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21 Scordelis, A. C., Elfgren, L. G., and Larsen, P. K. (1977) distortions as well as cross-sectional distortions can then be "Ultimate Strength of Curved RC Box-Girder Bridge," determined throughout the curved box-girder. The forces Journal of the Structural Division, Vol. 103, No. 8, that are determined include bending moment and flexural pp. 15251542. shear, pure torsion, warping torsion, and bi-moment. These forces, in addition to distortional functions, yield resulting Results obtained in a study of a large-scale curved two-span normal bending, normal warping, and normal distortional four-cell reinforced concrete box-girder bridge model are pre- stresses. The technique is then used to determine the dead sented. The model, which was a 1:2.82 scale replica of a proto- load and live load response of a series of typical curved box type, had overall plan dimensions of 72 ft (21 m) long by 12 ft beams. A study of the data has resulted in a series of empirical (3.7 m) wide. The radius of curvature was 100 ft (30.5 m). This design equations. represents the sharpest curvature normally used for bridges in the California highway system. Experimental and theoretical Abendroth, R. E., Klaiber, F. W., and Shafer, M. W. (1995) results are considered for reactions, steel and concrete strains, "Diaphragm Effectiveness in Prestressed-Concrete Girder deflections, and moments due to conditioning overloads pro- Bridges," Journal of Structural Engineering, Vol. 121, No. 9, ducing stress values as high as 2.5 times the nominal design pp. 13621369. stress. The loading to failure and the ultimate strength behav- ior is examined. The excellent live-load overload capacity of Each year many prestressed-concrete (P/C) girder bridges the bridge is evaluated and comparisons are made with the are damaged by overheight vehicles or vehicles transporting similar behavior of an earlier tested straight bridge model. overheight loads. The effects of this type of loading on P/C Conclusions appropriate for the design of this type of bridge bridge behavior were investigated for various types and loca- are given. tions of intermediate diaphragms. The research included a comprehensive literature review; a survey of design agencies; the testing of a full-scale, simple-span, P/C girder-bridge Design Issues model with eight intermediate diaphragm configurations, as Bearings well as a model without diaphragms; and the finite element analyses of the bridge model assuming both pinned- and fixed- Although several bearing failures consisting of uplift, over- end conditions. The vertical load distribution was deter- load, or binding have been experienced in curved box-girder mined to be essentially independent of the type and location bridges, no published research exclusively addressing this of the intermediate diaphragms, while the horizontal load issue was found. However, because an accurate 3-D analysis distribution was a function of the intermediate diaphragm will account for differences in bearing forces and displace- type and location. Construction details at the girder sup- ments, several references that deal with global analysis and ports produced significant rotational-end restraint for both laboratory experimentation deal with this issue (Aslam and vertical and horizontal loading. Both the vertical and hori- Godden, 1975; Scordelis et al., 1977; Choudhury and Scordelis, zontal load distributions were affected by the girder-end 1988; Sennah and Kennedy, 2002). This issue is also discussed restraint. A fabricated intermediate structural steel diaphragm in some textbooks (Menn, 1990). was determined to provide essentially the same type of re- sponse to lateral and vertical loads that was provided by the reinforced-concrete intermediate diaphragms currently used Diaphragms by the Iowa DOT. Diaphragms help prevent excessive distortions of the cross section, facilitate wheel load distribution, and distribute transverse load. The following two papers discuss research on Flexure and Flexural Shear determining the number and spacing of interior diaphragms Beyond the issue of global analysis, the mechanism for re- in box-girder bridges. sisting flexural and shear stresses in box-girders is important. The mechanisms of shear resistance and its interaction with Oleinik, J. C. and Heins, C. P. (1975) "Diaphragms for flexural stresses in reinforced and prestressed concrete have Curved Box-Girder Bridges," Journal of Structural Engi- been well researched (Marti, 1999, and Vecchio and Collins, neering, Vol. 101, No. 10, pp. 21612178. 1986). Also, the effectiveness of the deck and soffit slabs in resisting flexural compressive forces has been studied. This A finite difference procedure is used to determine the re- includes the phenomenon commonly known as shear lag. sponse of a single-span curved single box-beam bridge with Several published papers and reports have dealt with these any number of interior diaphragms. The bending and torsional issues. Some of these are discussed below.
