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Reaction allotment of continuous curved box girders
Thin-Walled Structures 11 (1991) 319-341
Reaction Allotment of Continuous Curved Box Girders
Shingeru Shimizu & Shunya Yoshida Department of Civil Engineering, Shinshu University, 500 Wakasato, Nagano 380 Japan
(Received 13 April 1989; revised version received 14 December 1989; accepted 16 February 1990)
ABSTRACT To design load-bearing diaphragms (or support diaphragms) the magnitude of reaction forces on the support must be known. This paper considers such reaction forces on the intermediate support of two-span continuous curved box girders. When a continuous curved box girder has two shoes in the section of its support, these two shoes are subjected to reaction forces which are generally different. In this paper, the reaction forces at such shoes are numerically analyzed by thefinite strip method, and the allotment of reaction forces to the two shoes is studied. The number or stiffness of intermediate diaphragms and the dimensions of the box section are used as parameters.
NOTATION a
Subtended angle between a line load and the end o f a model girder fiE D Parameter of effective diaphragm spacings, defined in I D R or BS5400 p Coefficient of reaction allotment (CRA), defined in eqn (2) b Width o f box sections d Spacing o f two shoes on the intermediate support h Depth (or height) of box sections Im Coefficient of torsion bending of box sections K Coefficient of pure torsion of box sections 319 Thin-Walled Structures 0263-8231/91/$03.50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
320 L L/R M
Q R S td
Xi. Xout
X, 2
Shigeru Shimizu, Shunya Yoshida
Length of girders Curvature of girders Torsional moment referred to in Ref. 9 Total reaction on the center support Radius of girders Dimensionless stiffness of intermediate diaphragms, defined in IDR or BS5400 Plate thickness of intermediate diaphragms Reaction of the interior shoe Reaction of the exterior shoe Reactions on the shoes
1 INTRODUCTION To design load-bearing diaphragms (or support diaphragms) the magnitude of reactions on their supports must be known. This paper deals with such reactions on the center-supports of two-span continuous curved box girders. Many studies have considered steel box girders or intermediate diaphragms which are included in box girders. Most of these studies dealt with the behavior of box girders, such as the distortion of box sections, or the behavior or contribution of intermediate diaphragms. Steel box girders also have support diaphragms (or load-bearing diaphragms) on their supports. Support diaphragms in steel box girders have been studied, for example Rockey and E1-Gaaly, ~ E1-Gaaly, 2 Dowling et al., 3 Puthli and Crisfield,4 Sawko and Simonian, 5 Shimizu and coworkers6.7 and Yoda et al., 8 and as a result, the behavior of box girder support diaphragms has been clarified. However, to design box girder support diaphragms, it is required to know not only their behavior but also the relevant reaction forces. Generally, a box girder bridge has one or two shoe(s) in the section of its support. This study considers box girders which have two shoes in each of their intermediate support sections (see Figs 1 and 2(a)). In a straight box girder bridge, when the shape and dimensions of the girder are geometrically symmetrical, these two shoes are subjected to reactions which are equal to each other under a symmetrical load. On the other hand, in a curved box girder bridge, the two reactions are different because of the torsional moment of the girder, even if the load, the shape and dimensions of the girder are geometrically symmetric. Thus, the 'reaction allotment problem', which involves allotting or apportioning the reactions to the two shoes, must be discussed.
Reaction allotment of continuous curved box girders
321
diatesup0rt
Y 2
L/2~ f L / 2 - - - ~
/
R Fig. 1. Analyzed model.
Miyawaki and Mori studied the 'reaction allotment problem' of curved beams in the design of substructures of bridges. 9They presented in their paper a formula for estimation of the two reactions acting on the two shoes from the torsional moment and shear of the beam. That is, using the total reaction force Q of the support considered (the shear of the section considered), the torsional moment M of the section and the distance d between two shoes of the support (the 'shoe spacing'), the reactions of the two shoes, XI and )(2, are expressed as 9 X~ M X2} = Q + d -
(1)
However, Miyawaki's study did not consider the contributions of sectional distortion or the existence of diaphragms of box girders in the reactions. The authors have made a series of numerical analyses of two-span continuous curved box girders, to study the reaction allotment to the two shoes on the center supports of girders. In these analyses, the number and stiffness of intermediate diaphragms, the shapes and dimensions of box sections, the distance between two shoes, etc., are adopted as parameters. The finite strip method (FSM), which is convenient for box girders, is used for the analysis. In this paper, the outline and results of the analyses are described.
Shigeru Shimizu, Shunya Yoshida
322
(a)
ta=2Omm /.~-L0ad
/
J
Bearing S t i f f e n e r s
,~ ~ / "
(A=25.Gem 2)
/
Stiffener
(A=62.4emz)
IOIBID
(b)
t
9GOmm ~k
t,=3.0,.
1GOmm (c)
t,=8.0a,
.= Manhole
'::'
400mm
Fig. 2. Diaphragms. (a) A support diaphragm; (b) an intermediate diaphragm without manhole; (c) an intermediate diaphragm with a manhole. 2 ANALYZED MODELS 2.1 General
A typical model a d o p t e d for F S M analyses is shown in Fig. 1. This model is a two-span c o n t i n u o u s curved box girder which has total length L, radius R, depth h a n d width b, a n d has two shoes at its center support. The distance between these two shoes is d. The angle s u b t e n d e d by both ends of the m o d e l girder is L/R. All analyzed models have a support d i a p h r a g m with load-bearing stiffeners on their intermediate supports. The support d i a p h r a g m has a thickness of 20 m m , a n d the load-bearing stiffeners attached to the d i a p h r a g m s have sections of 25.6 or 6 2 . 4 c m 2. The shapes a n d d i m e n s i o n s of the support d i a p h r a g m are shown in Fig. 2(a). These
Reaction allotment of continuous curved box girders
323
shapes and dimensions of the diaphragm and load-bearing stiffeners are chosen by reference to the test models used in an earlier experimental study by the authors. 7 Some of the models analyzed in this study also have one or more intermediate diaphragm(s) in each of their spans. Some of the intermediate diaphragms have a m a n h o l e and others have none. Typical intermediate diaphragms are illustrated in Fig. 2(b). The analyzed models have upper a n d lower flange plates of 20 m m thickness and web plates of 8 m m thickness. Some models are subjected to a uniform load of 98 k N / m 2 on their upper flanges. Other models are subjected to a line load of 49 k N / m at the location of the subtended angle a, as shown in Fig. 1. These two types ofloadings cover the line load and the uniform load specified in the JSHB (Japanese Specifications for Highway Bridges)) 5 The values for Young's modulus and Poisson's ratio adopted in the analysis are 208 GPa and 0.3 respectively. 2.2 Parameters
The following parameters are adopted in this study: (a) (b) (c) (d)
n u m b e r or stiffness of intermediate diaphragms; shapes and dimensions of box sections; curvature and length of curved girders; location of line load.
(a) Intermediate diaphragms Some of the analyzed models have no intermediate diaphragms. Others have one to three intermediate diaphragms(s) in each of their spans. On models with one intermediate diaphragm in each span, the stiffness of the diaphragms is adopted as a parameter, a n d is given by varying the thickness of the diaphragms from 0.5 to 8 mm. Of course, such values of diaphragm thickness are thinner than those required by the JSHB or other bridge design codes. In this study, the thinner plates are used to give diaphragms which are not stiff, a n d buckling of the plates and problems on their fabrications are not considered.
(b) Shapes and dimensions of box sections All models analyzed in this study have rectangular sections. A typical model has a depth h of 1440 m m and width b of 960 mm. Some models have b from 960 to 1920 mm, with h = 1440 mm, i.e. these models have their width as a parameter. For other models, h is adopted as a parameter, and is varied from 1440 to 2880 mm.
324
Shigeru Shimizu, Shunya Yoshida
Most models have their 'shoe spacings' (the distance between their two shoes at the observed supports), d, given by d = (2/3)b. Thus, a typical model, for which b = 960 mm, has d = 640 mm. In some models, the shoe spacing d is treated as a parameter.
(c) Curvature and length of model girders The lengths L of the models in this study are from 10 to 90 m. Therefore, span lengths of the models are from 5 to 45 m. The curvatures are expressed as L/R, where L is the length and R is the radius of the girder. Thus, L/R also denotes the subtended angle for both ends of the girder.
(d) Location of line load On models subjected to a line load on their upper flanges, the location of the line load is adopted as a parameter. The line load location is expressed by the subtended angle a between one end of the girder and the line load.
2.3 Method of analysis In the numerical analyses, the finite strip method (FSM) is adopted, together with the finite element method (FEM) on the diaphragms. This technique was adopted by the authors in Ref. 6. A similar combination of F S M / F E M was also developed by Graves Smith et al.'° In the analysis, a support diaphragm which is infinitely stiff in its own plane is assumed at each end-support of the girders. This assumption is related to the characteristics of the FSM. Intermediate diaphragms and intermediate support diaphragms are treated as being in an in-plane stress state, and 'truss elements' which have no bending stiffness are used for stiffeners.
