204/JOURNAL OF BRIDGE ENGINEERING/AUGUST1999JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/205ends.The stress state in each cylindrical strip was determined from the total potential energy of a nonlinear arch model using the Rayleigh-Ritz method.It was emphasized that the membrane stresses in the com-pression region of the curved models were less than those predicted by linear theory and that there was an accompanying increase in flange resultant force.The maximum web bending stress was shown to occur at 0.20h from the compression flange for the simple support stiffness condition and 0.24h for the fixed condition,where h is the height of the analytical panel.It was noted that 0.20h would be the optimum position for longitudinal stiffeners in curved girders,which is the same as for straight girders based on stability requirements.From the fixed condition cases it was determined that there was no significant change in the membrane stresses (from free to fixed)but that there was a significant effect on the web bend-ing stresses.Numerical results were generated for the reduc-tion in effective moment required to produce initial yield in the flanges based on curvature and web slenderness for a panel aspect ratio of 1.0and a web-to-flange area ratio of 2.0.From the results,a maximum reduction of about 13%was noted for a /R =0.167and about 8%for a /R =0.10(h /t w =150),both of which would correspond to extreme curvature,where a is the length of the analytical panel (modeling the distance be-tween transverse stiffeners)and R is the radius of curvature.To apply the parametric results to developing design criteria for practical curved girders,the deflections and web bending stresses that would occur for girders with a curvature corre-sponding to the initial imperfection out-of-flatness limit of D /120was used.It was noted that,for a panel with an aspect ratio of 1.0,this would correspond to a curvature of a /R =0.067.The values of moment reduction using this approach were compared with those presented by Basler (Basler and Thurlimann 1961;Vincent 1969).Numerical results based on this limit were generated,and the following web-slenderness requirement was derived:2D 36,500aa =1Ϫ8.6ϩ34(1)ͫͩͪͬt RRF w ͙ywhere D =unsupported distance between flanges;and F y =yield stress in psi.An extension of this work was published a year later,when Culver et al.(1973)checked the accuracy of the isolated elas-tically supported cylindrical strips by treating the panel as a unit two-way shell rather than as individual strips.The flange/web boundaries were modeled as fixed,and the boundaries at the transverse stiffeners were modeled as fixed and simple.Longitudinal stiffeners were modeled with moments of inertias as multiples of the AASHO (Standard 1969)values for straight ing analytical results obtained for the slenderness required to limit the plate bending stresses in the curved panel to those of a flat panel with the maximum allowed out-of-flatness (a /R =0.067)and with D /t w =330,the following equa-tion was developed for curved plate girder web slenderness with one longitudinal stiffener:D 46,000a a=1Ϫ2.9ϩ2.2(2)ͫͱͬt R f Rw ͙bwhere the calculated bending stress,f b ,is in psi.It was furtherconcluded that if longitudinal stiffeners are located in both the tension and compression regions,the reduction in D /t w will not be required.For the case of two stiffeners,web bending in both regions is reduced and the web slenderness could be de-signed as a straight girder panel.Eq.(1)is currently used in the ‘‘Load Factor Design’’portion of the Guide Specifications ,and (2)is used in the ‘‘Allowable Stress Design’’portion for girders stiffened with one longitudinal stiffener.This work wascontinued by Mariani et al.(1973),where the optimum trans-verse stiffener rigidity was determined analytically.During almost the same time,Abdel-Sayed (1973)studied the prebuckling and elastic buckling behavior of curved web panels and proposed approximate conservative equations for estimating the critical load under pure normal loading (stress),pure shear,and combined normal and shear loading.The linear theory of shells was used.The panel was simply supported along all four edges with no torsional rigidity of the flanges provided.The transverse stiffeners were therefore assumed to be rigid in their directions (no strains could be developed along the edges of the panels).The Galerkin method