ELSEVIER S0141-0296(96)00012-0

Engineering Structures, Vol. 18, No. 10, pp. 76%777, 1996 Copyright 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0141~)296/96 $15.00 + 0.00

Curved girders are special Dann H. Hall

Bridge Software Development International, Ltd (BSDI), PO Box 287, 562 Thomas Street, Coopersburg, PA 18036, USA

Horizontally curved I-girders have been used in highway bridges for over 30 years. Their structural behaviour is known to be quite different from straight girders because of the always present non- uniform torsion. Early studies of these members were based on a strength of materials approach. Modifications of these results were made to account for distortion and amplification effects. More recent investigations in Japan have involved inelastic finite element studies. From these studies various modifications of straight I-girder bending strength equations have been presented. Bending tests of curved beams are compared to these equations. It is suggested that current research may lead toward the thought that curved girders are the general case, and straight girders may be considered a special case. Copyright 1996 Elsevier Science Ltd.

Keywords: horizontally curved girders, bridges, beams

1. Introduction

Horizontally curved girders may be thought of as special straight girders. At times a girder may be designed as straight but brought into the world accidentally curved. To account for this case, design specifications may be based on an assumed out-of-straightness. It is possible also to think in the converse sense--straight girders are only spe- cial curved girders. Analysis of curved girders is different from straight girders in that torsion due to curvature is always present, whereas torsion may, or may not, be pre- sent in straight girders.

Curved girders are special in appearance, fabrication techniques and structural behaviour. The earliest curved girders were probably made from rolled shapes which were cold bent about their weak axis. When girder welding became accepted, curved girders grew in popularity. Hori- zontally curved girders are used in buildings, such as for balconies. However, the most widespread use of curved gir- ders is in the highway bridge market where high speeds require smooth changes of direction.

Dabrowski performed some of the earliest analytical study of curved girders t. His work was limited to strength- of-materials assumptions; however, he discussed other effects such as cross-section deformation. The earliest lab- oratory tests of steel curved girders were performed in the United States during the late 1960s and early 1970s. Pennsylvania Department of Transportation and the Federal Highway Administration financed much of this work 2. The FHWA financing was provided through the Consortium University Research Team (CURT) Project. This work resulted in the Guide spe.cifications for horizontally curved

highway bridges 3 which AASHTO printed in 1976. These provisions were based on working stress design. Later pro- visions based on load factor design were developed 4,5. These provisions have never been accepted as full AASHTO specifications and remain essentially unchanged at this time.

The Hanshin Expressway Corporation of Japan adopted guidelines for the design of horizontally curved girder bridges in 19886 . These provisions are based on working stress design although the factors of safety are applied to stresses determined from ultimate strength analyses. The Japanese have performed more recent studies of curved gir- ders including tests and finite element investigations.

There are no other specific provisions for curved girders. Often horizontally curved bridges are designed using vari- ations of straight girder design provisions.

The behaviour of curved I-girder structures is discussed in this paper. Bending moments from several predictor equations are compared to tests of single curved I-girders tested in bending. Finally, curved girder research underway in the United States is discussed. Although both curved I- girders (open) and box girders (closed) are utilized, this paper addresses only I-girders.

2. Structural system

Figures 1 and 2 each show a two-girder, 160-ft simple span structure. Girders in Figure 1 are straight, while those in Figure 2 are curved to a 400-fi radius. Figure 3 shows a similar structure except that the same offset from tangent accomplished in Figure2 with a 400-ft radius, is accomplished with a kink. The beams all have the same

769

770

Figure 1

Curved girders are special: D. H. Hall

Cross Frame Spacing - 20 feet

" ~ ~ ~ ~ p ~ppor ts - Vertical at ends of girders

spa.. ,e e, Radius - Infinite _ .

Girder Spacing - 10 feet

Girder Depth - 6 feet tt Straight two-girder structure

Figure2 Curved two-girder structure

properties. They are singly symmetrical because the top flange is smaller than the bottom flange.