2.4 Coefficient of reaction allotment The reaction forces on the two shoes obtained through the analyses are denoted as Xin for the interior shoe and Xou, for the exterior shoe. Xin + Xou, is equal to the total reaction force Q of the support. Here, the following dimensionless value is defined: Xou, _ Xou, # - Xi, +Xout Q
(2)
The dimensionless value # denotes the ratio of Xo~,, the reaction force on the exterior shoe, to the total reaction Q. Through this definition, it is
Reaction allotment of continuous curved box girders
325
clear that w h e n p -- 0.5, the reactions on both shoes are equal, and that w h e n p < 0.5, then Xout < Xi,. In this study, the dimensionless value p is denoted the 'coefficient of reaction allotment (CRA)'. From eqn (1), Sin and Xout are Xin,X.ut Q/2 + M/d, thus the CRA is expressed as =
1 M /z = ~ + Q.____d
(3)
From eqn (1) or (3), it is obvious that large shoe spacing d makes p approach 0.5.
3 M O D E L G I R D E R S SUBJECTED TO U N I F O R M LOAD 3.1 Contributions of girder curvature At first, relations between the CRA (p) and the curvature L/R of the girders are studied for the model girders subjected to a uniform load. These relations are illustrated in Figs 3-5. Figure 3 shows the relationships betweenp a n d L / R for models with no intermediate diaphragm, Fig. 4 for models with an intermediate diaphragm with no m a n h o l e and with a plate thickness of 3 m m in each span (so that this model has two
_ /
~
L=lom
I
0.5
0.4 L=20= 0.3
0.2
~
\
0.1
0.0 0.0
0.5
1.O
\
~ 1,5
L=30=
--L=60=
L=45= 2.0
L/R
Fig. 3. CRA-curvature curves (without intermediate diaphragm).
Shigeru Shimizu, Shunya Yoshida
326
L=lOm L=lSm 0.5 L=20m
L=30m
0.4
intermediate diaphragm without manhole
0.3
L=45m
L=60m 0.2
0.1
td=3mm
L:90m
f
0.0 0.
0.5
1.0
1.5
2.0
L/R
Fig. 4. CRA-curvature curves (with intermediate diaphragms).
L=lOm 0.5 L=20m
L=30m
0.4
intermediate diaphragm with a manhole
0.3
L=45m
L:60m 0.2
L 0.I
ta:SmB
I
0.0 O.
0.5
L=901 1.0
1.5
2.
L/R
Fig. 5. CRA-curvature curves (with intermediate diaphragms which have a manhole)
Reaction allotment of continuous curved box girders
327
intermediate diaphragms), a n d Fig. 5 shows the relationships o f p a n d L/R for models which have an intermediate d i a p h r a g m with a m a n h o l e a n d with a plate thickness of 8 m m in each span. The CRA, p, is shown on the vertical axes of these figures, a n d the curvature L/R on the horizontal axes. For these figures, the following d i m e n s i o n s of box girders are adopted: h = 1440 m m , b = 960 m m , d = 640 m m , flange thickness is 20 m m a n d web thickness is 6 m m . In Fig. 3,/.t-L/R curves for models with lengths of 10-60 m are plotted, a n d in Figs 4 a n d 5 curves for lengths from 10 to 90 m are plotted. These three figures show that the smaller values of curvature L / R give C R A values which are close to 0.5. This result is because a curved girder with a small curvature is approximately equal to a straight girder. Within the range s h o w n in these three figures, models with length of 10 m have C R A larger t h a n 0.5, a n d their C R A increases gradually with increasing curvature. In other words, in these models, the reactions of the exterior shoes are larger t h a n those of the interior shoes. These three figures also show that models with greater length L have smaller p. For example, from Fig. 3, it is f o u n d that the model with L = 20 m a n d with no intermediate d i a p h r a g m has p < 0.5 for curvature / J R exceeded ca. 0.8. T h a t is, in the range of L/R larger than ca. 0.8, the reaction of the interior shoe is larger t h a n that of the exterior shoe. Figure 3 also shows that the model with length 60 m a n d with no intermediate d i a p h r a g m has negative/~ for L/R > 1-6. In other words, in this range o f L/R, the m o d e l has a d o w n w a r d reaction at its exterior shoe. However, this m o d e l has extreme values of parameters, so that, in practical bridges, the p r o b l e m of such negative reactions does not Occur.
Figures 4 a n d 5 show that the intermediate d i a p h r a g m s are effective in giving more similar reactions at interior a n d exterior shoes. For example, the m o d e l with L = 20 m a n d L/R = 1.5 has its/~ of about 4.2 w h e n there is no intermediate d i a p h r a g m (Fig. 3); thus the m a g n i t u d e of the reaction at the exterior shoe, Xout, is ca. 72% o f that at the interior shoe, X~n. T h a t is, in this case the difference ofX~, andXout is about 28%. O n the other h a n d , for models with the same values of length a n d L/R, a n d with an intermediate d i a p h r a g m of/~ = 5.1, the difference ofXout andX~n is only 4%. 3.2 Number of intermediate diaphragms Relationship of the C R A values with the n u m b e r of intermediate d i a p h r a g m s is s h o w n in Fig. 6. For this figure, intermediate d i a p h r a g m s with a thickness t -- 3 m m a n d without a m a n h o l e are used. T h e lengths
Shigeru Shimizu, Shunya Yoshida
328
0.5
0.4
0.3
0.2
0.1
0.0
L:30m
f • R:30m (L/R:I.O) • R:ZOm (L/R:I.5) • R:15m (L/R:2.0)
L=45m
f o R:45m (L/R:I.0) [] R=30m (L/R=I.5)
3 Number of Diaphragms per Span
R:22.Sm (L/R:2.0)
Fig. 6. Contribution of the number of diaphragms.
and curvatures of models in this figure are L = 30 m and 45 m and L~ R = 1.0, 1.5 and 2-0. As described in Section 3.1, the existence of an intermediate diaphragm makes p approach 0.5, compared with models without an intermediate diaphragm. Figure 6 shows that models with two or more intermediate diaphragms per span have almost equal values of CRA as compared with those models with one intermediate diaphragm. Thus it is found from this figure that the existence of one intermediate diaphragm per span is enough to make CRA approach 0-5. For example, a model with L = 45 m a n d R = 22.5 m has/~ of 0.02 when it has no intermediate diaphragm, i.e. this model has a very small reaction at its exterior shoe. The existence of one intermediate diaphragm per span gives a/~ value of about 0-31, and models with two or three intermediate diaphragms also have/~ of about 0.31. Although it is not shown in this paper, the CRA values of models
329
Reaction allotment of continuous curved box girders
which have intermediate diaphragms with a m a n h o l e are similar to those for models with no manhole. For models with L -- 60 m or 90 m, numerical analysis is m a d e for only the case ofL/R = 1.0. For the models with L = 60 m, the CRA value is/J -- 0.44 for the model which has one intermediate diaphragm per span, and p = 0.46 a n d 0.47 for models with two or three intermediate diaphragms respectively. For models with L --- 90 m, the values of CRA for models with one, two or three intermediate diaphragm(s) are /J--0.40, 0.42 a n d 0.42, respectively. Thus, the above description is probably true for models with L = 60 m or L = 90 m.
3.3 Contribution of diaphragm stiffness The CRA curves of models with L = 30 m or 45 m are plotted in Figs 7 a n d 8 against thickness of intermediate diaphragms, t d. In these figures, the case with no intermediate diaphragm is treated as td = 0. F r o m these figures, it is lbund that, within a small range of diaphragm thickness td, /J increases rapidly with increasing t d. In the range of t o exceeding ca. 1 mm,/J increases very slowly. It is clear from these figures that the values of CRA do not converge to 0.5 with increasing diaphragm thickness, except for the model with L = 30 m a n d / J R = 1.0. In other words, in most models,Xi, and Xout are not equal, even if the models have stiff intermediate diaphragms. Sakai and coworkers studied the contribution of intermediate diaphragms to the sectional distortion o f box girders subjected to
R=30B (L/R=1.0)
0.5
R=201 (L/R=1.5)
R=15B (L/R=2.0) 0.4
0.3
/
0.2
0.1
0.0 1.0
2.0
3.0
4.0
Fig. 7. CRA-diaphragm thickness ~lationship (L = 30 m).
t~(n)
330
Shigeru Shimizu, Shunya Yoshida
0.5 R:45m (L/R=I.O) R:30m (L/R:I.5)
0.4
R=2215m (L/R=2.01
0.3
0.2
0.1
0.0
1.0
2.0
3.0
4.0
t~(m,)
Fig. 8. CRA-diaphragm thickness relationship (L = 45 m). torsional or distortional loads. 11.~2In Refs 11 a n d 12, the m a g n i t u d e of box distortion decreases rapidly with increasing stiffness of d i a p h r a g m s within a small d i a p h r a g m stiffness range, a n d the distortion decreases very slowly with large d i a p h r a g m stiffness. The results in Figs. 7 a n d 8 show that the contribution of d i a p h r a g m stiffness to the C R A is similar to those found by Sakai a n d coworkers.