was used to solve the governing differential equations,and minimum eigenvalues of the critical load were calculated and presented for a wide range of loading conditions (bedding,shear,and combined),aspect ratios,and curvatures.For all cases,it was demonstrated that the critical load is higher for curved panels over the comparable flat panel and increases with an increase in curvature.In 1980,Daniels et al.summarized the Lehigh University five-year experimental research program on the fatigue behav-ior of horizontally curved bridges and concluded that the slen-derness limits suggested by Culver were too severe.Equations for ‘‘Load Factor Design’’and for ‘‘Allowable Stress Design’’were developed (respectively)asD 36,500a =1Ϫ4Յ192(3)ͫͬt R F w ͙y D 23,000a =1Ϫ4Յ170(4)ͫͬt Rf w ͙bThe latter equation is currently used in the ‘‘Allowable Stress Design’’portion of the Guide Specifications for girders not stiffened longitudinally.Numerous analytical and experimental works on the subject have also been published by Japanese researchers since the end of the CURT project.Mikami and colleagues presented work in Japanese journals (Mikami et al.1980;Mikami and Furunishi 1981)and later in the ASCE Journal of Engineering Mechanics (Mikami and Furunishi 1984)on the nonlinear be-havior of cylindrical web panels under bending and combined bending and shear.They analyzed the cylindrical panels based on Washizu’s (1975)nonlinear theory of shells.The governing nonlinear differential equations were solved numerically by the finite-difference method.Simple support boundary condi-tions were assumed along the curved boundaries (top and bot-tom at the flange locations)and both simple and fixed support conditions were used at the straight (vertical)boundaries.The large displacement behavior was demonstrated by Mi-kami and Furunishi for a range of geometric properties.Nu-merical values of the load,deflection,membrane stress,bend-ing stress,and torsional stress were obtained,but no equations for design use were presented.Significant conclusions include that:(1)the compressive membrane stress in the circumfer-ential direction decreases with an increase in curvature;(2)the panel under combined bending and shear exhibits a lower level of the circumferential membrane stress as compared with the panel under pure bending,and as a result,the bending moment carried by the web panel is reduced;and (3)the plate bending stress under combined bending and shear is larger than that under pure bending.No formulations or recommendations for direct design use were made.Kuranishi and Hiwatashi (1981,1983)used the finite-ele-ment method to demonstrate the elastic finite displacement be-havior of curved I-girder webs under bending using models with and without flange rigidities.Rotation was not allowed (fixed condition)about the vertical axis at the ends of the panel (transverse stiffener locations).Again,the nonlinear distribu-206/JOURNAL OF BRIDGE ENGINEERING /AUGUST1999FIG. parison of Web Slenderness Requirements from AASHTO Guide Specifications and Nakaition of the membrane stress was noted but appears significant only for extreme curvature and slenderness.Based on this non-linear membrane stress distribution,an effective web height was demonstrated.Also,the reduction in bending moment re-sistance was demonstrated,but,for slenderness in the design range,only a small reduction was noted.No formulations or recommendations for direct design use were made.Fujii and Ohmura (1985)presented research on the nonlin-ear behavior of curved webs using the finite-element method.Models included simple support,fixed support,and flange ri-gidities at the flange/web boundaries.The large displacement behavior was demonstrated for loads beyond the elastic bifur-cation load.Also,the nonlinear membrane stress distribution was demonstrated,but the effect on resistance moment or flange stress increase was not mentioned.It was emphasized that the web panel model with no flange rigidity is inadequate in estimating the behavior of the curved panel under significant loading.No direct recommendations or formulations regarding the design of curved I-girder webs were made.Suetake et al.