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204/JOURNAL OF BRIDGE ENGINEERING/AUGUST1999

JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/205

ends.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 ?ange resultant force.The maximum web bending stress was shown to occur at 0.20h from the compression ?ange for the simple support stiffness condition and 0.24h for the ?xed 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 ?xed condition cases it was determined that there was no signi?cant change in the membrane stresses (from free to ?xed)but that there was a signi?cant 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 ?anges based on curvature and web slenderness for a panel aspect ratio of 1.0and a web-to-?ange 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 de?ections and web bending stresses that would occur for girders with a curvature corre-sponding to the initial imperfection out-of-?atness 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:

D 36,500a

a =1?8.6?34

(1)

???

t R

F w ?y

where D =unsupported distance between ?anges;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 ?ange/web boundaries were modeled as ?xed,and the boundaries at the transverse stiffeners were modeled as ?xed and simple.Longitudinal stiffeners were modeled with moments of inertias as multiples of the AASHO (Standard 1969)values for straight https://www.wendangku.net/doc/1a12530297.html,ing analytical results obtained for the slenderness required to limit the plate bending stresses in the curved panel to those of a ?at panel with the maximum allowed out-of-?atness (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

=1?2.9

?2.2

(2)

t R f R

w ?b

where the calculated bending stress,f b ,is in psi.It was further

concluded 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 Speci?cations ,and (2)is used in the ‘‘Allowable Stress Design’’portion for girders stiffened with one longitudinal stiffener.This work was

continued 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 ?anges 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 ?at panel and increases with an increase in curvature.

In 1980,Daniels et al.summarized the Lehigh University ?ve-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)as

D 36,500a =1?4?192(3)?

?t R F w ?y D 23,000a =1?4

?170

(4)

t R

f w ?b

The latter equation is currently used in the ‘‘Allowable Stress Design’’portion of the Guide Speci?cations 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 ?nite-difference method.Simple support boundary condi-tions were assumed along the curved boundaries (top and bot-tom at the ?ange locations)and both simple and ?xed 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,de?ection,membrane stress,bend-ing stress,and torsional stress were obtained,but no equations for design use were presented.Signi?cant 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 ?nite-ele-ment method to demonstrate the elastic ?nite displacement be-havior of curved I-girder webs under bending using models with and without ?ange rigidities.Rotation was not allowed (?xed condition)about the vertical axis at the ends of the panel (transverse stiffener locations).Again,the nonlinear distribu-

206/JOURNAL OF BRIDGE ENGINEERING /AUGUST

1999

FIG. https://www.wendangku.net/doc/1a12530297.html,parison of Web Slenderness Requirements from AASHTO Guide Speci?cations and Nakai

tion of the membrane stress was noted but appears signi?cant 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 ?nite-element method.Models included simple support,?xed support,and ?ange ri-gidities at the ?ange/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 ?ange stress increase was not mentioned.It was emphasized that the web panel model with no ?ange rigidity is inadequate in estimating the behavior of the curved panel under signi?cant loading.No direct recommendations or formulations regarding the design of curved I-girder webs were made.

Suetake et al.(1986)examined the in?uence of ?anges on the strength of curved I-girders under bending using the mixed ?nite-element approach.The ends of the panels (transverse stiffener locations)were modeled as simple supports,and var-ious width/thickness ratios for the ?anges 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 in?uence of the ?ange rigidity cannot be ignored.Also,observations were made on the torsional buckling behavior of the ?anges.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 ?nite-element method.The web plate panels were modeled with and without stiffnesses for the ?anges.Models without ?ange rigidity were modeled as ?xed 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 ?ange 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 ?ange-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 ?anges.It was also demon-strated that the maximum web de?ection occurs in the vicinity of 0.25D from the compression ?ange but that this transverse de?ection 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 plate

transverse de?ection,and plate bending stress,c c

?,?,max max would be limited to the maximum transverse de?ection and bending stress that would occur in the straight girder with the same dimensions but,instead of curvature,with a maximum initial de?ection,w 0,of D /250,which is the maximum allow-able initial de?ection stipulated in the Japanese design code (Speci?cations 1990a,b).From a comparison of the displace-ment results with the stress results,it was shown that criteria

based 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 ?0

?(5)

t w

a 1??0

for the case of no longitudinal stiffeners

??(6)

0t w

for the case of one or two longitudinal stiffeners and a /R ??0;and

a a ??????ε(7)

000

?????

t R

for the case of one or two longitudinal stiffeners and a /R >?0.