Table 1 gives selected results from finite element analy- ses of each structure for a downward load of one kip/ft on each girder. This loading can represent wet concrete of a bridge deck. Reactions, mid-span deflections and bending moments are presented in Table 1. Bending moments in the two straight girders are equal; however, the outside girder in the curved and kinked structures carries the entire load. The sum of the bending moments in the two girders is nearly equal in each case.

The interaction between the three pairs of girders differs in significant ways. The bracing members in Figure 1 only stabilize the girders, whereas they act as primary load carry- ing members in the curved and kinked structures. However, the offset, rather than the curvature, is the critical struc- tural effect.

Cross frames introduce restoring torques to the girders,

Table 1 Reactions, deflections and moments

Model girder Vertical Mid-span Mid-span reaction deflection moment (kips) (in) (K ft)

Straight Girder 1 80 3.18 Girder 2 80 3.18 Total 160

Curved Girder 1 -8 4.71 Girder 2 170 12.77 Total 162

Kinked Girder 1 14 6.10 Girder 2 148 6.63 Total 162

3292 3292 6584

-547 7318 6771

-1114 7862 6748

resulting in bimoments in the girders. The bimoments are manifest in equal but opposite lateral bending moments in the top and bottom flanges. Figure 4 gives lateral flange bending moments in the bottom flange of the outside girder for the two non-straight structures. When the girder has a uniform curvature, lateral flange bending is distributed along the girder in proportion to the vertical moment. When the offset is created by a kink in the girder, the restoring forces and lateral flange bending are concentrated near the kink.

Figure 3 Kinked two-girder structure

Curved girders are speciah D. H. Hall

3O0

~_. 2OO I,

I ~o0

0 i--- z LJ - I00 .~

0 ~_ -200

- 300

Figure 4

CURVD

--KINKFCI ~'~

I t SPAn

STRNG .IT \

'~ \ "", ,

,. ."

SPAN

Bottom flange lateral flange bending moment

2 13 23 30 32

0 -13 -24 / -30 -32 /

At eenterlin \ I 0 -1 ~ 4 145

1 0 2 -5 -145

Figure 5 Cross frame forces

Figure 5 shows the cross frame forces for the two offset structures. Note the extremely large cross frame forces at the kink. If the cross frame arrangement was modified such that there was no cross frame at the kink, the lateral flange bending moment at the kink would be extremely large while the maximum cross frame forces would be miti- gated substantially.

If deflections are small, an elastic small deflection analy- sis that considers appropriate boundary conditions, bracing members, girder bending and torsional stiffnesses can adequately predict structural behaviour of curved structural systems for design purposes. Figure 6 shows the exagger- ated deflected shape of the curved model. Differential top and bottom lateral deflections are often significant as well as differential vertical deflections between girders. Designs of curved bridges usually do not include consideration of lateral deflections or tw:[st of the girders.

3. Free body d iagrams

Curved I-girders may be either singly or doubly symmetri- cal. Testing and all of the theoretical research in the US has involved only doubly symmetrical I-girders having equal

771

tension and compression flanges. This family of I-girders has both the centre of gravity and shear centre at mid-depth of the web. This discussion is limited to doubly symmetri- cal sections, but the concept is equally applicable to singly symmetrical girders with different size flanges. Figure 7 shows a free body diagram of a doubly symmetrical hori- zontally curved I-girder between two torsional brace points subjected to a near constant vertical bending moment. Although the vertical bending moment effectively creates non-collinear axial forces in the flanges, equilibrium requires that torsion be created in the girder which in turn causes restoring forces in the cross frames. Vertical as well as lateral loads are created in the cross frames. The sum of the vertical components represents the load transferred from the inside girder to the outside girder required for equilibrium.

If the effect of the web is ignored, the flange lateral bend- ing moment, M~at, can be thought of as resulting from a virtual uniform lateral load equal to the axial flange force divided by the girder radius applied to the flanges. The flange is assumed supported laterally by the cross frames. Virtual loads are applied to top and bottom flanges in opposite directions. The virtual loads are inversely pro- portional to the girder depth.