4 G I R D E R S S U B J E C T E D TO A LINE L O A D For girders subjected to a line load, as shown in Fig. 1, the location of the line load should be a d d e d to the parameters. As illustrated in Fig. l, the location of a line load is expressed by the s u b t e n d e d angle a. Hereafter, /~-a curves are plotted with/~ on the vertical axis a n d a on the horizontal axis. A d o p t i n g this convention, the curves show the variation of C R A values with moving line load in the same way as the influence lines. Of course, such 'influence line-like' C R A curves are not real influence lines, so it should be noted that the Law of Superimposition is not applicable to the curves. Figure 9 shows such 'influence line-like' C R A curves for models with no intermediate diaphragm, for L = 15 m, 30 m, 45 m a n d 60 m. All models shown in this figure have C R A close to 0-5 w h e n load location a is close to 0.5 (i.e. the loads are near the intermediate supports of c o n t i n u o u s box girder models). For a = 0.5, all models shown in this figure have/1 = 0.54, that is, all models shown here have their exterior shoe reactions Xout larger by 17% t h a n interior reactions )(in, even if the
331
Reaction allotment of continuous curved box girders
L/R=1.0 0.5-
0.4
0.3
0.2
0.1
\
L=60=
0.0
-0"1 -0.2 t -0.3
0.1
I
I
0.2
I
0.3
I
I
0.4
I
0.5
tx
Fig. 9. CRA curves against model length (subjected to a line load). line loads are applied at the i n t e r m e d i a t e supports. T h e m o d e l with L = 15 m has a C R A w h i c h is larger t h a n 0.5 but close to 0.5 for the w h o l e range o f a. For the o t h e r m o d e l s s h o w n in the figure, C R A decreases g r a d u a l l y as the line loads m o v e to the ends o f models. T h e m o d e l with L = 60 m has C R A o f a b o u t 0.16 for a = 0.1. In Fig. 10, the C R A curves against a are plotted for various thicknesses o f i n t e r m e d i a t e d i a p h r a g m s , t d. D i a p h r a g m s without a m a n h o l e are used for this figure. T h e d i a p h r a g m thickness Id 0"0 in this figure denotes that the m o d e l has no i n t e r m e d i a t e d i a p h r a g m . F r o m this figure, it is f o u n d that m o d e l s with i n t e r m e d i a t e d i a p h r a g m s have a p p r o x i m a t e l y the s a m e C R A curves. This fact is similar to the description in Section 3 o f the c o n t r i b u t i o n o f d i a p h r a g m stiffness to the C R A o f m o d e l s subjected to a u n i f o r m l y distributed load. =
5 CONTRIBUTION OF SECTIONAL DIMENSIONS In the descriptions in Sections 3 a n d 4, the box m o d e l s a n a l y z e d have typical sectional d i m e n s i o n s o f h = 1440 m m a n d b = 960 m m , a n d d = (2/3) b - - 6 4 0 m m . In this section, m o d e l s w h i c h have various sectional d i m e n s i o n s a n d shoe spacings are dealt with. Figure 1 l(a) a n d (b) shows C R A curves for m o d e l s with h -- 1440 m m , L = 60 m and/_JR = 1.0. Box widths b are s h o w n in the figures. M o d e l s
332
Shigeru Shimizu, Shunya Yoshida L=60= L/R=I.O
/J
0,5 ¸ 0.4" 0.3 a= 0.2
x~
1_.0ram
ta=O.511
\
0.I-
0,0-
~--
ta=O. OiII
-0.i-
-0.2
-0.3 0,1
0.2
03
04
0.5
Fig. 10. CRA curves against model thickness (subjected to a line load). in Fig. 1 l(a) have shoe spacings d o f d -- b/3, a n d those in Fig. 1 l(b) have d = 2b/3. F r o m Fig. l l(a), it is f o u n d that the m o d e l with b = 9 6 0 m m (d = 320 m m ) h a s p = - 0.06 f o r a = 0.1; in such a case this m o d e l has a d o w n w a r d reaction at its exterior shoe. This figure also shows that, for a=0.1, the m o d e l has p = 0 . 0 with b = 1 2 0 0 m m , p = 0 . 0 6 with b = 1440 mm, p = 0.11 w i t h b = 1680 mm, and the model with b = 1920 m m has C R A o f p = 0.16. Thus, it is f o u n d from this figure that, for a = 0.1, increasing b o x width b b y 240 m m enlarges p b y a b o u t 0.05 or 0.06. Figure 12(a)-(c) shows C R A curves for the s a m e m o d e l s as those used in Fig. 11, b u t with different shoe spacings d. M o d e l s in Fig. 12(a)-(c) have fixed shoe spacings o f d = 640 mm, 800 m m and 960 m m respectively, in spite o f their width b. It is clear from Fig. 12 that C R A values are not c o n t r i b u t e d b y b o x widths b b u t are c o n t r i b u t e d b y shoe spacings d. These facts, o b t a i n e d from Figs 11 a n d 12, are p r e s u m a b l y related to the definition o f CRA,/1, s h o w n in eqns (1) or (3), in w h i c h d is in the denominator. In Fig. 13, C R A curves are plotted using b o x d e p t h h as a parameter. In this figure, values o f h = 1440 m m to h = 2880 m m are adopted. M o d e l s in Fig. 13(a) have b o x width b = 960 m m a n d shoe spacing d = 640 m m ; in Fig. 13(b) b = 1400 m m a n d d = 960 mm, a n d in Fig. 13(c) b = 1920 m m a n d d = 1280 mm.
Reaction allotment of continuous curved box girders
(a)
333
L=601 L/I~=I.O h=144c~,
.t.z 0.5 04-
0.30.2-
b=960mm(d=3201u) -0"I t -0.2 -0.3
(b)
0.1
I
r 0.2
I
r 0.3
I
I 0.4
I
I
I
0.5
L=601 L/R=I.O h=144c''
~,
0.40.30.20.10.0-
-0"1 t -0.2 -0.3
0.1
I
I
0.2
I
I
0.3
I
0.4
0.5
Fig. 11. CRA curves against box width (subjected to a line load). (a) Shoe spacing d = ( b ) shoe spacing d = 2b/3.
b/3:
From Fig. 13, CRA values for models with smaller depth are closer to 0.5 than are those for models with larger depth. Figure 13 also shows that CRA values for models with small box width (or with small shoe spacing) are more sensitive to variation of box depth. In Table 1, values of K and Ira, which are of course related to
Shigeru Shimizu, Shunya Yoshida
334
(a)
L=60= L/R=1.0
/z
0.5
h=144cm
f
0.4-
0.3-
0.2-
0.1
0.0 \
-°1t
k__ b=1200n
-0.2
-0.3
r
0.1
(b)
I 0.2
I 0.3
L:60m L/R=1.0
~'
r
I 0.4
(z
I
0.5
h=144cm
0.5 0.4-
0.3-
0.2-
0.1-
' \ '~
b=9GOmm
'\,~ \ ~-
b= 1680sn
I
I 0.4
b= 1440m=
0.0-
-0.1-
-0.2
-0.3
I
0.1
~ 0.2
I
I 0.3
I
0.5
Fig. 12. C R A curves against box width (subjected to a line load). (a) Shoe spacing d = 640 m: (b) shoe spacing d ---800 m:
Reaction allotment of continuous curved box girders L=60" L/R=I.O
(c)
335
h=144c,*
0.5
0.4-
0.3-
0.2-
0.1
0.0
-0.1
-0.2 -0.3
I
0.1
I 0.2
I
i 0.3
I 0.4
I
0.5
(c) shoe spacing d = 960 m. d i m e n s i o n s o f b o x sections, are listed with sectional dimensions. Table 1 a n d Figs 11-13 also indicate that the c o n s t a n t s o f pure b e n d i n g K or c o n s t a n t s o f torsion b e n d i n g Im o f b o x sections m a k e a small c o n t r i b u t i o n to CRA. It is o b v i o u s from Fig. 11 a n d Table 1 that b o x m o d e l s with different K or Im values have a p p r o x i m a t e l y equal C R A values. O n the o t h e r h a n d , these figures a n d the table d e n o t e that s o m e m o d e l s have different C R A values even if they have a p p r o x i m a t e l y equal K orlm. F o r example, Fig. 13(a) s h o w s that the C R A value for the m o d e l which h a s h = 2880 mm, b = 960 mm, d = 640 m m a n d K = 3747 × 106 cm 4 is/1 = - 0 . 1 7 for a = 0-1; the m o d e l with h = 1440 m m , b = 1440 m m , d = 640 m m a n d K = 3413 X 10 6 c m 4 has/1 = 0.12; a n d the m o d e l with h = 1440 m m , b = 1680 m m , d = 640 m m a n d K = 4.434 × 10 6 c m 4 has /.t = 0.13 (Fig. 12(a)). Thus, c o n s t a n t s o f pure torsion K or c o n s t a n t s o f torsion b e n d i n g Im are not suitable as indicators or p a r a m e t e r s o f C R A values.
6 REMARKS ON THE RESULTS OF ANALYSES In this paper, reaction allotments for two shoes o n the i n t e r m e d i a t e s u p p o r t o f t w o - s p a n c o n t i n u o u s curved b o x girders are studied. T h e reaction allotments are d i s c u s s e d b y using the d i m e n s i o n l e s s value/.t, or
Shigeru Shimizu, Shunya Yoshida
336
(a)
L=60m L/R=I.O
u
b=96cm
0.5
0.4
0.3
0.2" 0.1-
j//\\\~,:,,oo..
0.0-0.i -
~'~ h=2880111
-0.2
-0.3
J
(b)
F
I
1
0.2
0.
~
I
0.3
t:60m L / R : 1 . 0
I
I
0.4
0.5
b:144cz
0.4 0.3
0.2
o~
~
\\\~=~.o. ~
h:2160p
-0.I
-0.2
-0.3
I
O.