(1986)examined the influence of flanges on the strength of curved I-girders under bending using the mixed finite-element approach.The ends of the panels (transverse stiffener locations)were modeled as simple supports,and var-ious width/thickness ratios for the flanges were modeled.Ge-ometric nonlinear analyses were conducted.Conclusions in-cluded that the aspect ratio of the panel was of minor importance and that the influence of the flange rigidity cannot be ignored.Also,observations were made on the torsional buckling behavior of the flanges.No quantitative formulations for design use were recommended.Nakai et al.(1986)conducted analytical research on the elastic large displacement behavior of curved web plates sub-jected to bending using the finite-element method.The web plate panels were modeled with and without stiffnesses for the flanges.Models without flange rigidity were modeled as fixed and simple supports.The boundary conditions at the panel ends (transverse stiffener locations)were modeled as simple support.One and two levels of longitudinal stiffeners were also modeled.It was determined that inclusion of the flange stiffnesses is essential to extract reliable results for the behav-ior of the curved web panels;therefore,all parametric results provided are from the use of the flange-rigidity models.It was further shown that increasing curvature has little ef-fect on the resisting moment (less that 10%within the range of actual bridge parameters).This was attributed to the fact that the web contributes only a small portion to the resistance moment,as compared with the flanges.It was also demon-strated that the maximum web deflection occurs in the vicinity of 0.25D from the compression flange but that this transverse deflection is effectively eliminated when one or two longitu-dinal stiffeners are present.The effect of curvature on the plate bending stresses was also demonstrated with respect to the effect of the slenderness ratio (D /t w )and cur-vature (a /R ).Web slenderness requirements were formulated by Nakai and Yoo (1988)based on the effects of curvature on displace-ment and stress and proposed for adoption by the Hanshin Guidelines for the Design of Horizontally Curved Girder Bridges (Kitada et al.1986).It was suggested that limiting values should be established so that the curved web platetransverse deflection,and plate bending stress,c c␦,␴,max max would be limited to the maximum transverse deflection and bending stress that would occur in the straight girder with the same dimensions but,instead of curvature,with a maximum initial deflection,w 0,of D /250,which is the maximum allow-able initial deflection stipulated in the Japanese design code (Specifications 1990a,b).From a comparison of the displace-ment results with the stress results,it was shown that criteriabased on the stress requirement would result in a more con-servative design.A regression analysis was performed,and (5)–(7)resulted (Guidelines 1988;Nakai and Yoo 1988).Also,analytical investigations on the effects of longitudinal stiffen-ers resulted in a proposal for the design rigidity of the stiff-eners:D ␤0Յ(5)t wa 1ϩ␣0ͩͪRfor the case of no longitudinal stiffenersDՅ␤(6)0t wfor the case of one or two longitudinal stiffeners and a /R Յ␩0;and2Da a Յ␤␥Ϫ␦ϩε(7)000ͫͩͪͩͪͬt RRwfor the case of one or two longitudinal stiffeners and a /R >␩0.A plot of (5)–(7)along with the current Guide Specifica-tions requirements is presented in Fig.1,in which extreme disparity in the reduction due to curvature can be noted for the design of girders with longitudinal stiffeners.The portion of the design equations that represent the effects of curvature can be separated for comparison as2aa R =1Ϫ8.6ϩ34(8)C 1ͫͩͪͬRRJOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/207FIG. parison of Curvature Reduction Equations—No Longitudinal Stiffenersa aR =1Ϫ2.9ϩ2.2(9)C 2ͫͱͬR R aR =1Ϫ4(10)D ͫͬR1R =(11)N 1a ͫͬ1ϩ␣0ͩͪR 2a a R =␥Ϫ␦ϩε(12)N 200ͫͩͪͩͪͬRREq.(8)refers to the reduction on design D /t by Culver as shown in (1);(9)refers to the reduction by Culver as shown in (2);(10)refers to the reduction by Daniels as shown in (3)and (4);(11)refers to the reduction by Nakai as shown in (5);and (12)refers to the reduction by Nakai as shown in (7).Fig.2com-bines the available reduction factors (R C 1,R D ,R N 1)on design D /t w for girders without longitudinal stiffeners.The reduction equations for design of girders with longitudinal stiffeners (R C 2and R N 2)will be demonstrated in a subsequent paper.Nakai and coresearchers conducted other research pertaining to the behavior of curved I-girder webs,including a series of experimental