A plot of (5)–(7)along with the current Guide Speci?ca-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 as

a R =

1?8.6?34

(8)

C 1?

???

JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/

207

FIG. https://www.wendangku.net/doc/1a12530297.html,parison of Curvature Reduction Equations—No Longitudinal Stiffeners

a a

R =

1?2.9

?2.2

(9)

C 2?

R R a

R =1?4

(10)D ?

R =

(11)

N 1a ?

?1??0

??R 2

a a R =

????ε(12)

N 200

?????

Eq.(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-i?ed 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 ?nite-element analyses of ?at 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 METHODOLOGY

To 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 that

approximate the linear behavior of the system;and (2)the ?nite-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 veri?cation.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.RESULTS

Parametric Review

A 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 Behavior

To begin the investigation,the behavior of single web pan-els of various aspect ratios,curvatures,and cross-section di-mensions was analyzed using the ?nite-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 ?xed sup-ports,and those at the top and bottom of the panel were mod-eled as simple,?xed,and with ?ange 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 ?ange rigidity matching that of a 30.48?609.6mm (1.2?24in.)?ange.The displacement of the doubly symmetric cross sec-tion at a /2is shown in Fig.4.Note that the rotation direction of the ?ange 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 ?at 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 other

208/JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999

TABLE

1.Critical Stress Comparison—Curved versus

Straight

CRITICAL 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 =100ft

t f /t w =3(9)t f /t w =5(10)100100100200200200300300300

123123123

95.6795.6795.6723.9223.9223.9210.6310.6310.63

101.78101.58101.4925.4825.4225.4011.3411.3211.31

1.0011.0021.0031.0031.0061.0071.0061.0111.014

1.0141.0151.0171.0161.0191.0201.0191.0241.026

1.0031.0061.0101.0101.0211.0301.0221.0711.057

1.0161.0191.0241.0241.0331.0401.0361.0751.063

1.0681.1381.1691.2901.2971.3381.4281.4461.500

1.0811.1321.1831.2931.2951.3391.4301.4431.498

Note:1in.=25.4mm;1ft =0.305m;1ksi =6.895MPa.

FIG. 3.Finite-Element Model Description

FIG. 5.Membrane Stress Distribution through Panel Depth with Increase in Curvature and Increase in h /t w

FIG. 4.

Displacement of Cross Section

researchers,as described above.As a result of this reduction in web membrane stress,the ?anges carry a higher load;thus,even without considering warping stresses in the ?anges,the curved section would be unable to carry as much vertical moment as the comparable straight section before yielding of the ?anges initiates.However,for the most severe case considered here [h /t w =200,R =30.5m (100ft),a /h =3.0],the increase in ?ange 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 ?ange/web intersec-tion,as it would for ?at panels (Davidson 1996).Lateral Pressure Analogy

The amount of transverse or ‘‘bulging’’displacement can be approximated by using a ‘‘lateral pressure’’analogy.Con-

JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/

209

FIG.7.Theoretical Development for Flat Panel under Hydro-static Loading

FIG. 6.

Lateral Pressure Analogy Development

sider 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 girder

P ?t q =

=(13)

c R R

Since 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 ?at 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 https://www.wendangku.net/doc/1a12530297.html,ing this anal-ogy,the displacements and plate bending moments can be readily solved for the fourth (?ange)edge of the plate as sim-ple and ?xed support,similar to the hydrostatic solutions pro-vided by Timoshenko and Woinowsky-Krieger (1959).For the ?at rectangular plate with two opposite edges simply sup-ported,Levy (1899)suggested taking the solution in the form of a series:

?m ?x w =

Y sin

(14)

m ?

m =1

where w =component of displacement in the transverse di-rection;and Y m =function of y only and must be in a form that satis?es the boundary conditions and the governing dif-ferential equation

444?w ?w ?w q q x

0?2?==(15)

4224?x ?x ?y ?y D aD p p

Further simpli?cation can be made by taking the solution of

(15)in the form

w =w ?w (16)

where w 1=particular solution to the de?ection of a strip under a hydrostatic load represented as

q 3x 0

33w =

?10ax ?7a x 1??