I 0.2
I
I 0.3
i
I 0.4
OC
I
0.5
Fig. 13. C R A curves against box depth (subjected to a line load). (a) Shoe spacing d = 640 m; (b) shoe spacing d =960 m:
337
Reaction allotment of continuous curved box girders
(c)
L=601 L/R=I.O
/J
b=192cs
0.5
0.4-
0.30.2
0.1
0.0
-0.1
-0.2 -
-0.3
I
I
0.1
I
I
0.2
0.3
I
I
I
0.4
o¢
0.5
(c) shoe spacing d = 1280 m.
the coefficient of reaction allotment (CRA);/.t has a value of 0.5 when the shoes are subjected to equal reactions. At first, contributions of girder curvature to CRA are studied for models subjected to a uniformly distributed load. It should be obvious that small curvature gives CRA close to 0.5 because girders with such curvature are approximately equal to straight girders. From the analyses, TABLE 1 Constants of Pure Torsion and Torsion Bending h (mm)
b (ram) 960
1440 1800 2160 2520 2880
K (X Ira (X K (X /ra (X K (× Ira (X K (X 1m (X K (X Ira (X
106cm 4) 109 cm 6) 106 cm 4) 109 cm 6) 106cm 4) 109 cm 6) 106 cm 4) 109 cm 6) 106 cm 4) 109 cm 6)
1"676 0"8199 2' 187 1"757 2"704 3" 185 3'225 5"192 3"747 7"867
1440
3'413 1"326 4"524 3'209 5"658 6"226 6"805 10'59 7"962 16'51
1680
4'434 1"455 5"919 3'808 7'439 7'707 8"984 13"46 10"55 21'35
1920
5"539 1"473 7"442 4.244 9.399 9"019 11-39 16"22 13'41 26"24
338
Shigeru Shimizu, Shunya Yoshida
it was found that larger girder curvature gives C R A values larger t h a n 0.5 for short models a n d gives C R A smaller t h a n 0.5 for long models. That is, the results described here indicate that, in short girders, the exterior shoes are subjected to larger reactions t h a n those of interior shoes, a n d in long girders interior shoes are subjected to larger reactions. On models subjected to a line load, C R A values generally decrease as the line load moves from the intermediate support to the e n d of the girder. As predicted from eqn (1) or (3), the C R A value is contributed by the spacing of the two shoes on the intermediate support. The existence of an intermediate d i a p h r a g m per span results in the two reactions on the shoes on the intermediate support closer to each other than those of models with no intermediate diaphragm. However, in some models, especially in those with large curvature, even an intermediate d i a p h r a g m does not result in C R A values of 0-5. F r o m the analytical results in this paper, it is seen that the existence of two or more intermediate diaphragms makes only a small contribution to C R A values. Models with two or more intermediate d i a p h r a g m s per span have approximately equal C R A values to those of models with a diaphragm. Increase of the plate thickness of an intermediate d i a p h r a g m makes C R A values larger in the small thickness range, but the increase of thickness makes a small contribution after it exceeds about 1 ram. These results indicate that, in most cases, a curved two-span continuous box girder with two shoes on its intermediate support has reactions at the shoes which are different, even if the girder has a stiff intermediate d i a p h r a g m or has multiple diaphragms. In other words, one intermediate d i a p h r a g m with thickness td over 1 m m has a sufficient effect on the CRA. A typical intermediate d i a p h r a g m a d o p t e d in this study has a thickness of 3 ram, a n d is t h i n n e r than d i a p h r a g m s used in practical bridges. The JSHB requires the thickness of plates such as d i a p h r a g m s to be over 8 mm. Thus, intermediate d i a p h r a g m s which are designed in accordance with the JSHB have a sufficient effect on CRA. The stiffness of intermediate d i a p h r a g m s is also specified in IDW 3 (Clause 6.3.4) o r B8540014 (Clause B.3.4.). In these design rules, the required dimensionless stiffnesses S of d i a p h r a g m s are shown against /3LD (parameter for spacing of effective diaphragms). In Table 2, values of S and/3LD for models with typical sectional d i m e n s i o n s used in this study are shown. The specifications in I D R or BS5400 are for d i a p h r a g m s in box girders subjected to torque, so this table probably does not give valid information. However, the table predicts that intermediate d i a p h r a g m s designed in accordance with I D R or BS5400 also have sufficient effect on CRA.
339
Reaction allotment of continuous curved box girders TABLE 2 Dimensionless Stiffness of Diaphragms (S) and Diaphragm Spacing Parameters (LD)
L(m) 10 tiLt) S
0'26 2042
20 0"39 1362
30 0"52 1021
40
50
0"78 681
1"17 454
60 1'57 340
TABLE 3 Ratios of Xin and Xout CRA 0"60 Xn.t/Xin Xin/Xou !
1"50 0"67
0"55 1"22 0-82
0"50 1'00 1-00
0"45
0"40
0 " 8 2 0'67 1-22 1'50
0"35
0"30
0"25
0"20
0 " 5 4 0 " 4 3 0 " 3 3 0"25 1"85 2 " 3 3 3 " 0 0 4'00
It should be noted again that, in most cases, the two reactions at the shoes of the intermediate support are not equal, even if the intermediate diaphragms have a sufficient effect on CRA. Such diaphragms merely make the reactions closer to each other than those of models with no diaphragm. Table 3 indicates ratios o f Xin and Xo,t derived from the definition of CRA for some values ofp. This table shows that, for example, for the case ofp = 0.4, an interior shoe is subjected to a reaction which is 1.5 times that at the exterior shoe. Generally, a support diaphragm is subjected to a large reaction force, so that attention should be paid to the buckling behaviour or load-bearing capacity of such support diaphragms. To design a support diaphragm, the magnitude of reactions should be taken into account. Results described in this paper indicate that attention also should be paid to the allotment of reaction to the two shoes on each support of curved box girders. 7 CONCLUSION The behaviour of support diaphragms in box girder bridges has been studied by many researchers, and from these studies, the behavior and load-bearing capacities of support diaphragms have been clarified. However, to design support diaphragms, the magnitude of reaction forces must be known.
340
Shigeru Shimizu, Shunya Yoshida
In this paper, the reaction forces on curved two-span continuous box girders are studied. Although a limited n u m b e r of cases is dealt with, the results shown here give some suggestions on design. In addition, further study should be m a d e on this theme. In this paper, decreased thickness or larger spacings of intermediate diaphragms t h a n those used in practical bridges are adopted. However, the results shown in this paper indicate that such d i m e n s i o n s are sufficient to the contributions of diaphragms on the reaction allotment.
ACKNOWLEDGEMENT The numerical calculations in this study were carried out by using the C o m p u t e r Center of S h i n s h u University.
REFERENCES 1. Rockey, K. C. & El-Gaaly, M. A., Stability of load bearing trapezoidal diaphragms. Publ. IABSE, 32-II (1972) 155-72. 2. E1-Gaaly, M. A., Stability oforthogonally stiffened load bearing trapezoidal diaphragms. Publ. IABSE, 34-11 (1974) 73-89. 3. Dowling, P. J., Loe, J. A. & Dean, J. A., The behaviour up to collapse of load bearing diaphragms in rectangular and trapezoidal stiffened steel box girders. In Steel Box Girder Bridges. Proc. Int. Conf. ICE, London, 1973, pp. 95-117. 4. Puthli, R. S. & Crisfield, M. A., Strength of stiffened box girder diaphragms. TRRL Supplementary Report 353. Structural Department, Transport and Road Research Laboratory, Crowthorne, Berkshire, UK, 1977. 5. Sawko, F. & Simonian, W. S., Elastic and buckling analysis of trapezoidal support diaphragms. Proc. ICE, 3 (1978) 17-39. 6. Shimizu, S., Kajita, T. & Naruoka, M., The stresses near intermediate supports in box girders (in Japanese). Proc. JSCE, No. 276 (1978) 13-23. 7. Fukumoto, Y., Shimizu, S. & Furuta, H., Strength of steel box girder diaphragms (in Japanese). Proc. JSCE, No. 318 (1982) 149-61. 8. Yoda, T.,Hirashima, M., Nakada, T. & Kanda, K., On the simplified design method of support diaphragms (in Japanese). Proc. Annual Meeting, JSCE, No. 42 (1987) 440-1. 9. Miyawaki, T. & Mori, S., A consideration on characteristics of supportreaction in two-span continuous curved beams (in Japanese). Bridge Engng, 15(4) (1979) 46-51. 10. Graves Smith, T. R., Gierlinski, J. T. & Walker, B., A combined finite strip/ finite element method for analysing thin-walled structures. Thin-Walled Structures, 3 (1985) 163-80.
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11. Okumura, T. & Sakai, F., Cross-sectional deformations of box-girders and the influences of intermediate diaphragms (in Japanese). Proc. JSCE, No. 190 (1971) 23-36. 12. Sakai, F. & Nakamura, H., A method for numerical analysis of thin-walled curved beams consisting of plates and shells (in Japanese). Proc. JSCE, No. 235 (1975) 41-54. 13. Committee into the Basis of Design and Method of Erection of Steel Box Girder Bridges. Interim Design and Workmanship Rules (IDR). HMSO, London, 1973. 14. Code of Practicefor Design of Steel Bridges, Part 3 of BS5400, Steel, Concrete and Composite Bridges. British Standard Institution, London, 1982. 15. Japan Road Association, Japanese Specifications for Highway Bridges (JSHB). Maruzen, Tokyo, 1980.
