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23 cantilever slabs with horizontally distributed spring support Numerical examples are shown for several problems to in- along the cantilever root with the spring constant K depend- vestigate the effect on effective width of curved girder bridges. ing on dimensions and material properties of the deck. The values of the effective width obtained by the present theory The spline finite strip analysis was shown to be in close are compared with those of the straight girder bridges. Accord- agreement with actual model test results. Tests showed that ing to the results, it is acceptable to say the values of effective the moment along the cantilever root approaches zero as the width of curved girder bridges are the same (approximately) as longitudinal distance from the point load increases. The the values of the straight girder bridges. This is a very important maximum moment at the root is when the point load is at conclusion for NCHRP Project 12-71. the end of the cantilever and approximately zero when the The inner and outer effective widths are denoted as 1 and load is not on the cantilever. There is also sagging in the can- 2, respectively, for a curved girder. 0 is half of the effective tilever around the point load. Although sagging moment is width for a straight beam and 2b = width of the flange. Effec- only local, many point loads at the same longitudinal loca- tive width ratio is defined as the ratio of an effective width tion can cause significant sagging moment, so this should to the actual breadth of the flange. The parametric study in- not be ignored. volves calculating the effective width ratio for simply sup- Conclusions made in this paper are as follows: ported girders and comparing results of (1) point load at mid span versus uniformly distributed load, (2) present theory 1. Cantilever decks of thin-walled box-girder bridges can be versus folded plate theory, (3) inner and outer effective width treated as cantilever slabs with horizontally distributed ratio for curved girders versus effective width ratio for straight spring support along the cantilever root, taking into ac- girders, and (4) box cross section versus channel cross sec- count the influence of local distortion of the box section. tion. In the test, the curved beam has a radius of curvature 2. Spline finite strip results, based on the simplified idealiza- R = 4L and b/L = 0.1. In the case of a concentrated loading, tion of the cantilever decks of thin-walled box-girder the values of the effective width ratio are at minimum at the bridges, are in close agreement with test results. The sagging center of the span length and increase rapidly toward the moment at the cantilever root can be obtained in conjunc- supports. However, in the case of a uniformly distributed tion with information tabulated in the article. loading, the values are at maximum at the mid span and de- 3. In cantilever slabs of infinite length and large cantilever crease toward the supports. The values of effective width ratio length, sagging moment cannot be ignored. The sagging of the present theory are lower than the folded plate theory moment may be taken from information also provided in curves, and the inside and outside effective width ratio agree the article. with the values of the straight beams, irrespective of cross- sectional shapes or types of load distributions. Other than the general information provided for evaluat- The authors analyze the inner and outer effective widths at ing cantilever decks of straight box-girder bridges, there is no mid span relative to the curvature R/L. For a box-girder specific information pertaining to curved box-girders. under a distributed load, the effective widths remain fairly constant for b/L = 0.1 when R/L > 2 and only decreases Hasebe, K., Usuki, S., and Horie, Y. (1985) "Shear Lag slightly for b/L = 0.2. With a point load and b/L = 0.1, the Analysis and Effective Width of Curved Girder Bridges," effective width also remains fairly constant for R/L > 4. When J. Eng. Mech., Vol. 111, No. 1, pp. 8792. b/L = 0.2 with a point load however, the effective widths deviate significantly for small R/L. This paper develops guidelines and graphs for estimating For a curved girder, as b/L increases, the difference between the effective width of curved girder bridges. The methodol- the inner and outer effective widths becomes significant, ogy is formulated by substituting the flange stress derived especially for box-girders. For small values of b/L however, from present theory into the equation of effective width def- those two values are almost identical. inition for the curved girders. The required information in One needs to be careful when R/L is small, b/h is small, or formulating the effective width rule for design of curved b/L is large, especially under concentrated loads. Otherwise, it girder bridges is provided. The actual longitudinal stress is reasonable to assume the effective width of a curved girder distributions for the curved girders are evolved from the is equal to that of a straight girder for practical applications. present theory for shear lag in order to determine the effective width. The thin-walled curved girders used in this investi- Torsion gation are based on box and channel cross sections and are analyzed for a uniform lateral load and for a concentrated Torsion design is currently addressed by the AASHTO load. LRFD code and, in the case of box-girders, is integrated