360D a

4m ?12q a (?1)m ?x

sin ?

D ?m a

p m =1,3,5...

(17)which satis?es the boundary conditions at x =0and x =a of

2?w

w =0,

=0(18)

?x and w 2represents the homogeneous solution to (15)and is of the form

q a m ?y m ?y m ?y

0w =

A cosh

?B sinh 2m m ?

D a a a

m ?y m ?y m ?y ?C sinh

?D cosh m m ?

a a a

(19)

Noting that the last two terms of (19)are odd functions and

that 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 form

m ?1q a 2(?1)m ?y

0w =

?A cosh m

55D ?m a

m =1

m ?y m ?y m ?x

?B sinh sin

a a

(20)

From the boundary conditions,the constants can be ob-tained as

m ?1

(2??tanh ?)(?1)m m A =?

(21)m 55?m cosh ?m

m ?1

(?1)B =(22)

m 55

?m cosh ?m

where

m ?b ?=

(23)

m 2a

and,further noting that at y =0,sinh(?m )equals zero and cosh(?m )equals unity,(20)can be simpli?ed to

?4m ?1

q a 2(?1)m ?x 0(w )=?A sin

(24)

y =0

D ?m a

m =1

or in simpler terms

4?q a 0(w )=

(25)

y =0

D p

where

210/JOURNAL OF BRIDGE ENGINEERING /AUGUST

1999

FIG.10.?versus a /h c Plot for h /t w =

200

FIG.9.

?versus a /h c Plot for h /t w =

200

FIG.8.

?versus a /h c Plot for h /t w =100

?m ?1

2(?1)m ?x ?=

?A sin

(26)

?m a

m =1

and is a function of x and y and the plate aspect ratio only;and where

Et D =

(27)

p 212(1??)

The plate bending moments M x and M y can be derived as

22?w ?w

M =?D ??(28)x p ??

22?x ?y 22?w ?w

M =?D ??(29)

y p

?y ?x

After substitution of (20),(28)and (29)can be most simply written as

2(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 ?xed.For this case,the particular solution,w 1,is of the form

q x

04224w =

(16y ?24b y ?5b )

(32)

1384aD p

which satis?es 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 (?ange)edge of the model as simple support.In reality,of course,the ?ange will provide a torsional rigidity between that of ?xed and simple.For the maximum plate bending stress occurring at the top of the web (?ange/web juncture),the case where the fourth edge is ?xed will provide a conservative value of ?.Following this approach,the maximum displace-ment and bending stress of the panel,respectively,can then be approximated by

?h ?12(1??)c m ?=

(33)

max

2Et R

2?h t ?c

M =

(34)

b ?R

where ?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/?ange line resulting from vertical bending moment.Finite-element models were created to verify the applica-bility of the above formulation.Both ?at 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 ?ange/web edge was modeled as sim-ple,?xed,and with various ?ange 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 ?xed support at the top (?ange)boundary bound the values of ?where ?ange

rigidities 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 ?ange rigidities are bounded by the ?xed case,and the ?xed-?ange 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 the?ange/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 de?ection ampli?cation effects, will be demonstrated in a subsequent paper.

Elastic Buckling Behavior

As 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 without?ange rigidity included in the mathematical models.As veri?cation,eigen-values were extracted for the models described above.The critical stresses for the curved panels were normalized to that of the?at 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 the?at web panel with ?ange rigidity and is the basis for the web design parameters used in current design speci?cations:

??[0.8(k?k)?k](35)

cr?st fs ss ss22

12(1??)(h/t)w

where k fs and k ss represent the buckling constants for a plate under vertical bending stresses with?xed-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 the?at panels. The buckled modeshapes for the?at panel and the curved are practically identical(Ballance1996).Because this veri?es that the elastic buckling critical stresses of the curved panels are greater than that of the comparable?at,no further effort will be given to the eigenvalue analysis.

