Tensiones de deformación en puentes de viga en caja curva

Warping Stresses in Curved Box Girder Bridges: Case Study

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Ayman M. Okeil, M.ASCE,1 and Sherif El-Tawil, M.ASCE2 Abstract: This paper presents a detailed investigation of warping-related stresses in 18 composite steel-concrete box girder bridges. The bridge designs were adapted from blueprints of existing bridges in the state of Florida and encompass a wide range of parameters including horizontal curvature, cross-sectional properties, and number of spans. The bridges after which the analysis prototypes are modeled were designed by different firms and constructed at different times and are considered to be representative of current design practice. Forces are evaluated from analyses that account for the construction sequence and the effect of warping. Loading is considered following the 1998 AASHTO-LRFD provisions. Differences between stresses obtained taking warping into account and those calculated by ignoring warping are used to evaluate the effect of warping. Analysis results show that warping has little effect on both shear and normal stresses in all bridges. Current design provisions are discussed in light of the analysis results.

DOI: 10.1061/共ASCE兲1084-0702共2004兲9:5共487兲 CE Database subject headings: Bridges, box girder; Torsion; Warpage; Steel; Concrete; Stress; Bridge design.

Background Composite steel-concrete box girders are commonly used in curved bridges, interchanges, and ramps 共Fig. 1兲. Curved composite box girders have a number of unique qualities that make them suitable for such applications, such as 共1兲 their structural efficiency allows designers to build long slender bridges that have an aesthetically pleasing appearance; and 共2兲 composite box girders are particularly strong in torsion and can be easily designed to resist the high torsional demands created by horizontal bridge curvature and vehicle centrifugal forces.

Curved composite box girder bridges generally comprise one or more steel U-girders attached to a concrete deck through shear connectors. Diaphragms connect individual steel U-girders periodically along the length to ensure that the bridge system behaves as a unit. The cross section of a steel box is flexible 共i.e., can distort兲 in the cross-wise direction and must be stiffened with cross frames that are installed in between the diaphragms to prevent distortion. Web and bottom plate stiffeners are required to improve stability of the relatively thin steel plates that make up the steel box. During construction, overall stability and torsional rigidity of the girder are enhanced by using top bracing members.

These bracing members become unimportant once the concrete decks hardens, but are usually left in place anyway.

Analysis and design of curved composite box bridges is complicated by many factors, including composite interaction between the concrete deck and steel U-girder, local buckling of the thin steel walls making up the box, torsional warping, distortional 1

Assistant Professor, Dept. of Civil and Environmental Engineering, Louisiana State Univ., Baton Rouge, LA 70803. E-mail: aokeil@lsu.edu 2 Associate Professor, Dept. of Civil and Environmental Engineering, Univ. of Michigan, Ann Arbor, MI 48103-2125. E-mail: eltawil@ engin.umich.edu Note. Discussion open until February 1, 2005. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on August 6, 2002; approved on June 20, 2003. This paper is part of the Journal of Bridge Engineering, Vol. 9, No. 5, September 1, 2004. ©ASCE, ISSN 1084-0702/2004/5-487– 496/$18.00.

warping, interaction between different kinds of cross-sectional forces, and the effect of horizontal bridge curvature on both local and global behavior. The existing literature contains extensive information about the analysis, behavior, and design of horizontally curved composite box girder bridges. General theories can be found in textbooks 共e.g., Guohao 1987; Nakai and Yoo 1988兲 and a comprehensive survey of experimental and analytical work on curved steel girders 共including box girders兲 can be found in Zureick et al. 共1994兲 and Sennah and Kennedy 共2001, 2002兲.

Current codes pertaining to analysis and design of curved composite girders include the American Association of State Highway and Transportation Official’s 共AASHTO兲 LRFD Bridge Design Specifications 共AASHTO 1998兲 and Guide Specifications for Horizontally Curved Highway Bridges 共AASHTO 1997兲. Provisions in these specifications are mostly based on experimental and analytical research conducted over 30 years ago as part of project CURT 共Consortium of University Research Teams 1975兲, funded by the Federal Highway Administration 共FHWA兲. A more recent curved steel bridge research 共CSBR兲 project is currently nearing completion. It was initiated and conducted under the auspices of the FHWA with the following objectives 共Zureik et al.