research on the behavior of the curved I-girder web under bending,shear,and combined bending and shear (Nakai et al.1983,1984a–c,1985a,b).In 1983,Nakai et al.presented the results from eight experimental test specimens under pure bending with slenderness ratios of D /t w =178with no longitudinal stiffeners and one specimen with a longitudinal stiffener and a D /t w =250.Panel aspect ratios of 0.5and 1.0and radii of curvature of 10and 30m were used.It was ver-ified that the ideal buckling phenomenon does not occur in the curved panels,but rather that the out-of-plane displacement of the web plate gradually increases in accordance with applied bending moment.But even so,it was noted that a critical bend-ing moment could be clearly observed.A recent investigation by Frank and Helwig (1995)using elastic buckling finite-element analyses of flat panels resulted in suggested design equations for determining the buckling capacity of webs when the neutral axis varies during the var-ious stages of loading.Panels with and without longitudinal stiffeners were considered.GENERAL METHODOLOGYTo understand the behavior of the curved web and to de-velop predictor equations,a combined approach was used in-volving:(1)a theoretical development to derive equations thatapproximate the linear behavior of the system;and (2)the finite-element method to verify the applicability of the theo-retical equations and to investigate the elastic buckling and geometric nonlinear behaviors.This paper focuses on the the-oretical development and verification.In a subsequent paper,strength reduction equations are formulated and proposed as a possible starting point in developing equations for the design of curved plate girder webs.RESULTSParametric ReviewA parametric analysis was conducted to summarize the nu-merical range of parameters used in current designs and pre-vious numerical and experimental research by others.Many of the earlier investigations on the subject were conducted using exaggerated cross-section parameters and curvatures (not to mention loading and boundary conditions),which resulted in exaggerated and unrealistic results.Due to publication length restrictions,not all of the results can be presented here,but they can be found in the dissertation ‘‘Nominal Bending and Shear Strength of Horizontally Curved Steel I-Girder Bridges,’’by Davidson (1996).The dimensions presented in the parametric summaries include those used in the current analyses along with a brief selection extracted from a variety of sources representing actual designs (AISC 1992,1993a–d),design examples (‘‘V-load’’1984;Yadlosky 1993),planned tests as part of the FHWA Curved Steel Bridge Research Pro-ject,and tests [as summarized by Hall and Yoo (1996)]per-formed by Culver and coresearchers (Mozer et al.1975a,b),Daniels et al.(1979a,b),Fukumoto and Nishida (1981),and Nakai et al.(1983,1984a).Curved Web BehaviorTo begin the investigation,the behavior of single web pan-els of various aspect ratios,curvatures,and cross-section di-mensions was analyzed using the finite-element method (MSC/NASTRAN 1994).In general,the dimensions used in the models follow the dimensions described in Table 1.The boundary conditions at the ends of the panels (transverse stiff-ener locations)were modeled as both simple and fixed sup-ports,and those at the top and bottom of the panel were mod-eled as simple,fixed,and with flange rigidity included.In general,though,boundary conditions were used that would provide the most conservative results (maximum transverse displacement or maximum stress)with respect to developing criteria for design.Loading was applied at end node points to simulate bending moment.As an example of the resulting lin-ear-elastic behavior,consider the curved panel model de-scribed in Fig.3with a radius of curvature of R =30.5m (100ft),web height of h =2,032mm (80in.),panel aspect ratio a /h =2.0,thickness of t w =10.16mm (0.4in.),and flange rigidity matching that of a 30.48ϫ609.6mm (1.2ϫ24in.)flange.The displacement of the doubly symmetric cross sec-tion at a /2is shown in Fig.4.Note that the rotation direction of the flange is opposite that assumed by linear torsion theory and also note the ‘‘bulging’’out displacement of the web.This ‘‘bulging’’displacement will obviously result in plate bending stresses with respect to both the vertical (z )and circumferential (␪)directions,which would not occur in the