The flange lateral moment is a function of the bracing spacing squared. An approximate equation for the lateral bending moment at brace points is given as equation (1). This equation comes from the V-load method7:

MvL 2 Mlat- IODR (1)

where: My = vertical moment, L = unbraced length, D = girder depth and R = girder radius.

The lateral flange moments at brace points tend to

+M t I

Figure 7 Curved I-girder free body

+ML

_ML -ML

Figure 6 Deflected shape of curved two-girder structure

772 Curved girders are special: D H Hall

. . o . . . . . . . . . . . .

Figure 8 Curved compression flange free body

restrain bowing of the flange due to the virtual load. The net effect of the non-uniform torsion due to curvature is always to increase curvature of the compression flange as seen in Figure 8.

Figure 9 shows a tension flange free body, again ignor- ing web effects. The virtual load tends to reduce curvature while the lateral flange bending moment at the brace point tends to restore it. The net effect on the flange can be to either increase or decrease curvature of the tension flange, depending on flange stiffness and curvature. Dashed lines indicate deflected shapes in Figures 8 and 9.

3.1. Secondary effects If a strength-of-materials analysis were performed, it would yield lateral bending stresses similar to those of equation (1). However, there are important secondary effects in curved girders.

A curved girder subjected to bending tends to deform radially, causing both flanges to undergo increased bowing. These deformations tend to amplify the curvature effect 8. This in turn, causes a further increase in the lateral flange bending in the compression flange and a decrease in the lateral flange bending in the tension flange. As a result of the radial effect, the magnitude of the lateral flange bending stresses in the two flanges are no longer equal. The change in curvature of the flanges due to the radial effect causes the lateral bending moments in the compression flange to be amplified. The radial effect causes the cross frame forces to be different on the compression and tension flanges of a doubly symmetrical beam. The radial effect is evident as soon as loading commences.

The lateral flange moments are also amplified when the curvature is increased due to lateral bending in the flanges. The resulting increased lateral deflection leads to a larger external moment. This effect has been observed experimen- tally and theoretically to commence at approximately 30% of the ultimate capacity of the section 8. These secondary effects are thought to have an increasingly important influ- ence on the strength of horizontally curved I-girders as the curvature and flexibility of the girder increases 8.

Figure9 Curved tension flange free body

OUTSIDE OF COMPRESSION

Figure 10

Split curved compression flange free body

If the cross-section is assumed to retain its shape, a warp- ing analysis could be performed using the strength-of- materials method. In the case of bridge girders, such an analysis would yield bimoments and resulting lateral flange stresses similar to those based on the virtual lateral loading.

Figure 10 shows a free body diagram of both halves of a curved compression flange considering the web. The web exerts a lateral force tending to restrain the flange from bowing. It also restrains the web from buckling into the plane of the web. The stress gradient in each half of the flange is different. The lateral moment varies such that the extreme fibre of the flange on the outside of the curve is largest at brace points. On the inside of the curve, it is largest mid-way between brace points.

Forces can result in flange instability different from that of a straight girder flange. Local buckling of curved flanges is related to the stress gradient across the flange as well as the average stress, curvature and width-to-thickness ratio of the flange.

3.2. Web A free body diagram of an unstiffened horizontally curved I-girder web is shown in Figure 11. The compression edge of the web bows outward while the tension edge of the web is flattened if the potential effect of the tension flange is ignored. In addition to vertical bending, the web undergoes distortion about an axis parallel to the flanges as the flanges attempt to move in opposite directions.

J Figure 11

J Curved web free body

Curved girders are special: D. H. Hall 773

The addition of transverse web stiffeners tends to reduce web bending about a longitudinal axis. They are not likely to have much effect on bowing about a vertical axis. Attaching transverse stiffeners to the flanges has been found to increase the ultimate bending strength of curved I-girders 9.