2000兲: 共1兲 to gain a better understanding of the behavior of curved steel girders through large-scale tests and numerical modeling; and 共2兲 to update existing design provisions. Although the CSBR project is expected to provide much-needed information on behavior, analysis, and design of curved composite bridges, its focus is on I-girders rather than on box girders.

Motivation and Objectives A complicated state of forces develops in curved girders when they are loaded. The forces that are developed include bending moments, shear forces, pure 共i.e., St. Venant兲 torsion, warping 共i.e., nonuniform兲 torsional moments, and bimoments. Torsional moments and bimoments due to cross-section distortion also develop. However, distortion-related effects can be easily reduced to insignificant levels by providing an adequate number of cross frames 共Oleinik and Heins 1975兲.

When a section is twisted, plane sections will generally warp, i.e., will not remain plane 共as shown in Fig. 2兲, and Bernoulli’s

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Fig. 1. Curved composite box interchange

beam theory is therefore violated. If restrained, these out-of-plane deformations create additional normal and shear stresses, which when integrated over the cross section yield the bimoment and warping torsional moment, respectively. In general, the torsional moments acting on curved girders are larger than those encountered in straight girders.

Calculating warping related stresses is not a straightforward process. The current curved box bridge specification 共AASHTO 1997兲 insists that the effects of nonuniform torsion must be considered in design, but does not provide any assistance or guidance on how to do so. The only guidance given to determine when warping stresses could be important falls under a section in the commentary dealing with top bracing and appears to pertain to this particular section only 共i.e., during construction兲. This guidance is adapted from the work of Nakai and Heins 共1977兲, who investigated a variety of curved bridge types and proposed criteria based on cross-section properties, bridge length, and subtended angle that would allow engineers to determine when warping is significant. The study by Nakai and Heins 共1977兲 has several limitations including 共1兲 idealized loading and boundary conditions were assumed; 共2兲 although normal stresses due to warping

were considered, shear stresses due to warping were ignored; and 共3兲 the effects of centrifugal forces were not accounted for. Centrifugal forces always occur in curved bridges and can be substantial.

Other researchers have also tried to quantify the significance of warping in curved box girders. Turkstra and Fam 共1978兲 investigated the effect of diaphragms on the behavior of curved box girder bridges. They conducted a parametric study using finiteelement models and investigated for simple load cases the ratio between stresses calculated from the finite-element model and corresponding stresses obtained from idealized beam models.

Both concrete and composite single box girders were considered.

As expected, the results showed that diaphragms improve load distribution and positively influence stress ratios. A wide range of radii was covered in the study. Girders that showed large stress ratios belong to a group with small radii, which fall into an impractical range 关 R⬍30.48 m 共100 ft兲兴. Shear stresses were also not investigated in this study.

More recently, Waldron 共1988兲 investigated the effect of warping on normal stresses in single box girders. Forces were calculated by deriving closed form solutions of the fundamental equation governing torsion and warping for special loading cases.

Using concrete box examples, it was shown that warping could increase normal stresses by as much as 29%. This high stress ratio corresponds to a theoretical loading condition where a single concentrated load acts on one of the webs at midspan. For truck loads 共following the British code兲, stress ratios drop to around 5%.

Based on the study, it was concluded that the width-to-depth ratio significantly impacts the normal warping stress ratio.

The studies summarized previously all suffer from a number of common drawbacks. They did not address warping shear stresses and were based on idealized loading and boundary conditions. They also did not address the construction sequence of composite box girders. This paper addresses all of these issues and presents a detailed investigation of warping stresses in curved composite box girders. The study is conducted on 18 prototypes that were modeled after existing bridges in the state of Florida.