SUMMARY AND CONCLUSIONS

Although 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 Speci?cations 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.ACKNOWLEDGMENTS

The 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.REFERENCES

Abdel-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. AISC Marketing,Inc.(1993b).‘‘Bridges,a preliminary design study:NH Rte.27over Rte.51,Hampton,New Hampshire.’’PDS92/067,Chi-cago.

AISC Marketing,Inc.(1993c).‘‘Bridges,a preliminary design study: Ramp14,Florida Ave.Bridge over IHNC,Orleans Parish,Louisiana.’’PDS93/009,Chicago.

AISC Marketing,Inc.(1993d).‘‘Bridges,a preliminary design study: Westbound ramp over Independence Boulevard Mecklenburg Co., North Carolina.’’PDS93/029,Chicago.

Ballance,S.R.(1996).‘‘The behavior of horizontally curved I-girder webs under pure bending,’’MS thesis,Auburn University,Auburn,Ala. Basler,K.,and Thurliman,B.(1961).‘‘Strength of plate girders in bend-ing.’’J.Struct.Div.,ASCE,87(6),153–181.

Brogan,D.(1972).‘‘Bending behavior of cylindrical web panels,’’MS thesis,Carnegie-Mellon University,Pittsburgh.

Culver,C.G.(1972).‘‘Design recommendations for curved highway bridges.’’Project68-32,Commonwealth of Pennsylvania Department of Transportation,Harrisburg,Pa.

Culver,C.G.,Dym,C.,and Brogan,D.(1972a).‘‘Bending behaviors of cylindrical web panels.’’J.Struct.Div.,ASCE,98(10),2201–2308. Culver,C.G.,Dym,C.,and Brogan,D.(1972b).‘‘Instability of horizon-tally curved members—bending behaviors of cylindrical web panels.’’Rep.Prepared for PENDOT,Carnegie-Mellon University,Pittsburgh. Culver,C.G.,Dym,C.L.,and Uddin,T.(1973).‘‘Web slenderness re-quirements for curved girders.’’J.Struct.Div.,ASCE,99(3),417–430. Daniels,J.H.,Zettlemoyer,N.,Abraham, D.,and Batcheler,R.P. (1979a).‘‘Fatigue of curved steel bridge elements—analysis and de-sign of plate girder and box girder test assemblies.’’DOT-FH-11-8198.1,Lehigh University,Bethlehem,Pa.

Daniels,J.H.,Fisher,J.W.,Batcheler,R.P.,and Maurer,J.K.(1979b).‘‘Fatigue of curved steel bridge elements—ultimate strength tests of horizontally curved plate and box girders.’’DOT-FH-11-8198.7,Le-high University,Bethlehem,Pa.

Daniels,J.H.,Fisher,J.W.,and Yen,B.T.(1980).‘‘Fatigue of curved steel bridge elements,design recommendations for fatigue of curved plate girder and box girder bridges.’’Rep.No.FHWA-RD-79-138,Of-?ces of Res.and Devel.Struct.and Appl.Mech.Div.,Federal Highway Administration,Washington,D.C.

Davidson,J.S.(1996).‘‘Nominal bending and shear strength of curved steel I-girder bridge systems,’’PhD dissertation,Auburn University, Auburn,Ala.

Frank,K.H.,and Helwig,T.A.(1995).‘‘Buckling of webs in unsym-metric plate girders.’’Engrg.J.,32(3),43–53.

Fujii,K.,and Ohmura,H.(1985).‘‘Nonlinear behavior of curved girder web considering?ange rigidities.’’Proc.,JSCE,Struct.Engrg./Earth-quake Engrg.,Tokyo,2(1).

