box-Girders Using Corrugated Steel Webs and Balanced Cantilever Construc-tion, FIP Symposium Kyoto, 1993.10.
6) Kamihira, K., Yagishita, F., Ebina, T. and Sonoda, K. : A study on the method of connection of steel plates of prestressed concrete box girder bridges with corrugated steel plate webs, Proceedings of Concrete Research and Technology, Vol. 9, pp. 9-17, July 1998. (in Japanese)
7) Yamaguchi, K., Yamaguchi, T. and Ikeda, N. : A study on the mechanical behavior of composite prestressed concrete girders with corrugated steel plate webs, Proceeding of Concrete Research and Technology, Vol. 8, pp. 24-41, 1997. (in Japanese)
8) Taniguchi, N. and Yoda, T. : A study on a simple flexural analysis method for composite girders with corrugated steel plate webs, Proceeding of the Japan Society of Civil Engi-neers, No. 557/I-41, pp. 107-120, October 1997. (in Japa-nese)
9) Shiromizu, A., Sano, Y. and Oshita, S. : A study of analyti-cal method using the "constant shear panels" for composite bridges with corrugated steel webs, JSCE Journal of Struc-tural Engineering, Vol. 46A, pp. 1667-1674, March 2000. (in Japanese)
10) Expressway Technology Center: Technical assessment re-lated to the increase of span of steel-concrete composite bridges with corrugated steel plate webs, Report, March 2001. (in Japanese)
11) Kato, H., Kawabata, A. and Nishimura, N. : Practical meth-ods for calculating the displacement and stress resultant of steel-concrete composite bridges with corrugated steel plate webs, Proceeding of the Japan Society of Civil Engineers, No. 703/I-59, pp. 293-300, April 2002. (in Japanese)
12) Komatsu, S. : Structural Analysis I and II, Maruzen Co., Ltd., 1982. (in Japanese)
13) Weaver, W. Jr. and Gere, J. M. : Matrix Analysis of Framed Structures, D.Van Nostrand Company, 1980.
(Received January 28, 2004)