Reaction Allotment of Continuous Curved Box Girders
Shingeru Shimizu & Shunya Yoshida Department of Civil Engineering, Shinshu University, 500 Wakasato, Nagano 380 Japan
(Received 13 April 1989; revised version received 14 December 1989; accepted 16 February 1990)
ABSTRACT To design load-bearing diaphragms (or support diaphragms) the magnitude of reaction forces on the support must be known. This paper considers such reaction forces on the intermediate support of two-span continuous curved box girders. When a continuous curved box girder has two shoes in the section of its support, these two shoes are subjected to reaction forces which are generally different. In this paper, the reaction forces at such shoes are numerically analyzed by thefinite strip method, and the allotment of reaction forces to the two shoes is studied. The number or stiffness of intermediate diaphragms and the dimensions of the box section are used as parameters.
NOTATION a
Subtended angle between a line load and the end o f a model girder fiE D Parameter of effective diaphragm spacings, defined in I D R or BS5400 p Coefficient of reaction allotment (CRA), defined in eqn (2) b Width o f box sections d Spacing o f two shoes on the intermediate support h Depth (or height) of box sections Im Coefficient of torsion bending of box sections K Coefficient of pure torsion of box sections 319 Thin-Walled Structures 0263-8231/91/$03.50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
320 L L/R M
Q R S td
Xi. Xout
X, 2
Shigeru Shimizu, Shunya Yoshida
Length of girders Curvature of girders Torsional moment referred to in Ref. 9 Total reaction on the center support Radius of girders Dimensionless stiffness of intermediate diaphragms, defined in IDR or BS5400 Plate thickness of intermediate diaphragms Reaction of the interior shoe Reaction of the exterior shoe Reactions on the shoes
1 INTRODUCTION To design load-bearing diaphragms (or support diaphragms) the magnitude of reactions on their supports must be known. This paper deals with such reactions on the center-supports of two-span continuous curved box girders. Many studies have considered steel box girders or intermediate diaphragms which are included in box girders. Most of these studies dealt with the behavior of box girders, such as the distortion of box sections, or the behavior or contribution of intermediate diaphragms. Steel box girders also have support diaphragms (or load-bearing diaphragms) on their supports. Support diaphragms in steel box girders have been studied, for example Rockey and E1-Gaaly, ~ E1-Gaaly, 2 Dowling et al., 3 Puthli and Crisfield,4 Sawko and Simonian, 5 Shimizu and coworkers6.7 and Yoda et al., 8 and as a result, the behavior of box girder support diaphragms has been clarified. However, to design box girder support diaphragms, it is required to know not only their behavior but also the relevant reaction forces. Generally, a box girder bridge has one or two shoe(s) in the section of its support. This study considers box girders which have two shoes in each of their intermediate support sections (see Figs 1 and 2(a)). In a straight box girder bridge, when the shape and dimensions of the girder are geometrically symmetrical, these two shoes are subjected to reactions which are equal to each other under a symmetrical load. On the other hand, in a curved box girder bridge, the two reactions are different because of the torsional moment of the girder, even if the load, the shape and dimensions of the girder are geometrically symmetric. Thus, the 'reaction allotment problem', which involves allotting or apportioning the reactions to the two shoes, must be discussed.
Reaction allotment of continuous curved box girders
321
diatesup0rt
Y 2
L/2~ f L / 2 - - - ~
/
R Fig. 1. Analyzed model.
Miyawaki and Mori studied the 'reaction allotment problem' of curved beams in the design of substructures of bridges. 9They presented in their paper a formula for estimation of the two reactions acting on the two shoes from the torsional moment and shear of the beam. That is, using the total reaction force Q of the support considered (the shear of the section considered), the torsional moment M of the section and the distance d between two shoes of the support (the 'shoe spacing'), the reactions of the two shoes, XI and )(2, are expressed as 9 X~ M X2} = Q + d -
(1)
However, Miyawaki's study did not consider the contributions of sectional distortion or the existence of diaphragms of box girders in the reactions. The authors have made a series of numerical analyses of two-span continuous curved box girders, to study the reaction allotment to the two shoes on the center supports of girders. In these analyses, the number and stiffness of intermediate diaphragms, the shapes and dimensions of box sections, the distance between two shoes, etc., are adopted as parameters. The finite strip method (FSM), which is convenient for box girders, is used for the analysis. In this paper, the outline and results of the analyses are described.
Shigeru Shimizu, Shunya Yoshida
322
(a)
ta=2Omm /.~-L0ad
/
J
Bearing S t i f f e n e r s
,~ ~ / "
(A=25.Gem 2)
/
Stiffener
(A=62.4emz)
IOIBID
(b)
t
9GOmm ~k
t,=3.0,.
1GOmm (c)
t,=8.0a,
.= Manhole
'::'
400mm
Fig. 2. Diaphragms. (a) A support diaphragm; (b) an intermediate diaphragm without manhole; (c) an intermediate diaphragm with a manhole. 2 ANALYZED MODELS 2.1 General
A typical model a d o p t e d for F S M analyses is shown in Fig. 1. This model is a two-span c o n t i n u o u s curved box girder which has total length L, radius R, depth h a n d width b, a n d has two shoes at its center support. The distance between these two shoes is d. The angle s u b t e n d e d by both ends of the m o d e l girder is L/R. All analyzed models have a support d i a p h r a g m with load-bearing stiffeners on their intermediate supports. The support d i a p h r a g m has a thickness of 20 m m , a n d the load-bearing stiffeners attached to the d i a p h r a g m s have sections of 25.6 or 6 2 . 4 c m 2. The shapes a n d d i m e n s i o n s of the support d i a p h r a g m are shown in Fig. 2(a). These
Reaction allotment of continuous curved box girders
323
shapes and dimensions of the diaphragm and load-bearing stiffeners are chosen by reference to the test models used in an earlier experimental study by the authors. 7 Some of the models analyzed in this study also have one or more intermediate diaphragm(s) in each of their spans. Some of the intermediate diaphragms have a m a n h o l e and others have none. Typical intermediate diaphragms are illustrated in Fig. 2(b). The analyzed models have upper a n d lower flange plates of 20 m m thickness and web plates of 8 m m thickness. Some models are subjected to a uniform load of 98 k N / m 2 on their upper flanges. Other models are subjected to a line load of 49 k N / m at the location of the subtended angle a, as shown in Fig. 1. These two types ofloadings cover the line load and the uniform load specified in the JSHB (Japanese Specifications for Highway Bridges)) 5 The values for Young's modulus and Poisson's ratio adopted in the analysis are 208 GPa and 0.3 respectively. 2.2 Parameters
The following parameters are adopted in this study: (a) (b) (c) (d)
n u m b e r or stiffness of intermediate diaphragms; shapes and dimensions of box sections; curvature and length of curved girders; location of line load.
(a) Intermediate diaphragms Some of the analyzed models have no intermediate diaphragms. Others have one to three intermediate diaphragms(s) in each of their spans. On models with one intermediate diaphragm in each span, the stiffness of the diaphragms is adopted as a parameter, a n d is given by varying the thickness of the diaphragms from 0.5 to 8 mm. Of course, such values of diaphragm thickness are thinner than those required by the JSHB or other bridge design codes. In this study, the thinner plates are used to give diaphragms which are not stiff, a n d buckling of the plates and problems on their fabrications are not considered.
(b) Shapes and dimensions of box sections All models analyzed in this study have rectangular sections. A typical model has a depth h of 1440 m m and width b of 960 mm. Some models have b from 960 to 1920 mm, with h = 1440 mm, i.e. these models have their width as a parameter. For other models, h is adopted as a parameter, and is varied from 1440 to 2880 mm.
324
Shigeru Shimizu, Shunya Yoshida
Most models have their 'shoe spacings' (the distance between their two shoes at the observed supports), d, given by d = (2/3)b. Thus, a typical model, for which b = 960 mm, has d = 640 mm. In some models, the shoe spacing d is treated as a parameter.
(c) Curvature and length of model girders The lengths L of the models in this study are from 10 to 90 m. Therefore, span lengths of the models are from 5 to 45 m. The curvatures are expressed as L/R, where L is the length and R is the radius of the girder. Thus, L/R also denotes the subtended angle for both ends of the girder.
(d) Location of line load On models subjected to a line load on their upper flanges, the location of the line load is adopted as a parameter. The line load location is expressed by the subtended angle a between one end of the girder and the line load.
2.3 Method of analysis In the numerical analyses, the finite strip method (FSM) is adopted, together with the finite element method (FEM) on the diaphragms. This technique was adopted by the authors in Ref. 6. A similar combination of F S M / F E M was also developed by Graves Smith et al.'° In the analysis, a support diaphragm which is infinitely stiff in its own plane is assumed at each end-support of the girders. This assumption is related to the characteristics of the FSM. Intermediate diaphragms and intermediate support diaphragms are treated as being in an in-plane stress state, and 'truss elements' which have no bending stiffness are used for stiffeners.