The bridges are carefully selected to cover a wide range of design parameters including horizontal curvature, cross-sectional properties, and number of spans. The bridges were designed by different firms and constructed at different times and are considered to be representative of current design practice. A summary of the main properties of the bridges is given in Table 1 and further information can be found in Okeil et al. 共2000兲 and Okeil and El-Tawil 共2002兲. The 1998 AASHTO load resistance factor design 共AASHTO-LRFD兲 load provisions are considered in the study, and the effect of warping on both normal and shear stresses is quantified. The implications of the analysis results with respect to current design provisions are then discussed.

Analysis Method

Fig. 2. Warping of closed thin-walled cross section

Analysis methods for curved box girders can be classified into two broad categories. The first is a macroapproach, such as the plane grid method or space frame method. In this technique, the bridge system is discretized into a number of beam-column or grid elements and the focus is on forces rather than on stresses.

The elements can be straight or curved. The second category of analysis techniques is the micromodel approach, examples of which are the finite-element method or the finite-strip method. In these methods, the bridge system is discretized into a number of continuum elements, and the emphasis is on stresses and corresponding strains.

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Table 1. Summary Data for Analyzed Existing Bridges Span lengths

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Bridge 390 521 525 528 537 538a 538b 538c 538d 539 540 541a 541b 542a 542b 598 606 607

Spans

Lanes

Minimum 共m兲

5 3 2 5 5 5 5 4 7 5 6 5 5 6 6 4 3 3

1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 2 2

54.86 23.16 41.54 48.16 30.48 37.19 46.33 50.90 36.88 39.62 23.77 38.25 32.16 43.89 34.75 44.35 56.08 46.94

Radius of curvature

Maximum 共m兲

Minimum 共m兲

Maximum 共m兲

Finite-element length 共mm兲

Number of finite elements

Number of load cases

71.17 36.58 49.59 58.52 64.92 52.43 64.31 64.01 64.01 55.93 54.25 63.09 61.26 62.18 59.13 52.27 80.37 66.85

188.98 1,758.90 Straight 1,746.38 218.30 436.60 436.60 873.19 436.60 218.60 290.76 250.40 250.40 349.58 349.58 107.56 438.88 870.91

188.98 1,758.90 Straight 1,746.38 218.30 3,033.17 873.19 873.19 873.19 218.60 290.76 431.51 645.52 349.03 8,732.18 211.68 875.00 Straight

152.4 101.6 101.6 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 152.4 101.6 101.6

1,923 816 897 1,658 1,390 1,585 1,762 1,508 2,486 1,539 1,661 1,686 1,604 2,096 1,834 1,895 1,310 1,582

74 35 37 68 58 65 69 61 100 63 69 69 60 84 74 51 69 60

Micromodels are more rational than macromodels, and provide more detailed information. However, they are difficult to set up and analyze, and they are usually used for research purposes or to validate new designs. Macromodeling techniques, on the other hand, are simple to implement and, because they yield reasonable results, are commonly used by practitioners and researchers. In fact, the plane grid method was used to calibrate the load distribution factors in the current AASHTO-LRFD specifications 共AASHTO 1998兲. The analysis method used in this paper is the space frame approach, which falls under the macromodel category.

Geometric Properties of Closed Cross Sections Accurate cross-sectional properties are essential for conducting a successful analysis. This research involves the use of two major cross-sectional properties, namely, the warping constant, I ␻ , and the sectoral area, S ␻ (s). Both quantities are determined with the knowledge of a warping function, ␻(s). In this study, these quantities were calculated based on expressions that can be found in the literature and textbooks 共e.g., Guohao 1987; Nakai and Yoo 1988兲. The general expressions are applied to closed and open cross sections differently. In both cases, the shear center is first determined. The details of the procedure can be found in Okeil et al. 共2000兲.