flat panel under pure vertical bending moment.Also,the radius of curvature was varied for several sections,and the membrane stresses at the lengthwise center of the panel (a /2)were analyzed.From Fig.5,it can be noted that,as curvature (and panel slenderness h /t w )is increased,the membrane stress distribution becomes increasing nonlinear through the depth of the section.This has been noted by other208/JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999TABLE1.Critical Stress Comparison—Curved versusStraightCRITICAL STRESS RATIO (CURVED/STRAIGHT)h =80in.;b f =24in.h /t (1)a /h (2)Critical Stress (ksi)Theoretical (k =36.5)(3)Finite-element result (4)R =1,000ft t f /t w =3(5)t f /t w =5(6)R =500ft t f /t w =3(7)t f /t w =5(8)R =100ftt f /t w =3(9)t f /t w =5(10)10010010020020020030030030012312312395.6795.6795.6723.9223.9223.9210.6310.6310.63101.78101.58101.4925.4825.4225.4011.3411.3211.311.0011.0021.0031.0031.0061.0071.0061.0111.0141.0141.0151.0171.0161.0191.0201.0191.0241.0261.0031.0061.0101.0101.0211.0301.0221.0711.0571.0161.0191.0241.0241.0331.0401.0361.0751.0631.0681.1381.1691.2901.2971.3381.4281.4461.5001.0811.1321.1831.2931.2951.3391.4301.4431.498Note:1in.=25.4mm;1ft =0.305m;1ksi =6.895MPa.FIG. 3.Finite-Element Model DescriptionFIG. 5.Membrane Stress Distribution through Panel Depth with Increase in Curvature and Increase in h /t wFIG. 4.Displacement of Cross Sectionresearchers,as described above.As a result of this reduction in web membrane stress,the flanges carry a higher load;thus,even without considering warping stresses in the flanges,the curved section would be unable to carry as much vertical moment as the comparable straight section before yielding of the flanges initiates.However,for the most severe case considered here [h /t w =200,R =30.5m (100ft),a /h =3.0],the increase in flange normal stress resultant is less than 6%based on the linear anal-ysis.This seems reasonable,since,for this section,the web contributes less than 18%of the total moment of inertia.Also of interest is the location of the absolute maximum stresses,since,in the design of the curved I-girder webs,it is desired to limit the maximum stress below a certain allowable stress,generally based on the yield stress.Because the mem-brane stress dominates,the maximum combined stresses in the panel occur at the top of the panel at the flange/web intersec-tion,as it would for flat panels (Davidson 1996).Lateral Pressure AnalogyThe amount of transverse or ‘‘bulging’’displacement can be approximated by using a ‘‘lateral pressure’’analogy.Con-JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/209FIG.7.Theoretical Development for Flat Panel under Hydro-static LoadingFIG. 6.Lateral Pressure Analogy Developmentsider a virtual width strip,dh ,of the curved web panel of thickness t and radius R under vertical bending stress,as shown in Fig.6.The resultant force,P ,due to vertical bending moment on the virtual strip of the panel is non-collinear.Be-cause of this non-collinearity,a lateral ‘‘virtual’’distributed load results along the unit strip,which,after considering that the radius is very large with respect to the panel length,can be viewed as a virtual pressure through the depth of the girderP ␴t q ==(13)c R RSince the distortion of the cross section results from an ap-plied vertical bending moment and the transverse displacement of the deformed cross section will cross the undeformed ver-tical axis at the neutral axis,the displacement behavior is anal-ogous to that of a flat plate of length a ,thickness t ,and width h c ,simply supported on the bottom edge,with a linearly in-creasing transverse load,as shown in ing this anal-ogy,the displacements and plate bending moments can be readily solved for the fourth (flange)edge of the plate as sim-ple and fixed support,similar to the hydrostatic solutions pro-vided by Timoshenko and Woinowsky-Krieger (1959).For the flat rectangular plate with two opposite edges simply sup-ported,Levy (1899)suggested taking the solution in the form of a series:ϱm ␲x w =Y sin(14)m ͸am =1where w =component of displacement in the transverse di-rection;and Y m =function of y only and must be in a form that satisfies the boundary conditions and the governing dif-ferential equation444Ѩw Ѩw Ѩw q q x0ϩ2ϩ==(15)4224Ѩx Ѩx Ѩy Ѩy D aD p pFurther simplification can be made by taking the solution of(15)in the formw =w ϩw (16)12where w 1=particular solution to the deflection of a strip under a hydrostatic load represented as5q 3x 033w =Ϫ10ax ϩ7a x 1ͩͪ360D apϱ4m ϩ12q a (Ϫ1)m ␲x0=sin ͸55D ␲m ap m =1,3,5...