The behaviour of longitudinal web stiffeners on curved webs has not been studied in the US. However, they are permitted according to the AASHTO Guide Specification. Their stiffness must be increased over that required for a straight girder web.

4. Stability

4.1. Single girder Resistance to lateral torsional buckling of a straight girder can be thought of as tile ability of the girder to resist a slight lateral deflection. As long as the internal lateral bend- ing moment in the compression flange is larger than the externally applied lateral moment, the girder has not reached bifurcation (buckling load).

When a curved 1-girder bridge is erected, a single girder is sometimes set. During this time it may be torsionally restrained by external means such as lateral restraint at the bearings and lateral ties of the top flange at the bearings. This arrangement is adequate if the compression flange is stable. Otherwise, additional supports, either torsional, ver- tical, or both are required. A method of determination of the stability beyond those limits given in the AASHTO Guide Specification is not available at this time.

4.2. Multi-girder Figure 2 shows a simple span system consisting of two curved girders connected by cross bracing and supported by bearings at their ends. This structural system is stable when the bracings between girders and the bearings are capable of equilibrium. Stability is achieved from the inter- action between girders since neither girder alone would be stable. Failure of one of the girders would cause a signifi- cant reduction in the stability of the system and probably cause collapse.

The results reported are based on an elastic, small deflection analysis. Since the two girders are the same size, the outside girder is likely to yield first. As the outside beam commences to yiel!d, its stiffness decreases causing a drop in the portion of load it receives from the inside girder. Thus, the rate of change in moment of the inside girder becomes greater than in the outside one. The system may have significant reserve strength beyond the total load determined by elastic analysis to cause plastification of the outside beam.

5. Bending strength

Most structural design specifications provide strength pre- dictor equations for individual members comprising the entire structure. Horizontally curved I-girders are different from typical structural raembers in that their behaviour is dependent on the boundary conditions provided by bracing members and attachment of these bracing members to other curved members. However, predictor equations for curved beams are based on member behaviour as for other types of structural members.

Tests of the curved beams have been limited to doubly symmetrical members and the predictor models also have this limitation. Many of the tests are of single curved beams with rigidly connected torsional braces. Application of the test predictor equations based on these investigations requires that assumptions be made regarding singly sym- metrical members and the effect of boundary conditions other than those studied. There have been several types of curved beams tested, but each test arrangement has led to its own set of problems when applying them to practical structures.

Practical curved bridge beams are usually singly sym- metric with different size tension and compression flanges. Such girders do not have coincident centres of shear and gravity. Composite multi-girder cross-sections even have a shear centre above the deck. The applicability of the research limited to doubly symmetrical girders to more practical girders needs further investigation.

Upon application of vertical load, the compression flange commences to bend outward between brace points. Thus, the spacing between brace points is always greater than the effective bracing length of the curved compression flange. Typically, the effective distance between braces of a straight girder, K, equals 1.0 because the flange can buckle in an S shape with nodes at the brace points. Equation (2) gives an estimate of gamma used to determine the effective unbraced length for a partially braced curved beam flange from the Hanshin Expressway Guidelines 6. The effective unbraced length equals gamma times the spacing between cross frames, L.

~/= 1.0-1.97qbl/3+4.25qb--26.3~b3 (2)

l where: qb = ~, where: l

radius.

= unbraced length and R = girder

5.1. Web behaviour The compression side of a curved web tends to bow, caus- ing an increase in the lateral flange bending stress. Further, the tendency of the flanges to bend in opposite directions tends to cause the web to distort such that the flanges no longer remain parallel to each other. Nakai 9 found that attaching transverse stiffeners to the compression flange improves the pure bending strength. Both the Hanshin Guidelines and the AASHTO Guide Specification limit transverse stiffener spacing to the girder depth. However, the AASHTO Guide Specification 3 permits transverse stiff- eners to be eliminated when the web meets slenderness and shear stress limits.