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24 with shear design. Recent studies (Rahal and Collins, 2003) Rahal, K. N., and Collins, M. P. (2003) "Experimental Eval- indicate that this approach yields good results when the uation of ACI and AASHTO-LRFD Design Provisions for correct value of is used. Several other papers on the sub- Combined Shear and Torsion," ACI Structural Journal, ject of torsional resistance have been reviewed and it is con- Vol. 100, No. 3, pp. 277282. cluded that the current design methods are acceptable. Only minor clarifications of the current specifications and The experimental results from four large nonprestressed guidelines for applying them to box-girders are required. specimens loaded in combined flexural shear and torsion are Following is a summary of the papers that were reviewed on used to evaluate the torsion design procedures of ACI 318-02 torsion design. and AASHTO-LRFD. Both sets of procedures calculate the required amounts of hoop reinforcement for torsion based Collins, M. P., and Mitchell, D. (1980) "Shear and Torsion on a space truss model with compression diagonals inclined Design of Prestressed and Non-Prestressed Concrete at an angle of to the longitudinal axis. It is shown that the Beams," Journal of the Prestressed Concrete Institute, Vol. 25, ACI provisions give very conservative results if the recom- No. 5, pp. 321000. mended value of 45 degrees is used for . If the lower limit of 30 degrees is used, however, some unconservative results are This document presents design proposals for flexural possible. The AASHTO-LRFD provisions predicted values of shear and torsion of prestressed and non-prestressed con- of approximately 36 degrees for these specimens and gave crete beams based on the compression field theory. A rational accurate estimates of the strengths. method is proposed which addresses members subject to flexural shear, torsion, combined flexural shear and flexure, Fu, Chung C, and Yang, Dailli (1996) "Designs of Concrete and combined torsion, flexural shear and flexure. Early de- Bridges with Multiple Box Cells due to Torsion Using Soft- sign procedures using truss models are also presented. The ened Truss Model" ACI Structural Journal, Vol. 93, No. 6, compression field theory, a development of the traditional pp. 696702. truss model for flexural shear and torsion, considers in addition to the truss equilibrium conditions, geometric Using a softened truss model, this paper presents a compatibility conditions and material stress-strain rela- method for torsional design of multicell concrete bridges. tionships. The compression field theory can predict the fail- Earlier researchers have successfully shown the development ure load as well as the complete load-deformation response. for solving single-cell torsion by combining the equalibrium, Measured and predicted response of numerous members is compatibility, and the softened constitutive laws of con- presented. The proposed design recommendations are pro- crete. By solving the simultaneous equations based on the vided in code format. Comparisons with the provisions of membrane analogy, multicontinuous or separate cells can be the ACI 318-77 and CEB codes are provided. Numerous solved. worked examples are provided that demonstrate the pro- posed method. Hsu, T. T. C. (1997) "ACI Shear and Torsion Provisions for Prestressed Hollow Girders," ACI Structural Journal, Rahal, K. N., and Collins, M. P. (1996) "Simple Model for Vol. 94, No. 6, pp. 787799. Predicting Torsional Strength of Reinforced and Prestressed Concrete Sections," ACI Structural Journal, Vol. 93, No. 6, New torsion design provisions have been proposed for the pp. 658666. 1995 ACI Building Code. As compared with the 1989 provi- sions, these generalized 1995 provisions have three advan- A noniterative method for calculating the ultimate tor- tages: First, they are applicable to closed cross sections of sional strength and the corresponding deformations of re- arbitrary shapes. Second, they are applicable to prestressed inforced and concentrically prestressed concrete sections is concrete. Third, they are considerably simplified by deleting presented. This method, based on the truss model, avoids the "torsional concrete contribution" and its interaction with the need for iterations by making simplifying assumptions flexural shear. These new provisions are suitable for applica- about the thickness of the concrete diagonal, the softening tion to concrete guideways and bridges because these large of the concrete due to diagonal cracking, and the principal structures are always prestressed and are often chosen to have compressive strains at ultimate conditions. A simple check hollow box sections of various shapes. This paper discusses on the spalling of the concrete cover is implemented. The the background of the new code provisions, suggests modifi- calculated torsional capacities of 86 beams are compared cations to code formulas, and illustrates the application of the with the experimental results and very good agreement is code provisions to prestressed hollow girders by way of a obtained. guideway example.