(17)which satisfies the boundary conditions at x =0and x =a of2Ѩww =0,=0(18)2Ѩx and w 2represents the homogeneous solution to (15)and is of the form4q a m ␲y m ␲y m ␲y0w =A coshϩB sinh 2m m ͩD a a apm ␲y m ␲y m ␲y ϩC sinhϩD cosh m m ͪa a a(19)Noting that the last two terms of (19)are odd functions andthat the displacement of the plate about the x -axis must be symmetric due to symmetric boundary conditions,C m and D m must therefore be zero.The total solution is now of the formϱ4m ϩ1q a 2(Ϫ1)m ␲y0w =ϩA cosh m͸ͫ55D ␲m apm =1m ␲y m ␲y m ␲xϩB sinh sinmͬa aa(20)From the boundary conditions,the constants can be ob-tained asm ϩ1(2ϩ␣tanh ␣)(Ϫ1)m m A =Ϫ(21)m 55␲m cosh ␣mm ϩ1(Ϫ1)B =(22)m 55␲m cosh ␣mwherem ␲b ␣=(23)m 2aand,further noting that at y =0,sinh(␣m )equals zero and cosh(␣m )equals unity,(20)can be simplified toϱ4m ϩ1q a 2(Ϫ1)m ␲x 0(w )=ϩA sin(24)y =0m͸ͫͬ55D ␲m apm =1or in simpler terms4␣q a 0(w )=(25)y =0D pwhere210/JOURNAL OF BRIDGE ENGINEERING /AUGUST1999FIG.10.␤versus a /h c Plot for h /t w =200FIG.9.␣versus a /h c Plot for h /t w =200FIG.8.␣versus a /h c Plot for h /t w =100ϱm ϩ12(Ϫ1)m ␲x ␣=ϩA sin(26)m͸ͫͬ55␲m am =1and is a function of x and y and the plate aspect ratio only;and where3Et D =(27)p 212(1Ϫ␯)The plate bending moments M x and M y can be derived as22Ѩw ѨwM =ϪD ϩ␯(28)x p ͩͪ22Ѩx Ѩy 22Ѩw ѨwM =ϪD ϩ␯(29)y pͩͪ22Ѩy ѨxAfter substitution of (20),(28)and (29)can be most simply written as2(M )=␤q a (30)x y =0x 02(M )=␤q a (31)y y =0y 0where ␤x and ␤y result from the summation of terms similarly to ␣in (26)and,again,are functions of x and y and the plate aspect ratio only.This same procedure can be followed for the case of three simply supported edges and the fourth edge fixed.For this case,the particular solution,w 1,is of the formq x04224w =(16y Ϫ24b y ϩ5b )(32)1384aD pwhich satisfies the differential equation (15)and the boundary conditions.Again,␣and ␤constants can be extracted.With respect to the lateral pressure analogy for the bending of the curved web panel,these boundary conditions were cho-sen to provide bounds with respect to the lateral displacements and stresses.For the maximum ‘‘bulging’’transverse displace-ment,the value of ␣is derived by considering the fourth (flange)edge of the model as simple support.In reality,of course,the flange will provide a torsional rigidity between that of fixed and simple.For the maximum plate bending stress occurring at the top of the web (flange/web juncture),the case where the fourth edge is fixed will provide a conservative value of ␤.Following this approach,the maximum displace-ment and bending stress of the panel,respectively,can then be approximated by42␣h ␴12(1Ϫ␯)c m ␦=(33)max2Et R2␤h t ␴cmM =(34)b ␪Rwhere ␣and ␤are constants depending on the location of the displacement or moment,respectively,and the aspect ratio only;h c =height of the web in compression;and ␴m =stress at the web/flange line resulting from vertical bending moment.Finite-element models were created to verify the applica-bility of the above formulation.Both flat plate models and curved web models under stress due to the vertical bending moment were used for comparison.For the curved plate mod-els,the ␣and ␤results are averaged for various curvatures (Davidson 1996).The flange/web edge was modeled as sim-ple,fixed,and with various flange rigidities.Figs.8–10clearly demonstrate that the lateral pressure analogy closely models the linear elastic displacement behavior of the curved web panel,within reasonable limits of curvature.From Figs.8–10,it can be noted that the simple support and fixed support at the top (flange)boundary bound the values of ␣where flangerigidities were included,up to an aspect ratio (in compression)a /h c =6.The values for the simple support condition converge to approximately ␣=0.00651with increasing aspect ratio (a /h c ).Similarly,the values of ␤with flange rigidities are bounded by the fixed case,and the fixed-flange values con-verge to ␤=0.0667.It should be noted though,that the lateral pressure analogy becomes less accurate for higher curvatures.This is because the analogy assumes a linear distribution of the membrane stress through the depth of the web in com-pression,which becomes less true for increasing curvatures. But,with regard to predicting