Two conditions are considered for each box girder. The first condition represents the cross section during construction, i.e., before the concrete deck hardens. At this stage, diagonal bracing between the top steel flanges is provided to ensure stability. The cross section cannot be considered as an open cross section because of the top bracing, and, as the section is not a closed section either, it is considered as quasiclosed, i.e., with a fictitious top plate to represent the effect of the top bracing. Several expressions for the thickness of the equivalent plate have been proposed in the past, of which the one proposed by Kollbrunner and Basler 共1969兲 is used in this study

t eq ⫽

E ab G d3 2 a3 ⫹ Fd 3 Fo

(1)

where E⫽modulus of elasticity; G⫽shear modulus; a⫽spacing between cross frames; b⫽distance between flanges; d⫽length of bracing member; F o ⫽area of top flange; and F d ⫽area of bracing member.

After hardening, the concrete deck becomes an integral part of the cross section and its contribution is accounted for. The equivalent plate’s contribution becomes negligible as compared to the deck and is ignored. In accounting for the concrete deck, it is first transformed into an equivalent steel plate using an appropriate modular ratio: E c /E s . This transformation is justified when the deck is completely in compression, but is approximate when the deck or parts of it are under tension. Because 共1兲 it is not clear when the deck will be completely in tension due to the combined effect of moment and torsion; and 共2兲 the contribution of the top bracing is being ignored, the use of the uncracked properties is deemed reasonable in this work. Other researchers have also made use of this approximation in the past, including Johnson and Mattock 共1967兲, whose work forms the basis of several provisions in AASHTO-LRFD 共AASHTO 1998兲. Figs. 3 and 4 show a few of the geometric functions for one of the bridges considered before and after integration of the deck in the cross section.

For simple geometries 共regular spans, curvatures, etc.兲 and loadings 共uniform torque, single concentrated torque兲, girder forces are a function of a dimensionless parameter, ␬, which is given 共Nakai and Yoo 1988兲 as ␬⫽L

冑

GK EI ␻

(2)

where L⫽span length; G⫽shear modulus; K⫽torsional constant; E⫽modulus of elasticity; and I ␻ ⫽warping constant. A large ␬ implies that the contribution of warping to stiffness is relatively small and that warping-related stresses are therefore low. On the

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Fig. 3. Geometric properties of quasiclosed cross section 共dimensions in mm; ␻ in mm2; S x ,S y in mm3; S ␻ in mm4兲: 共a兲 Quasi-closed cross section; 共b兲 S x ; 共c兲 S y ; 共d兲 ␻; and 共e兲 S ␻

Fig. 5. ␬ values for quasiclosed and closed cases: 共a兲 Spans with critical normal stresses and 共b兲 spans with critical shear stresses

other hand, a small ␬ implies that the warping contribution to stiffness is large and that warping-related stresses could be relatively high.

The reliance of member forces on ␬ is true only for certain idealized loading, geometric, and boundary conditions. In real bridges, the cross section and radius of horizontal curvature both

vary along each span as well as from span-to-span, which renders the use of Eq. 共2兲 impossible. To account for these variations, a weighted ␬ 共calculated for each span兲 is used as follows. The relationship between this weighted ␬ parameter and warping stresses is discussed later in the paper. ␬⫽

冕冑 L

GK dL EI ␻

(3)

Two ␬ values are computed for each span of each bridge corresponding to quasiclosed and closed conditions. Fig. 5 shows a plot of these ␬ values versus the average L/R ratio for the corresponding spans. The ␬ values in Fig. 5 are calculated for spans where the critical normal stresses 关Fig. 5共a兲兴 and shear stresses 关Fig. 5共b兲兴 take place, as described later. As expected, Fig. 5 clearly shows that closed cross sections possess better torsional qualities than open cross sections, i.e., higher ␬ values. Although there is some scatter, it is clear that current design practices yield ␬ values with an average of about 38 for quasiclosed cross sections and 114 for closed cross sections. A slight increase in the trend of ␬ is also observed with higher L/R, which reflects the more efficient designs for spans with a sharper radius of curvature.