5.2. Predictor equations

5.2.1. Culver-McManus. Predictor equations for compact and non-compact sections that were developed are used in the AASHTO Guide Specifications 3. The working stress design provisions permit only non-compact sections, in which the strength of these sections is defined as when the section reaches first yield. The load factor design pro- visions permit either non-compact or compact sections. The strength of a compact section is defined as full plasticiz- ation of the cross-section. A flange can be treated as com- pact without the web being compact. Vertical bending moments and bimoments are to be determined by analysis

774 Curved girders are special: D. H. Hall

of the entire superstructure. Bimoments need be determined only at brace points. Only the compression flange is con- sidered because in the development of the equations, dou- bly symmetrical sections were considered. The tension flange is never critical in these sections.

The lateral flange stress is limited to 0.5 times the verti- cal bending stress. The unbraced length divided by the radius is limited to 0.1 for both non-compact and com- pact sections.

The strength equations are based on modifications of the equation for lateral torsional buckling given in equation (3), used by the twelth edition of the AASHTO Bridge Design Specifications m:

Fbs = Fy (1-3A 2) (3)

where: Fy = yield stress of flange and:

Equation (4) shows the modifications to equation (3) used to determine the average critical flange stress, Fbc, for a curved compact section.

Fbc = FbsPbPw (4)

where:

- 1 job = 001;

03

Pw = 0.95 + 18 0.1 - -t fb pb(FbJFy)

l = unbraced length, b = width of compression flange, R = radius of curvature, fw = lateral flange bending stress, and fb = vertical bending stress.

Equation (5) shows the modifications to equation (3) used to determine the average critical flange stress, Fbc, for a curved non-compact section.

Fbc = Fb,mpw (5)

where:

Pb =

PW 1 - -

Pw2 -

1

l l I+ - - -

R bf

'] 1--~b 1 -7~ f

1/b 0.95 +

30+8000(0.1 _~)2 1 + 0.6 (fw/fb)

When fwlfb >-0: Pw = Pw~ or Pw2, whichever is smaller; when fw/fb < 0; pw = Owl.

5.2.2. Nakai. NakaV l shows equation (6) for pre- dicting the bending strength of curved girders, Mu. This equation modifies the lateral buckling equation and web bend buckling of straight girders found in the Japanese Bridge Specification ~2. Since the Japanese specification is based on working stress design, Nakai's equation is modi- fied by removal of the factor of safety.

[ L f] Mu = 1.92 + 0.357 Ma (6) where: L = unbraced length, Ma = moment ratio based on the Japanese Bridge Specification, R = radius and b e = compression flange width.

5.2.3. Fukumoto. Fukumoto 13 developed equation (7) for the prediction of the strength of curved I-girders. This quartic equation must be solved using an iterative pro- cedure and is thus difficult for designers to use.

A484 - 1 + ~ A4+ 1 82

- 8 + 1 = 0

(7)

where:

Mu

and Pe ---- elastic buckling load of section, Mp = plastic capacity, Me = elastic moment, d = section depth, tcf = comp. flange thickness, bcf = width of comp. flange, L = unbraced length, and R = girder radius.

5.2.4. Hanshin. The Hanshin Expressway Corpor- ation developed Guidelines 6. Equation (8) is presented in these Guidelines to predict the strength of curved I-girders. It is a linear interaction equation between vertical and lat- eral bending stresses. Since the Hanshin Guidelines, as well as the Japanese Bridge Specifications, are working stress design based, equation (8) has been modified by removal of the safety factors. Lateral flange moments are specified to be computed by the V-load method.

L fw - - + < 1 .0 (8 ) (Fba) c F~ao--

where: fb = normal flexural stress, (Fba) = critical normal bending stress, fw = lateral flange bending stress, and Fbao = yield stress.

The unbraced length divided by the radius is limited to 0.2. The sections must be compact. However, the definition of compact flanges in the Japanese Bridge Specifications is more liberal than that in the AASHTO Guide Specification.