2.4 Coefficient of reaction allotment The reaction forces on the two shoes obtained through the analyses are denoted as Xin for the interior shoe and Xou, for the exterior shoe. Xin + Xou, is equal to the total reaction force Q of the support. Here, the following dimensionless value is defined: Xou, _ Xou, # - Xi, +Xout Q
(2)
The dimensionless value # denotes the ratio of Xo~,, the reaction force on the exterior shoe, to the total reaction Q. Through this definition, it is
Reaction allotment of continuous curved box girders
325
clear that w h e n p -- 0.5, the reactions on both shoes are equal, and that w h e n p < 0.5, then Xout < Xi,. In this study, the dimensionless value p is denoted the 'coefficient of reaction allotment (CRA)'. From eqn (1), Sin and Xout are Xin,X.ut Q/2 + M/d, thus the CRA is expressed as =
1 M /z = ~ + Q.____d
(3)
From eqn (1) or (3), it is obvious that large shoe spacing d makes p approach 0.5.
3 M O D E L G I R D E R S SUBJECTED TO U N I F O R M LOAD 3.1 Contributions of girder curvature At first, relations between the CRA (p) and the curvature L/R of the girders are studied for the model girders subjected to a uniform load. These relations are illustrated in Figs 3-5. Figure 3 shows the relationships betweenp a n d L / R for models with no intermediate diaphragm, Fig. 4 for models with an intermediate diaphragm with no m a n h o l e and with a plate thickness of 3 m m in each span (so that this model has two
_ /
~
L=lom
I
0.5
0.4 L=20= 0.3
0.2
~
\
0.1
0.0 0.0
0.5
1.O
\
~ 1,5
L=30=
--L=60=
L=45= 2.0
L/R
Fig. 3. CRA-curvature curves (without intermediate diaphragm).
Shigeru Shimizu, Shunya Yoshida
326
L=lOm L=lSm 0.5 L=20m
L=30m
0.4
intermediate diaphragm without manhole
0.3
L=45m
L=60m 0.2
0.1
td=3mm
L:90m
f
0.0 0.
0.5
1.0
1.5
2.0
L/R
Fig. 4. CRA-curvature curves (with intermediate diaphragms).
L=lOm 0.5 L=20m
L=30m
0.4
intermediate diaphragm with a manhole
0.3
L=45m
L:60m 0.2
L 0.I
ta:SmB
I
0.0 O.
0.5
L=901 1.0
1.5
2.
L/R
Fig. 5. CRA-curvature curves (with intermediate diaphragms which have a manhole)
Reaction allotment of continuous curved box girders
327
intermediate diaphragms), a n d Fig. 5 shows the relationships o f p a n d L/R for models which have an intermediate d i a p h r a g m with a m a n h o l e a n d with a plate thickness of 8 m m in each span. The CRA, p, is shown on the vertical axes of these figures, a n d the curvature L/R on the horizontal axes. For these figures, the following d i m e n s i o n s of box girders are adopted: h = 1440 m m , b = 960 m m , d = 640 m m , flange thickness is 20 m m a n d web thickness is 6 m m . In Fig. 3,/.t-L/R curves for models with lengths of 10-60 m are plotted, a n d in Figs 4 a n d 5 curves for lengths from 10 to 90 m are plotted. These three figures show that the smaller values of curvature L / R give C R A values which are close to 0.5. This result is because a curved girder with a small curvature is approximately equal to a straight girder. Within the range s h o w n in these three figures, models with length of 10 m have C R A larger t h a n 0.5, a n d their C R A increases gradually with increasing curvature. In other words, in these models, the reactions of the exterior shoes are larger t h a n those of the interior shoes. These three figures also show that models with greater length L have smaller p. For example, from Fig. 3, it is f o u n d that the model with L = 20 m a n d with no intermediate d i a p h r a g m has p < 0.5 for curvature / J R exceeded ca. 0.8. T h a t is, in the range of L/R larger than ca. 0.8, the reaction of the interior shoe is larger t h a n that of the exterior shoe. Figure 3 also shows that the model with length 60 m a n d with no intermediate d i a p h r a g m has negative/~ for L/R > 1-6. In other words, in this range o f L/R, the m o d e l has a d o w n w a r d reaction at its exterior shoe. However, this m o d e l has extreme values of parameters, so that, in practical bridges, the p r o b l e m of such negative reactions does not Occur.
Figures 4 a n d 5 show that the intermediate d i a p h r a g m s are effective in giving more similar reactions at interior a n d exterior shoes. For example, the m o d e l with L = 20 m a n d L/R = 1.5 has its/~ of about 4.2 w h e n there is no intermediate d i a p h r a g m (Fig. 3); thus the m a g n i t u d e of the reaction at the exterior shoe, Xout, is ca. 72% o f that at the interior shoe, X~n. T h a t is, in this case the difference ofX~, andXout is about 28%. O n the other h a n d , for models with the same values of length a n d L/R, a n d with an intermediate d i a p h r a g m of/~ = 5.1, the difference ofXout andX~n is only 4%. 3.2 Number of intermediate diaphragms Relationship of the C R A values with the n u m b e r of intermediate d i a p h r a g m s is s h o w n in Fig. 6. For this figure, intermediate d i a p h r a g m s with a thickness t -- 3 m m a n d without a m a n h o l e are used. T h e lengths
Shigeru Shimizu, Shunya Yoshida
328
0.5
0.4
0.3
0.2
0.1
0.0
L:30m
f • R:30m (L/R:I.O) • R:ZOm (L/R:I.5) • R:15m (L/R:2.0)
L=45m
f o R:45m (L/R:I.0) [] R=30m (L/R=I.5)
3 Number of Diaphragms per Span
R:22.Sm (L/R:2.0)
Fig. 6. Contribution of the number of diaphragms.
and curvatures of models in this figure are L = 30 m and 45 m and L~ R = 1.0, 1.5 and 2-0. As described in Section 3.1, the existence of an intermediate diaphragm makes p approach 0.5, compared with models without an intermediate diaphragm. Figure 6 shows that models with two or more intermediate diaphragms per span have almost equal values of CRA as compared with those models with one intermediate diaphragm. Thus it is found from this figure that the existence of one intermediate diaphragm per span is enough to make CRA approach 0-5. For example, a model with L = 45 m a n d R = 22.5 m has/~ of 0.02 when it has no intermediate diaphragm, i.e. this model has a very small reaction at its exterior shoe. The existence of one intermediate diaphragm per span gives a/~ value of about 0-31, and models with two or three intermediate diaphragms also have/~ of about 0.31. Although it is not shown in this paper, the CRA values of models
329
Reaction allotment of continuous curved box girders
which have intermediate diaphragms with a m a n h o l e are similar to those for models with no manhole. For models with L -- 60 m or 90 m, numerical analysis is m a d e for only the case ofL/R = 1.0. For the models with L = 60 m, the CRA value is/J -- 0.44 for the model which has one intermediate diaphragm per span, and p = 0.46 a n d 0.47 for models with two or three intermediate diaphragms respectively. For models with L --- 90 m, the values of CRA for models with one, two or three intermediate diaphragm(s) are /J--0.40, 0.42 a n d 0.42, respectively. Thus, the above description is probably true for models with L = 60 m or L = 90 m.
3.3 Contribution of diaphragm stiffness The CRA curves of models with L = 30 m or 45 m are plotted in Figs 7 a n d 8 against thickness of intermediate diaphragms, t d. In these figures, the case with no intermediate diaphragm is treated as td = 0. F r o m these figures, it is lbund that, within a small range of diaphragm thickness td, /J increases rapidly with increasing t d. In the range of t o exceeding ca. 1 mm,/J increases very slowly. It is clear from these figures that the values of CRA do not converge to 0.5 with increasing diaphragm thickness, except for the model with L = 30 m a n d / J R = 1.0. In other words, in most models,Xi, and Xout are not equal, even if the models have stiff intermediate diaphragms. Sakai and coworkers studied the contribution of intermediate diaphragms to the sectional distortion o f box girders subjected to
R=30B (L/R=1.0)
0.5
R=201 (L/R=1.5)
R=15B (L/R=2.0) 0.4
0.3
/
0.2
0.1
0.0 1.0
2.0
3.0
4.0
Fig. 7. CRA-diaphragm thickness ~lationship (L = 30 m).
t~(n)
330
Shigeru Shimizu, Shunya Yoshida
0.5 R:45m (L/R=I.O) R:30m (L/R:I.5)
0.4
R=2215m (L/R=2.01
0.3
0.2
0.1
0.0
1.0
2.0
3.0
4.0
t~(m,)
Fig. 8. CRA-diaphragm thickness relationship (L = 45 m). torsional or distortional loads. 11.~2In Refs 11 a n d 12, the m a g n i t u d e of box distortion decreases rapidly with increasing stiffness of d i a p h r a g m s within a small d i a p h r a g m stiffness range, a n d the distortion decreases very slowly with large d i a p h r a g m stiffness. The results in Figs. 7 a n d 8 show that the contribution of d i a p h r a g m stiffness to the C R A is similar to those found by Sakai a n d coworkers.