Bridge Models

Fig. 4. Geometric properties of closed cross section 共dimensions in mm; ␻ in mm2; S x ,S y in mm3; S ␻ in mm4兲: 共a兲 Closed cross section; 共b兲 S x ; 共c兲 S y ; 共d兲 ␻; and 共e兲 S ␻

All 18 box girders considered in this study are modeled using a spatial beam-column element that accounts for warping. The element has seven degrees of freedom per node, in which the seventh DOF represents warping 共Fig. 6兲. The analyses are conducted using the computer program ABAQUS 共ABAQUS 1997兲.

Although several of the Florida bridges have more than one box, only one box girder is modeled per prototype to reduce the modeling and computational effort. Each box girder is assumed to support a slab that is wide enough to accommodate at least one

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Fig. 6. Two-node 7 DOF beam element used in bridge analyses

traffic lane. Small elements 关101.6 mm 共4 in.兲 or 152.4 mm 共6 in.兲 in length兴 are used to model the bridges, resulting in hundreds of elements per girder. This is necessary in order to accurately calculate the warping-related forces. For example, the bimoment experiences large spikes at supports, which are difficult to capture unless small elements are used. Several comparisons between the proposed model and detailed finite-element models, along with other comparisons with theoretical solutions, confirm that the chosen element sizes are sufficient to provide acceptable accuracy. Readers are referred to Okeil et al. 共2000兲 for more details about these validation studies.

Intermediate support locations are restrained against vertical and transverse translations and twisting rotations. The warping degree of freedom is not restrained anywhere, because diaphragms are not capable of providing significant resistance to out-of-plane deformations. However, continuity over intermediate supports leads to the development of bimoments and torsional warping moments, which is not the case at end supports where the developed bimoment is zero. Table 1 lists the number of elements used for each bridge, which ranged from 816 –2,486. The table also lists other details for the bridges covered in this study.

Fig. 7. Positioning of live loads for single-lane bridge 共dimensions in mm兲

live load factored 共Strength I limit state兲 envelopes for warpingrelated forces obtained from the analysis of an idealized threespan pilot bridge.

Because ABAQUS does not provide results for the warping torsional moment, T ␻ , the fact that it is the derivative of the bimoment 共i.e., T ␻ ⫽M ⬘␻ ) is used to obtain these results. A numerical differentiation scheme 共Greenspan and Casulli 1988兲 of the bimoment, M ␻ , is utilized. To enhance the quality of the numerical results, three points are used to numerically differentiate M ␻ at any point—the point under consideration and the points before and after. At points where abrupt changes take place, such as supports, a forward 共point under consideration and two following points兲 or a backward 共point under consideration and two previous points兲 differentiation scheme is used so that the spikes can be accurately captured.

Stress Calculations Bridge Loading and Resulting Forces Dead loads are estimated based on the dimensions and materials given in blueprints provided by the Florida Department of Transportation. Live loads are considered according to AASHTOLRFD 共AASHTO 1998兲. Following the HL-93 loading, a uniform lane load of 9.3 N/mm, a uniformly distributed load over a 3,000 mm width, is considered in addition to a tandem load 共two 110 kN axles兲. Transversely, the loads are positioned at the outermost possible location to generate the maximum torsional effects 共Fig. 7兲, and centrifugal forces are taken into account for curved segments of the bridges. The tandem load is swept along the longitudinal direction of the bridge axis, while the uniform lane load is positioned on each span separately. Analysis of these load cases is used to generate envelopes for the live load force distributions.

Each set includes M x , M y , M ␻ , V x , V y , T, T s , and T ␻ , which are described in the next section. Fig. 8 shows an example of the

Normal stresses—including the effect of warping—are calculated as follows 共Nakai and Yoo 1988兲: ␴ exact⫽

Mx My M␻ M␻ y⫹ x⫹ ␻⫽␴ approx⫹ ␻ Ix Iy I␻ I␻

(4)

where I x and I y ⫽moments of inertia about the x and y axes; and x and y⫽distances from the centroid of the cross section. Eq. 共4兲 shows that the exact normal stress, ␴ exact , is generated by the bending moments (M x ,M y ) and bimoment (M ␻ ) which causes warping. The sum of the first two terms is from classical beam theory, which does not account for warping, and will be referred to as ␴ approx . The third term is a function of the warping constant, I ␻ , and the warping function, ␻.