5.2.5. Comparisons The predicted bending strength of Nakai's Specimen M2 is shown in Figure 12 by AASHTO, Hanshin and Fukumoto are compared in Figure l3. Since the flange is non-compact by the AASHTO Guide Specification, only the first yield moments are considered. If the AASHTO plastic moment capacity were used, the strength would be much higher.

Curved girders are speciah D. H. Hall

0.472" I t

~.1.5"

I 0.472" I

7.15"

w = 0.177 in.

R = 1157.5 in. Fyf = 56.1 ksi Fyw= 46.2 ksi L = 78.7 in.

Figure 12 Cross-section for specimen 'M2'

Figure 13 shows a plot comparing the bending strength versus slenderness. The abscissa is the unbraced length div- ided by radius (L/R). The solid line plots are for varying L while R is constant at the value of the specimen. The dashed line plots are for varying R while holding L constant at the value of the specimen. Of course, the plots cross at the L/R value of the specimen. The ultimate moment capacity is non-dimensionalized by the plastic moment capacity of a straight girder of the same scantlings and yield strength. The lateral flange bending stress is computed by the V-load method to be 0.56 times the vertical bending stress in the test specimen.

The AASHTO solid line is terminated where L/R equals 0.1. The AASHTO dashed line is terminated where the V- load lateral flange stress divided by the vertical bending stress is 1.0. The Hanshin plots are terminated where L/R=0.2. The Fukumoto plots extend over the entire range. In no case is the strength a pure function of the variable L/R. However, all these curves behave similarly and they each underestimate significantly the test speci- men strength.

r 0.9

0.111

0.7

0.6

: 05

0.4

0.3

H AASHTO - vanat)Je t.

O-- O AASHTO - Vanaole R H Harlsnm - Varla01e L

O Q Hanshm - Vanaole R Fukumoto - Variable L

. " l FukumOW. Variable R

4, Test Value

"~t .

~"7. 7, "

Q "~ "L''O''I3. "/3

02 ] Is,~,,~,,,o,~,t o.1 [ t/R-ooeo j

0.0 J"

O.0S 0.10 0.15 0.20 02S

L IR

Figure 13 Bending strength predictons versus L/R for speci- men 'M2'

775

5.3. Component bending tests Figure 14 shows a typical test arrangement used by Mozer et al) 4A5, to test curved girders in pure bending. Figure 15 shows a typical arrangement used by Nakai et al. 9, to per- form similar tests. Mozer's tests are in positive bending while Nakai's are in negative bending. The bending span is longer than the torsional unbraced length in Mozer's tests. The test fixture restrains the girder torsionally and vertically in Nakai's tests.

Table 2 gives a comparison of the test results for pure bending compared to the Culver-McManus predictors for both first yield and plastic moments. Some of the specimens had rather slender flanges that would be defined by Culver- McManus as non-compact. These specimens would be designed by AASHTO for first yield. It is seen that the plastic moment better predicts the strength of the test speci- mens.

Table 3 gives a comparison of the same test results com- pared to predictions using Hanshin, Nakai and Fukuoto. The Culver-McManus provisions are limited to L/R less than 0.1 and Hanshin provisions are limited to 0.2. It is possible that the Japanese tests have excessive torsional restraint that may cause the apparent high values compared to the predictor equations.

All but the Culver-McManus predictors apply to only compact sections. However, the definition of compactness is more liberal than that used by AASHTO.

6. Shear

There have been no unique mathematical predictor models for shear in curved girder webs. The total shear stress is composed of bending and torsional shear. The web tends to bow as well as buckle as in straight girders. Thus, additional rigidity of the web stiffeners is required for curved webs.

Culver and others report that postbuckling strength can develop in curved webs. However, there is some fear that deflections will be large compared to straight girders. As a result, the elastic buckling value is used in the AASHTO and Hanshin design specifications.

6.1. Bending-shear interaction Interaction between shear and bending is ignored in the AASHTO curved girder provisions because it is ignored in the straight girder provisions when elastic buckling limits are used for the web. Research has shown that the typical moment-shear interaction equations seem to give rather good results for component tests of moment and shear ~.