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22 Chang, S. T., and Zheng, F. Z. (1987) "Negative Shear Lag in from the webs. At a cross section where negative shear lag is Cantilever Box-Girder with Constant Depth," Journal of significant, the bending stress away from the webs is greater Struct. Eng., Vol. 113, No. 1, pp. 2035. than the stress near the webs. This paper is only indirectly applicable to this project be- This paper addresses the classical phenomenon of shear lag cause the paper does not deal specifically with curved girders. in box-girders and draws attention to distinguishing between However, given that shear lag effects are an important con- positive and negative shear lag. The effects of shear lag and sideration in developing analysis and design strategies, the negative shear lag in cantilever box-girders are analyzed conclusions in this paper, and the theoretical solutions are through a variation approach and finite element techniques. noteworthy. In short, the relevant conclusions are Expressions are derived to determine the region of negative shear lag effect with the interrelation of span/width parame- 1. Positive shear lag may occur under both point and uni- ters involved. The theoretical results obtained are compared form load, but negative shear lag occurs only under uni- with a Plexiglas model test. Finally, conclusions are drawn form load. with regard to further study and research. 2. Negative shear lag also depends on the ratio of L/b, where Positive shear lag is the phenomenon in which, near the b is the net width of the box section. The smaller the ratio, support of a cantilever, flange longitudinal stresses near the the more severe are the effects of positive and negative web are larger than away from the web. But for a cantilever shear lag. box-girder with constant depth under a uniform load, away 3. Negative shear lag depends on the boundary condition of from the support, the bending stress in the deck near the displacement as well as on the external force applied to the webs is smaller than the stress away from the webs. This is a girder. result of negative shear lag. Using the principle of minimum 4. In cantilever box-girders, although the negative shear potential energy, following Reissner's procedure with slight lag yield in the region of the bending stress is small, the modifications, shows that the additional moment created by relative additional stress induced by this effect is often flange shear deformation plays an important role in both considerably greater. It cannot be neglected. It should positive and negative shear lag. For a single point load at the never be believed that in all cases only positive shear lag is free end of the cantilever, only positive shear lag is created. produced. When there is a uniformly distributed load along the full span of the cantilever box-girder however, negative shear lag Chang, S. T., and Gang, J. Z. (1990) "Analysis of Cantilever occurs. The region of the cantilever affected by negative Decks of Thin-Walled Box-Girder Bridges," Journal of shear lag is from the free end to more than 3/4 of the cantilever Structural Eng., Vol. 116, No. 9, pp. 24102418. length from the free end. Negative shear lag affects a larger region than positive shear lag. This paper, which addresses the cantilever decks ("wings") With a finite element model analysis, three load cases were of single-cell box-girder bridges, does not make any distinc- considered; a distributed load, a point load, and a combina- tion about the effects of horizontal curves, but it does present tion of a downward point load and an upward distributed some useful qualitative information about cantilever deck force. This analysis showed that negative shear lag occurred evaluation, in general. only with the first load case of a distributed load. This model The paper reports on a spline finite strip approach used to was consistent with the results from the minimum potential analyze the cantilever decks. Effects of distortion of thin- energy method. walled box sections are taken into account by treating the Negative shear lag depends on not only the load case but cantilever deck as a slab with horizontally distributed spring also the boundary conditions. The ratio of the length of the supports along the cantilever root. Perspex model tests were cantilever to the width of the box-girder affects the amount conducted in the model structural laboratory at Tong Ji Uni- of moment caused by shear. As the ratio increases, both versity. The results based on the spline finite strip method are positive and negative shear lag decrease. compared with those of the model test. Simplified solutions Actual testing using a Plexiglas model confirmed the the- are also given for the distribution of transverse moment along oretical results. When a uniform load is applied, not only is the cantilever root. positive shear lag more severe compared with a point load, A Plexiglas model of a single-cell box-beam was evaluated. but negative shear lag is also present. A cross-sectional analy- As a point load moved transversely across the box-girder, the sis of shear stress in the flange is taken at several locations. bending stress and membrane stress at the root of the over- Near the fixed end where shear lag is greatest, the bending hang of the deck were obtained. From this analysis, it was stress near the web is much larger than the stress away observed that it is reasonable to treat the cantilever decks as