The approximate shear stress, ␶ approx , is calculated from classical beam theory 共i.e., by ignoring the effects of warping兲 using the following equation:

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Fig. 8. Envelope of warping-related straining actions due to live loads for idealized bridge

␶ approx⫽

Vy Vx T S 共 s 兲⫹ S 共 s 兲⫹ tI x x tI y y 2tA c

(5)

where V x and V y ⫽shear forces; and T⫽torsional moment acting on the cross section. The other terms in the expression represent the geometric properties of the closed cross section, namely, wall thickness 共t兲, moments of area 关 S x (s) and S y (s)], and enclosed cross-sectional area (A c ). The third term assumes that the entire torsional moment is St. Venant torsion. If warping is considered, the torsional moment must be split into its two constituent terms 共pure torsion, T s , and warping torsion, T ␻ ), and the shear stresses can be calculated as follows: ␶ exact⫽

Vy Vx Ts T␻ S 共 s 兲⫹ S 共 s 兲⫹ ⫹ S 共s兲 tI x x tI y y 2tA c tI ␻ ␻

(6)

The geometric properties I ␻ 共warping constant兲 and S ␻ (s) 共sectoral area兲 are used to calculate the shear stress due to warping torsional moments 关the last term in Eq. 共6兲兴.

Warping Stress Ratios „WSR…

WSR⫺N⫽

␴ exact⫺␴ approx ␴ approx

(7)

WSR⫺S⫽

␶ exact⫺␶ approx ␶ approx

(8)

Results The WSRs for all cases summarized in Table 2 are calculated using the procedures described earlier. Two sets of plots are generated in an attempt to identify trends in the results. Figs. 10 and 11 show the WSR for normal and shear stresses plotted versus the average L/R ratios for corresponding spans. The WSR are also plotted versus ␬, as shown in Figs. 12 and 13. Each of the four figures provides three plots corresponding to the ratios due to dead, live, and total 共combined dead and live兲 loads. The dashed line in each plot represents the average level. In spite of significant scatter in some of the plots, several observations are evident from Figs. 10–13. These observations are discussed next.

Normal Stresses

One of the main goals of this research is to determine the effect of warping on both shear and normal stresses. This is achieved by investigating the ratio between approximate stresses calculated using classical beam theory 共i.e., ignoring warping兲 and exact stresses, which are defined herein to include the effect of warping.

For each bridge, ratios pertaining to both normal and shear stresses are calculated at selected key points in critical sections with the highest exact stresses 关calculated according to Eqs. 共4兲 and 共6兲兴. The key points considered in each cross section are shown in Fig. 9. The warping stress ratios for both normal stresses (WSR⫺N) and shear stresses (WSR⫺S) are then calculated as follows. The results are given in Table 2.

During construction 共i.e., quasiclosed cross section under dead load兲, ignoring warping implies that normal stresses are underes-

Fig. 9. Keypoints considered for stress calculations

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Table 2. Definitions of Warping Stress Ratio 共WSR兲

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Exact stress calculations

Approximate stress calculations

Stress ratios

␴ DL,exact⫽(M xDL /I xO )y ⫹(M yDL /I yO )x⫹(M ␻DL /I ␻O )␻

␴ DL,approx⫽(M xDL /I xO )y⫹(M ydL /I yO )x

␴ ⫽ 关 (M ␻DL WSRDL

␴ LL,exact⫽(M xLL /I xC )y ⫹(M yLL /I yC )x⫹(M ␻LL /I ␻C )␻

␴ LL,approx⫽(M xLL /I xC )y⫹(M yLL /I yC )x

␴ ⫽ 关 (M ␻LL /I ␻C )␻ 兴 /␴ LL,approx WSRLL

␴ T,exact⫽␴ DL,exact⫹␴ LL,exact

␴ T,approx⫽␴ DL,approx⫹␴ LL,approx

␶ DL,exact⫽(V yDL /tI xO )S xO (s) ⫹(V xDL /tI yO )S yO (s)⫹(T sDL /2tA cO ) ⫹(T ␻DL /tI ␻O )S ␻O (s)