7. Current research

The National Cooperative of Highway Research Program (NCHRP) of the Transportation Research Board initiated a project (NCHRP 12-38) to update the present AASHTO curved girder provisions in the Guide Specification for Horizontally Curved Bridges. This work is limited to appli- cation of the present state of the art. Special emphasis is placed on the construction of curved girder bridges. A sur- vey of curved girder bridges fabricated across the United States is underway. Results from this survey will provide insight into the range of curvature experienced as well as how the bridges have been detailed. This work, including a new specification ready for ballot by AASHTO Bridge Committee, is scheduled to be completed in 1996.

776 Curved girders are special: D. H. Hall

Figure 14

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'il' F TEST SPECIMEN

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Support

Loading and supporting device for bending test.

Figure 15 Nakai bending test arrangement

Table 3 Comparison of predictors to test data

Hanshin Nakai Fukumoto Mu test

(K ft) Pre- Ratio Pre- Ratio Pre- Ratio dictor dictor dictor

L1A 1830 1107 0.60 1999 1.09 1509 0.82 L2A 1830 1136 0.62 1993 1.09 1540 0.84 GI5 1377 651 0.47 1279 0.93 991 0.72 G08 2120 1167 0.55 2213 1.04 1923 0.91 M1 8098 7428 0.92 5657 0.70 7989 0.99 M2 7752 4653 0.60 6508 0.84 6169 0.80 M3 6131 3498 0.57 5356 0.87 5145 0.84 M4 7203 2618 0.36 7972 1.11 4261 0.59 M5 5902 2291 0.39 6790 1.15 3589 0.61 M6 6287 2311 0.37 6649 1.06 3558 0.57 M7 6547 2366 0.36 7176 1.10 3827 0.58 M8 2935 * 1451 0.49 1069 0.36 M9 5548 * 6934 0.25 3126 0.56

Table 2 data

Comparison of Culver-McManus predictors to test *Equation not applicable (central angle exceeds allowable). Mx- - Reference 15; Lx - Reference 3; Gx- Reference 14.

Mo test (Kft)

First-yield Plastic

Predictor Ratio Predictor Ratio

L1A 1830 843 0.46 1716 0.94 L2A 1830 872 0.48 1749 0.96 GI5 1377 453 0.33 1058 0.77 G08 2120 931 0.44 1995 0.94 M1 8098 6930 0.85 7844 0.97 M2 7752 3692 0.48 7062 0.91 M3 6131 2620 0.43 5718 0.93 M4 7203 * 4774 0.66 M5 5902 * 3995 0.68 M6 6287 * 4024 0.64 M7 6547 * 4338 0.66 M8 2935 * * M9 5548 * 2575* 0.46

*Equation not applicable (central angle exceeds allowable). Mx- Reference 15; Lx - Reference 3; Gx- reference 14.

The Federal Highway Administration has financed a five- year basic research effort. This work is to assist in the development of an AASHTO Load and Resistance Factor Design (LRFD) code for horizontally curved bridges. The project includes both analytical and experimental studies. Open and closed sections are included.

The experimental work involves laboratory testing of components and testing of I-girder and box girder bridges. The I-girder bridge is planned to be a full-scale simple span structure. Full-scale tests eliminate the need for compensa- tory dead loads and are able to reflect accurately fabrication influences such as initial distortions and residual stresses.

The proposed steel framing for the I-girder tests is shown in Figure 16. This framing may be used to test some girder components in pure bending and shear with bending. By using a bridge as a test frame for components, ideal bound- ary conditions can be obtained. Other component tests may

Curved girders are special: D. H. Hall 777

go'-O ALONG GIRDER 2

t ~ " 2 ~ ~ R I~LI" I11 I I -- r~auIUSI / " " ,~ l~, . .~ I l

.< ,,5-/

Uuffs ~ DECK - - '~ / " ~ AT 16 E j ~ /

PLAN

Figure 16 Three-girder test frame

be performed on single girder specimens which will have flexible lateral braces aT: various locations.