4 G I R D E R S S U B J E C T E D TO A LINE L O A D For girders subjected to a line load, as shown in Fig. 1, the location of the line load should be a d d e d to the parameters. As illustrated in Fig. l, the location of a line load is expressed by the s u b t e n d e d angle a. Hereafter, /~-a curves are plotted with/~ on the vertical axis a n d a on the horizontal axis. A d o p t i n g this convention, the curves show the variation of C R A values with moving line load in the same way as the influence lines. Of course, such 'influence line-like' C R A curves are not real influence lines, so it should be noted that the Law of Superimposition is not applicable to the curves. Figure 9 shows such 'influence line-like' C R A curves for models with no intermediate diaphragm, for L = 15 m, 30 m, 45 m a n d 60 m. All models shown in this figure have C R A close to 0-5 w h e n load location a is close to 0.5 (i.e. the loads are near the intermediate supports of c o n t i n u o u s box girder models). For a = 0.5, all models shown in this figure have/1 = 0.54, that is, all models shown here have their exterior shoe reactions Xout larger by 17% t h a n interior reactions )(in, even if the
331
Reaction allotment of continuous curved box girders
L/R=1.0 0.5-
0.4
0.3
0.2
0.1
\
L=60=
0.0
-0"1 -0.2 t -0.3
0.1
I
I
0.2
I
0.3
I
I
0.4
I
0.5
tx
Fig. 9. CRA curves against model length (subjected to a line load). line loads are applied at the i n t e r m e d i a t e supports. T h e m o d e l with L = 15 m has a C R A w h i c h is larger t h a n 0.5 but close to 0.5 for the w h o l e range o f a. For the o t h e r m o d e l s s h o w n in the figure, C R A decreases g r a d u a l l y as the line loads m o v e to the ends o f models. T h e m o d e l with L = 60 m has C R A o f a b o u t 0.16 for a = 0.1. In Fig. 10, the C R A curves against a are plotted for various thicknesses o f i n t e r m e d i a t e d i a p h r a g m s , t d. D i a p h r a g m s without a m a n h o l e are used for this figure. T h e d i a p h r a g m thickness Id 0"0 in this figure denotes that the m o d e l has no i n t e r m e d i a t e d i a p h r a g m . F r o m this figure, it is f o u n d that m o d e l s with i n t e r m e d i a t e d i a p h r a g m s have a p p r o x i m a t e l y the s a m e C R A curves. This fact is similar to the description in Section 3 o f the c o n t r i b u t i o n o f d i a p h r a g m stiffness to the C R A o f m o d e l s subjected to a u n i f o r m l y distributed load. =
5 CONTRIBUTION OF SECTIONAL DIMENSIONS In the descriptions in Sections 3 a n d 4, the box m o d e l s a n a l y z e d have typical sectional d i m e n s i o n s o f h = 1440 m m a n d b = 960 m m , a n d d = (2/3) b - - 6 4 0 m m . In this section, m o d e l s w h i c h have various sectional d i m e n s i o n s a n d shoe spacings are dealt with. Figure 1 l(a) a n d (b) shows C R A curves for m o d e l s with h -- 1440 m m , L = 60 m and/_JR = 1.0. Box widths b are s h o w n in the figures. M o d e l s
332
Shigeru Shimizu, Shunya Yoshida L=60= L/R=I.O
/J
0,5 ¸ 0.4" 0.3 a= 0.2
x~
1_.0ram
ta=O.511
\
0.I-
0,0-
~--
ta=O. OiII
-0.i-
-0.2
-0.3 0,1
0.2
03
04
0.5
Fig. 10. CRA curves against model thickness (subjected to a line load). in Fig. 1 l(a) have shoe spacings d o f d -- b/3, a n d those in Fig. 1 l(b) have d = 2b/3. F r o m Fig. l l(a), it is f o u n d that the m o d e l with b = 9 6 0 m m (d = 320 m m ) h a s p = - 0.06 f o r a = 0.1; in such a case this m o d e l has a d o w n w a r d reaction at its exterior shoe. This figure also shows that, for a=0.1, the m o d e l has p = 0 . 0 with b = 1 2 0 0 m m , p = 0 . 0 6 with b = 1440 mm, p = 0.11 w i t h b = 1680 mm, and the model with b = 1920 m m has C R A o f p = 0.16. Thus, it is f o u n d from this figure that, for a = 0.1, increasing b o x width b b y 240 m m enlarges p b y a b o u t 0.05 or 0.06. Figure 12(a)-(c) shows C R A curves for the s a m e m o d e l s as those used in Fig. 11, b u t with different shoe spacings d. M o d e l s in Fig. 12(a)-(c) have fixed shoe spacings o f d = 640 mm, 800 m m and 960 m m respectively, in spite o f their width b. It is clear from Fig. 12 that C R A values are not c o n t r i b u t e d b y b o x widths b b u t are c o n t r i b u t e d b y shoe spacings d. These facts, o b t a i n e d from Figs 11 a n d 12, are p r e s u m a b l y related to the definition o f CRA,/1, s h o w n in eqns (1) or (3), in w h i c h d is in the denominator. In Fig. 13, C R A curves are plotted using b o x d e p t h h as a parameter. In this figure, values o f h = 1440 m m to h = 2880 m m are adopted. M o d e l s in Fig. 13(a) have b o x width b = 960 m m a n d shoe spacing d = 640 m m ; in Fig. 13(b) b = 1400 m m a n d d = 960 mm, a n d in Fig. 13(c) b = 1920 m m a n d d = 1280 mm.
Reaction allotment of continuous curved box girders
(a)
333
L=601 L/I~=I.O h=144c~,
.t.z 0.5 04-
0.30.2-
b=960mm(d=3201u) -0"I t -0.2 -0.3
(b)
0.1
I
r 0.2
I
r 0.3
I
I 0.4
I
I
I
0.5
L=601 L/R=I.O h=144c''
~,
0.40.30.20.10.0-
-0"1 t -0.2 -0.3
0.1
I
I
0.2
I
I
0.3
I
0.4
0.5
Fig. 11. CRA curves against box width (subjected to a line load). (a) Shoe spacing d = ( b ) shoe spacing d = 2b/3.
b/3:
From Fig. 13, CRA values for models with smaller depth are closer to 0.5 than are those for models with larger depth. Figure 13 also shows that CRA values for models with small box width (or with small shoe spacing) are more sensitive to variation of box depth. In Table 1, values of K and Ira, which are of course related to
Shigeru Shimizu, Shunya Yoshida
334
(a)
L=60= L/R=1.0
/z
0.5
h=144cm
f
0.4-
0.3-
0.2-
0.1
0.0 \
-°1t
k__ b=1200n
-0.2
-0.3
r
0.1
(b)
I 0.2
I 0.3
L:60m L/R=1.0
~'
r
I 0.4
(z
I
0.5
h=144cm
0.5 0.4-
0.3-
0.2-
0.1-
' \ '~
b=9GOmm
'\,~ \ ~-
b= 1680sn
I
I 0.4
b= 1440m=
0.0-
-0.1-
-0.2
-0.3
I
0.1
~ 0.2
I
I 0.3
I
0.5
Fig. 12. C R A curves against box width (subjected to a line load). (a) Shoe spacing d = 640 m: (b) shoe spacing d ---800 m:
Reaction allotment of continuous curved box girders L=60" L/R=I.O
(c)
335
h=144c,*
0.5
0.4-
0.3-
0.2-
0.1
0.0
-0.1
-0.2 -0.3
I
0.1
I 0.2
I
i 0.3
I 0.4
I
0.5
(c) shoe spacing d = 960 m. d i m e n s i o n s o f b o x sections, are listed with sectional dimensions. Table 1 a n d Figs 11-13 also indicate that the c o n s t a n t s o f pure b e n d i n g K or c o n s t a n t s o f torsion b e n d i n g Im o f b o x sections m a k e a small c o n t r i b u t i o n to CRA. It is o b v i o u s from Fig. 11 a n d Table 1 that b o x m o d e l s with different K or Im values have a p p r o x i m a t e l y equal C R A values. O n the o t h e r h a n d , these figures a n d the table d e n o t e that s o m e m o d e l s have different C R A values even if they have a p p r o x i m a t e l y equal K orlm. F o r example, Fig. 13(a) s h o w s that the C R A value for the m o d e l which h a s h = 2880 mm, b = 960 mm, d = 640 m m a n d K = 3747 × 106 cm 4 is/1 = - 0 . 1 7 for a = 0-1; the m o d e l with h = 1440 m m , b = 1440 m m , d = 640 m m a n d K = 3413 X 10 6 c m 4 has/1 = 0.12; a n d the m o d e l with h = 1440 m m , b = 1680 m m , d = 640 m m a n d K = 4.434 × 10 6 c m 4 has /.t = 0.13 (Fig. 12(a)). Thus, c o n s t a n t s o f pure torsion K or c o n s t a n t s o f torsion b e n d i n g Im are not suitable as indicators or p a r a m e t e r s o f C R A values.
6 REMARKS ON THE RESULTS OF ANALYSES In this paper, reaction allotments for two shoes o n the i n t e r m e d i a t e s u p p o r t o f t w o - s p a n c o n t i n u o u s curved b o x girders are studied. T h e reaction allotments are d i s c u s s e d b y using the d i m e n s i o n l e s s value/.t, or
Shigeru Shimizu, Shunya Yoshida
336
(a)
L=60m L/R=I.O
u
b=96cm
0.5
0.4
0.3
0.2" 0.1-
j//\\\~,:,,oo..
0.0-0.i -
~'~ h=2880111
-0.2
-0.3
J
(b)
F
I
1
0.2
0.
~
I
0.3
t:60m L / R : 1 . 0
I
I
0.4
0.5
b:144cz
0.4 0.3
0.2
o~
~
\\\~=~.o. ~
h:2160p
-0.I
-0.2
-0.3
I
O.