␶ DL,approx⫽(V yDL /tI xO )S xO (s) ⫹(V xDL /tI yO )S yO (s)⫹(T DL /2tA cO )

␶ WSRDL ⫽(␶ DL,approx⫺␶ DL,exact)/␶ DL,approx

␶ LL,exact⫽(V yLL /tI xC )S xC (s) ⫹(V xLL /tI yC )S yC (s)⫹(T sLL /2tA cC ) ⫹(T ␻LL /tI ␻C )S ␻C (s)

␶ LL,approx⫽(V yLL /tI xC )S xC (s) ⫹(V xLL /tI yC )S yC (s)⫹(T LL /2tA cC )

␶ WSRLL ⫽(␶ LL,approx⫺␶ LL,exact)/␶ LL,approx

␶ T,exact⫽␶ DL,exact⫹␶ LL,exact

␶ T,approx⫽␶ DL,approx⫹␶ LL,approx

Fig. 10. Normal warping stress ratios versus L/R due to: 共a兲 Dead loads; 共b兲 live loads; and 共c兲 total loads

/I ␻O )␻ 兴 /␴ DL,approx

WSRT␴ ⫽ 关 (M ␻DL /I ␻O )␻ ⫹(M ␻LL /I ␻C )␻ 兴 /␴ T,approx

WSRT␶ ⫽(␶ T,approx⫺␶ T,exact)/␶ T,approx

Fig. 11. Shear warping stress ratios versus L/R due to: 共a兲 Dead loads; 共b兲 live loads; and 共c兲 total loads

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Fig. 12. Normal warping stress ratios versus ␬ due to: 共a兲 Dead loads; 共b兲 live loads; and 共c兲 total loads

timated by an average of 0.84%. The most critical case is underestimated by 1.90%. In designing for live loads 共closed section兲, the average effect of warping is an additional 0.25%, with the most severe case being 1.88%. The average WSR for total stress effect 共closed section subjected to dead and live loads兲 is 0.34%, with a maximum of 3.16%. In evaluating these numbers, readers should keep in mind that the locations of the maximum WSRs for dead, live, and combined loading are different. Although the plots in Figs. 10 and 12 do not show any conclusive trends because of scatter in the data, it is clear that the magnitudes of the WSR are quite small 共less than 3.16% for all cases兲.

Shear Stresses Although the WSRs pertaining to shear are somewhat higher than those corresponding to normal stresses, the ratios are still small.

For example, the calculations show that the average dead load shear stresses are underestimated by 1.57%, with a maximum of 7.42%. After casting the concrete deck, the cross section’s properties are greatly enhanced, and ignoring warping is actually on the conservative side with a tendency to overestimate stresses by an average of 1.52%. For combined loading 关Fig. 11共c兲兴, total stresses are underestimated if warping is not taken into account by an average of 1.37%.

Fig. 13. Shear warping stress ratios versus ␬ due to: 共a兲 Dead loads; 共b兲 live loads; and 共c兲 total loads

Effect of ␬ As previously discussed, Fig. 5 indicates that ␬ improves slightly as L/R increases, reflecting the greater torsional resistance provided by designers for spans with a sharper radius of curvature.

Figs. 12 and 13 show, respectively, the relationship between ␬ and WSR⫺N and between ␬ and WSR⫺S. Because ␬ is a measure of the contribution of warping to stiffness, it is logical to see some correlation in Figs. 12 and 13. However, the data in both figures is scattered and does not appear to have a specific trend. There are several reasons that can be put forward to explain this: 共1兲 Although spans with a larger ␬ are torsionally stronger, they are probably subjected to larger demands; 共2兲 the bridges were designed by different firms and hence the level of conservatism in design as well as the design models and tools vary from one bridge to another; and 共3兲 ␬ calculated according to Eq. 共3兲 is a weighted value that may not accurately reflect the vulnerability of a critical cross section to warping as well as ␬ calculated from Eq. 共2兲 would for idealized conditions.