Testing of a single curved member that derives part of its strength from the entire system is complex. Further, curved members in a stringer bridge receive load directly and may receive load through the: bracing members from other strin- gers. Analytical studies indicate that as one girder com- mences to yield, its stiffness is reduced sufficiently to affect the load distribution in the girders. The cross frame forces do not necessarily increase concomitantly with load under this condition. Instead of adding load to the exterior girder, the sense of force in the cross frames may change and load would be transferred to the next interior girder.

When the component tests are completed, a concrete deck will be placed on the steel frame and the bridge will be tested to failure.

8. Conclusions Curved I-girder researc!~ has been directed toward making adjustments to the straight girder predictor equations to pre- dict curved girder strength. This approach has not been very successful as evidenced by the wide range of predicted values for the test specimens. To date, there has been no comprehensive theory presented which explains the inelas- tic behaviour of curved I-girders. Most work being done at present revolves around inelastic finite element analyses.

The AASHTO and Nakai predictors involve the straight girder as a basis. The Hanshin provisions are indirectly based on straight girder provisions but deviate significantly from them via the interaction equation for vertical and lat- eral bending stresses.

Fukumoto's equation is most unique and seems to predict test results rather well. However, it is unwieldy from a designer's view. It may be more fruitful to develop a pre- dictor equation for curved girder strength that can be reduced to that for a straight girder when the radius becomes infinite. Such .an equation may include parameters such as Fukumoto's equation.

Such an approach would then permit straight girders to be considered as special curved girders.

References

1 Dabrowski, R. 'The analysis of curved thin-walled girders of open section', Stahlbau 1964, 33 (12), 364-372

2 Culver, C. G. 'Design recommendations for curved highway bridges', Final Report for Researcher Project 68-32, PENNDOT, Civil Engin- eering Department, Carnegie-Mellon University, June 1972

3 AASHTO 'Guide specifications for horizontally curved highway bridges' American Association of State Highway and Transportation Officials, Washington, DC, 1993

4 Galambos, T. V. 'Tentative load factor design criteria for curved steel bridges', Research Report No. 50, Washington University, Depart- ment of Civil Engineering, May 1978

5 Stegmann, T. H. and Galambos, T. V. 'Load factor design criteria for curved steel girders of open sections', Research Report No 43, Washington University, April 1976

6 The Hanshin Expressway Public Corporation, 'Guidelines for the design of horizontally curved girder bridges (draft)', October 1988

7 V-Load analysis, USS Highway Structures Design Handbook, Vol- ume 1, July 1981

8 McManus, P. F. 'Lateral buckling of curved plate girders', PhD thesis Carnegie-Mellon University, Pittsburgh, PA, 1971

9 Nakai, H., Kitada, T. and Ohminama, R., 'Experimental study of bending strength of web plate of horizontally curved girder bridges', Proc. JSCE 1983, 15, 19-28

1O Standard specifications for highway bridges, 12th Edn, American Association of State Highway and Transportation Officials, Wash- ington, DC, 1977

11 Nakai, H., Kitada, T. and Ohminama, R., 'Experimental study on ultimate strength of web panels in horizontally curved girder bridges subjected to bending, shear and their combination', 1984 Annual Technical Session - - Stability Under Seismic Loading, SSRC, San Francisco, CA, 10-11 April 1984, pp. 91-102

12 Specification for highway bridges, Japan Road Association, March 1984

13 Fukumoto, Y. and Nishida, S. 'Ultimate load behavior of curved I- beams', J. Engng Mech., ASCE 1981, 109 (1), 192-214

14 Mozer, J. D., Cook, J. and Culver, C. G. Horizontally curved highway bridges stability of curved plate girders - - P3, Federal Highway Administration, August 1975

15 Mozer, J., Ohlson, R. and Culver, C. Stability of curved plate gir- ders -- P2, Federal Highway Administration, August 1975