I 0.2
I
I 0.3
i
I 0.4
OC
I
0.5
Fig. 13. C R A curves against box depth (subjected to a line load). (a) Shoe spacing d = 640 m; (b) shoe spacing d =960 m:
337
Reaction allotment of continuous curved box girders
(c)
L=601 L/R=I.O
/J
b=192cs
0.5
0.4-
0.30.2
0.1
0.0
-0.1
-0.2 -
-0.3
I
I
0.1
I
I
0.2
0.3
I
I
I
0.4
o¢
0.5
(c) shoe spacing d = 1280 m.
the coefficient of reaction allotment (CRA);/.t has a value of 0.5 when the shoes are subjected to equal reactions. At first, contributions of girder curvature to CRA are studied for models subjected to a uniformly distributed load. It should be obvious that small curvature gives CRA close to 0.5 because girders with such curvature are approximately equal to straight girders. From the analyses, TABLE 1 Constants of Pure Torsion and Torsion Bending h (mm)
b (ram) 960
1440 1800 2160 2520 2880
K (X Ira (X K (X /ra (X K (× Ira (X K (X 1m (X K (X Ira (X
106cm 4) 109 cm 6) 106 cm 4) 109 cm 6) 106cm 4) 109 cm 6) 106 cm 4) 109 cm 6) 106 cm 4) 109 cm 6)
1"676 0"8199 2' 187 1"757 2"704 3" 185 3'225 5"192 3"747 7"867
1440
3'413 1"326 4"524 3'209 5"658 6"226 6"805 10'59 7"962 16'51
1680
4'434 1"455 5"919 3'808 7'439 7'707 8"984 13"46 10"55 21'35
1920
5"539 1"473 7"442 4.244 9.399 9"019 11-39 16"22 13'41 26"24
338
Shigeru Shimizu, Shunya Yoshida
it was found that larger girder curvature gives C R A values larger t h a n 0.5 for short models a n d gives C R A smaller t h a n 0.5 for long models. That is, the results described here indicate that, in short girders, the exterior shoes are subjected to larger reactions t h a n those of interior shoes, a n d in long girders interior shoes are subjected to larger reactions. On models subjected to a line load, C R A values generally decrease as the line load moves from the intermediate support to the e n d of the girder. As predicted from eqn (1) or (3), the C R A value is contributed by the spacing of the two shoes on the intermediate support. The existence of an intermediate d i a p h r a g m per span results in the two reactions on the shoes on the intermediate support closer to each other than those of models with no intermediate diaphragm. However, in some models, especially in those with large curvature, even an intermediate d i a p h r a g m does not result in C R A values of 0-5. F r o m the analytical results in this paper, it is seen that the existence of two or more intermediate diaphragms makes only a small contribution to C R A values. Models with two or more intermediate d i a p h r a g m s per span have approximately equal C R A values to those of models with a diaphragm. Increase of the plate thickness of an intermediate d i a p h r a g m makes C R A values larger in the small thickness range, but the increase of thickness makes a small contribution after it exceeds about 1 ram. These results indicate that, in most cases, a curved two-span continuous box girder with two shoes on its intermediate support has reactions at the shoes which are different, even if the girder has a stiff intermediate d i a p h r a g m or has multiple diaphragms. In other words, one intermediate d i a p h r a g m with thickness td over 1 m m has a sufficient effect on the CRA. A typical intermediate d i a p h r a g m a d o p t e d in this study has a thickness of 3 ram, a n d is t h i n n e r than d i a p h r a g m s used in practical bridges. The JSHB requires the thickness of plates such as d i a p h r a g m s to be over 8 mm. Thus, intermediate d i a p h r a g m s which are designed in accordance with the JSHB have a sufficient effect on CRA. The stiffness of intermediate d i a p h r a g m s is also specified in IDW 3 (Clause 6.3.4) o r B8540014 (Clause B.3.4.). In these design rules, the required dimensionless stiffnesses S of d i a p h r a g m s are shown against /3LD (parameter for spacing of effective diaphragms). In Table 2, values of S and/3LD for models with typical sectional d i m e n s i o n s used in this study are shown. The specifications in I D R or BS5400 are for d i a p h r a g m s in box girders subjected to torque, so this table probably does not give valid information. However, the table predicts that intermediate d i a p h r a g m s designed in accordance with I D R or BS5400 also have sufficient effect on CRA.
339
Reaction allotment of continuous curved box girders TABLE 2 Dimensionless Stiffness of Diaphragms (S) and Diaphragm Spacing Parameters (LD)
L(m) 10 tiLt) S
0'26 2042
20 0"39 1362
30 0"52 1021
40
50
0"78 681
1"17 454
60 1'57 340
TABLE 3 Ratios of Xin and Xout CRA 0"60 Xn.t/Xin Xin/Xou !
1"50 0"67
0"55 1"22 0-82
0"50 1'00 1-00
0"45
0"40
0 " 8 2 0'67 1-22 1'50
0"35
0"30
0"25
0"20
0 " 5 4 0 " 4 3 0 " 3 3 0"25 1"85 2 " 3 3 3 " 0 0 4'00
It should be noted again that, in most cases, the two reactions at the shoes of the intermediate support are not equal, even if the intermediate diaphragms have a sufficient effect on CRA. Such diaphragms merely make the reactions closer to each other than those of models with no diaphragm. Table 3 indicates ratios o f Xin and Xo,t derived from the definition of CRA for some values ofp. This table shows that, for example, for the case ofp = 0.4, an interior shoe is subjected to a reaction which is 1.5 times that at the exterior shoe. Generally, a support diaphragm is subjected to a large reaction force, so that attention should be paid to the buckling behaviour or load-bearing capacity of such support diaphragms. To design a support diaphragm, the magnitude of reactions should be taken into account. Results described in this paper indicate that attention also should be paid to the allotment of reaction to the two shoes on each support of curved box girders. 7 CONCLUSION The behaviour of support diaphragms in box girder bridges has been studied by many researchers, and from these studies, the behavior and load-bearing capacities of support diaphragms have been clarified. However, to design support diaphragms, the magnitude of reaction forces must be known.
340
Shigeru Shimizu, Shunya Yoshida
In this paper, the reaction forces on curved two-span continuous box girders are studied. Although a limited n u m b e r of cases is dealt with, the results shown here give some suggestions on design. In addition, further study should be m a d e on this theme. In this paper, decreased thickness or larger spacings of intermediate diaphragms t h a n those used in practical bridges are adopted. However, the results shown in this paper indicate that such d i m e n s i o n s are sufficient to the contributions of diaphragms on the reaction allotment.
ACKNOWLEDGEMENT The numerical calculations in this study were carried out by using the C o m p u t e r Center of S h i n s h u University.
REFERENCES 1. Rockey, K. C. & El-Gaaly, M. A., Stability of load bearing trapezoidal diaphragms. Publ. IABSE, 32-II (1972) 155-72. 2. E1-Gaaly, M. A., Stability oforthogonally stiffened load bearing trapezoidal diaphragms. Publ. IABSE, 34-11 (1974) 73-89. 3. Dowling, P. J., Loe, J. A. & Dean, J. A., The behaviour up to collapse of load bearing diaphragms in rectangular and trapezoidal stiffened steel box girders. In Steel Box Girder Bridges. Proc. Int. Conf. ICE, London, 1973, pp. 95-117. 4. Puthli, R. S. & Crisfield, M. A., Strength of stiffened box girder diaphragms. TRRL Supplementary Report 353. Structural Department, Transport and Road Research Laboratory, Crowthorne, Berkshire, UK, 1977. 5. Sawko, F. & Simonian, W. S., Elastic and buckling analysis of trapezoidal support diaphragms. Proc. ICE, 3 (1978) 17-39. 6. Shimizu, S., Kajita, T. & Naruoka, M., The stresses near intermediate supports in box girders (in Japanese). Proc. JSCE, No. 276 (1978) 13-23. 7. Fukumoto, Y., Shimizu, S. & Furuta, H., Strength of steel box girder diaphragms (in Japanese). Proc. JSCE, No. 318 (1982) 149-61. 8. Yoda, T.,Hirashima, M., Nakada, T. & Kanda, K., On the simplified design method of support diaphragms (in Japanese). Proc. Annual Meeting, JSCE, No. 42 (1987) 440-1. 9. Miyawaki, T. & Mori, S., A consideration on characteristics of supportreaction in two-span continuous curved beams (in Japanese). Bridge Engng, 15(4) (1979) 46-51. 10. Graves Smith, T. R., Gierlinski, J. T. & Walker, B., A combined finite strip/ finite element method for analysing thin-walled structures. Thin-Walled Structures, 3 (1985) 163-80.
Reaction allotment of continuous curved box girders
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11. Okumura, T. & Sakai, F., Cross-sectional deformations of box-girders and the influences of intermediate diaphragms (in Japanese). Proc. JSCE, No. 190 (1971) 23-36. 12. Sakai, F. & Nakamura, H., A method for numerical analysis of thin-walled curved beams consisting of plates and shells (in Japanese). Proc. JSCE, No. 235 (1975) 41-54. 13. Committee into the Basis of Design and Method of Erection of Steel Box Girder Bridges. Interim Design and Workmanship Rules (IDR). HMSO, London, 1973. 14. Code of Practicefor Design of Steel Bridges, Part 3 of BS5400, Steel, Concrete and Composite Bridges. British Standard Institution, London, 1982. 15. Japan Road Association, Japanese Specifications for Highway Bridges (JSHB). Maruzen, Tokyo, 1980.