Design Implications Although no trends are evident in Figs. 10 and 11, one important observation can be made; i.e., the effect of warping on both nor-

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mal and shear stresses in all 18 bridges is quite small 共as quantified in the previous paragraphs兲. Because each point plotted in the figures represents the largest ratio at the most critical point of the most critical span of each bridge, it appears that the effect of warping on overall behavior is rather insignificant.

The vertical lines in Figs. 10 and 11 indicate the critical L/R ratio, as defined by AASHTO-LRFD 共AASHTO 1998兲, below which bridges may be analyzed and designed as straight. Bridge designers prefer to deal with straight bridges because they are easier to analyze and design. Straight bridges must still be designed to resist some torsional forces, which result from eccentrically placed loading. However, torsional demands—and corresponding warping effects—in straight box girder bridges are quite small and are negligible in many cases. There are two limits permitted by AASHTO-LRFD 共AASHTO 1998兲. For open cross sections 共defined as quasiclosed in this research because of the top bracing兲, three span bridges with a subtended angle less than 5° (L/R⫽0.087) can be treated as straight. For closed cross sections the critical L/R is 0.21.

It is clear that a significant portion of the bridges fall outside these limits. For these bridges, AASHTO 共1997兲 becomes the applicable design code. This implies that the effect of horizontal curvature and the resulting torsional demands could be significant and should be taken into consideration. Although AASHTO 共1997兲 clearly states that nonuniform torsion should be explicitly considered, this study shows that the effect of warping is still negligible in all these bridges. This is true even in bridges where L/R is greater than twice the limit permitted by AASHTO-LRFD for closed cross sections, and more than four times for quasiclosed sections.

Summary and Conclusions A detailed investigation is presented on the effects of warping on the stress levels in 18 box girder bridges adapted from the Florida Department of Transportation inventory. The bridges cover a wide range of design parameters, including horizontal curvature, crosssectional properties, and number of spans. Forces are evaluated from analyses that account for the construction sequence and the effect of warping. Loading is considered following the 1998 AASHTO-LRFD provisions. By considering the differences between stresses obtained taking into account warping and those calculated by ignoring warping, it is shown that warping has little effect on both shear and normal stresses in the limited sample of bridges considered.

The results presented herein should not be construed to imply that warping is not important. Rather, this work points out that there could be a large subset of bridges where the warping effect is small enough to be ignored in structural calculations. This is particularly useful to designers, because warping calculations are complicated and time-consuming. Additional work is needed to define relevant parameters that can be used to identify bridges where warping calculations are warranted. The writers also believe that there is a need for a validated approximate design method that accounts for the effect of warping, without which it is hard to envision designers performing detailed analyses such as those presented here.

Acknowledgment This study is part of a project at the University of Central Florida that is investigating the behavior and design of curved box girder

bridges. Financial support for this research was provided in part by the Florida Department of Transportation 共Contract BC-421兲 and the University of Central Florida. The opinions stated here are those of the writers and do not necessarily represent the opinions of the sponsoring agencies.

Notation The following symbols are used in this paper: A c ⫽ enclosed cross-sectional area; a ⫽ spacing between cross frames; b ⫽ spacing distance between flanges; d ⫽ length of bracing members; E ⫽ modulus of elasticity; F d ⫽ area of bracing member; F o ⫽ area of top flange; G ⫽ shear modulus; I x ,I y ⫽ moment of inertia about x-axis and y-axis, respectively; I ␻ ⫽ warping constant; K ⫽ torsional constant; L ⫽ span length; M x ,M y ⫽ moment about x axis and y axis, respectively; M ␻ ⫽ bimoment; S x ,S y ⫽ moments of area about x axis and y axis, respectively; S ␻ ⫽ sectoral area; t ⫽ plate thickness; V x ,V y ⫽ shear in x axis and y axis directions, respectively; ␬ ⫽ warping stiffness parameter; ␴ ⫽ normal stress; and ␶ ⫽ shear stress.

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