the‘‘bulging’’displacement and the resulting plate bending stresses at theflange/web juncture for design,the inaccuracy is in the conservative direction,i.e., the nonlinear membrane stress distribution will cause the bulg-ing displacement and plate bending stresses predicted by the lateral load analogy to be greater than the actual.The accuracy and applicability of the approach with respect to curved I-girder web design,including deflection amplification effects, will be demonstrated in a subsequent paper.Elastic Buckling BehaviorAs mentioned in the‘‘Background’’section,an extensive investigation on the bifurcation load for curved web panels under both bending and shear and their combinations was done as early as1973by Abdel-Sayed,but withoutflange rigidity included in the mathematical models.As verification,eigen-values were extracted for the models described above.The critical stresses for the curved panels were normalized to that of theflat panel with the same dimensions and are presented in Table1.Also,the theoretical approximation shown as(35) is presented for reference purposes.Eq.(35)represents the approximate elastic buckling stress for theflat web panel with flange rigidity and is the basis for the web design parameters used in current design specifications:2␲E␴Ϸ[0.8(kϪk)ϩk](35)crϪst fs ss ss2212(1Ϫ␯)(h/t)wwhere k fs and k ss represent the buckling constants for a plate under vertical bending stresses withfixed-simple and simple-simple boundary conditions,respectively.From the results pre-sented in Table1,it can be noted that the critical loads of the panels with curvature are indeed higher than theflat panels. The buckled modeshapes for theflat panel and the curved are practically identical(Ballance1996).Because this verifies that the elastic buckling critical stresses of the curved panels are greater than that of the comparableflat,no further effort will be given to the eigenvalue analysis.SUMMARY AND CONCLUSIONSAlthough a number of researchers(mostly Japanese)have investigated the behavior of horizontally curved I-girder webs, very little useful design information has been presented.The current web slenderness requirements presented in the AASHTO Guide Specifications for Horizontally Curved High-way Bridges are based upon analytical research by Culver that was conducted as part of the CURT project and upon experi-mental research by Daniels in the1970s.The only other known formulations for curved web slenderness requirements are those suggested by the Japanese researcher Nakai.The reductions in required web slenderness for all of these for-mulations,though,represent regression curves of analytical data where the curvature of the panel,a/R,is the only param-eter affecting the design.There is great disparity between the reduction presented by Culver and that presented by Nakai. Furthermore,the research on which these reduction equations were based involved only doubly symmetric sections and lim-ited panel aspect ratios.It is doubtful that engineers using these design equations are aware of these limitations.The present research takes a more methodical and theory-based approach.A‘‘lateral pressure’’analogy is developed, and it is shown that this analytical model can be conservatively applied to approximate the‘‘bulging’’transverse displace-ments and plate bending stresses.This model will subse-quently be enhanced to include nonlinear effects and to de-velop equations that represent the reduction in strength due to curvature.ACKNOWLEDGMENTSThe research presented here was conducted at Auburn University and is supported as part of the Federal Highway Administration Contract No. DTFH61-92-C-00136Curved Steel Bridge Research Project.Auburn Uni-versity is a subcontractor to HDR Engineering,Inc.,who is the prime contractor for this project.The opinions and conclusions expressed or implied in the paper are those of the writers and are not necessarily those of the Federal Highway Administration.APPENDIX I.REFERENCESAbdel-Sayed,G.(1973).‘‘Curved webs under combined shear and nor-mal stresses.’’J.Struct.Div.,ASCE,99(3),511–525.AISC Marketing,Inc.(1992).‘‘Bridges,a preliminary design study: Ramp B,Okeechobee Rd.to SB S.R.826,Dade County Florida.’’PDS 92/043,Chicago.AISC Marketing,Inc.(1993a).‘‘Bridges,a preliminary design study:Fly-over Ramp DB,Charlotte,North Carolina.’’PDS93/005,Chicago. 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