Fukumoto,Y.,and Nishida,S.(1981).‘‘Ultimate load behavior of curved I-beams.’’J.Engrg.Mech.Div.,ASCE,107(2),367–385. Guidelines for the design of horizontally curved girder bridges(draft). (1988).Steel Structure Study Committee,Hanshin Expressway Public Corporation,Osaka,Japan.

Guide speci?cations for horizontally curved highway bridges.(1993). American Association of State Highway and Transportation Of?cials, Washington,D.C.

Hall,D.H.,and Yoo,C.H.(1996).‘‘Curved girder design and construc-tion current practice.’’Second Interim Rep.,Nat.Cooperative Hwy. Res.Program12–38,Auburn University/BSDI,Auburn,Ala. Hiwatashi,S.,and Kuranishi,S.(1984).‘‘The?nite displacement behav-ior of horizontally curved elastic I-section plate girders under bend-ing.’’Proc.,JSCE,Struct.Engrg./Earthquake Engrg.,Tokyo,1(2),59–69(in Japanese).

JOURNAL OF BRIDGE ENGINEERING/AUGUST1999/211

Kitada,T.,Ohminami,R.,and Nakai,H.(1986).‘‘Criteria for designing web and?ange plates of horizontally curved plate girders.’’Proc., SSRC1986Annu.Tech.Session,Structural Stability Research Council, Bethlehem,Pa.

Kuranishi,S.,and Hiwatashi,S.(1981).‘‘Elastic behavior of web plates of curved plate girders in bending.’’Proc.,JSCE,Tokyo,315,1–11 (in Japanese).

Kuranishi,S.,and Hiwatashi,S.(1983).‘‘Nonlinear behavior of elastic web plates of curved plate girders in bending.’’Proc.,3rd Int.Collo-quium on Stability of Metal Struct.,Paris,France,305–312.

Levy,M.(1899).Comp.Rend.,129,535–539.

Mariani,N.,Moser,J.D.,Dym,C.L.,and Culver,C.G.(1973).‘‘Trans-verse stiffener requirements for curved webs.’’J.Struct.Div.,ASCE, 97(4),757–771.

Mikami,I.,and Furunishi,K.(1981).‘‘Nonlinear behavior of cylindrical web panels under bending and shear.’’Theoretical and applied me-chanics,V ol.29,University of Tokyo Press,Tokyo,65–72. Mikami,I.,and Furunishi,K.(1984).‘‘Nonlinear behavior of cylindrical web panels.’’J.Engrg.Mech.,ASCE,110(2),230–251.

Mikami,I.,Furunishi,K.,and Yonezawa,H.(1980).‘‘Nonlinear behavior of cylindrical web panels under bending.’’Proc.,JSCE,Tokyo,299, 23–34(in Japanese).

Mozer,J.,Ohlson,R.,and Culver,C.G.(1975a).‘‘Stability of curved plate girders.’’Rep.No.P2,Federal Highway Administration,Wash-ington,D.C.

Mozer,J.,Ohlson,R.,and Culver,C.G.(1975b).‘‘Horizontally curved highway bridges,stability of curved plate girders.’’Rep.No.P3,Fed-eral Highway Administration,Washington,D.C.

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Nakai,H.,Kitada,T.,and Ohminami,R.(1984a).‘‘Experimental study on ultimate strength of web panels in horizontally curved girder bridges subjected to bending,shear,and their combinations.’’Proc.,SSRC 1984,Annu.Tech.Session and Meeting,Structural Stability Research Council,Bethlehem,Pa.,91–102.

Nakai,H.,Kitada,T.,and Ohminami,R.(1985a).‘‘Experimental study on buckling and ultimate strength of web panels in horizontally curved girder bridges subjected to combined loads of bending and shear.’’Proc.,JSCE,Tokyo,356/I-3,445–454(in Japanese).

Nakai,H.,Kitada,T.,and Ohminami,R.(1985b).‘‘Proposition for de-signing intermediate transverse stiffeners in web plate of horizontally curved girders.’’Proc.,JSCE,Tokyo,362/I-4,249–257(in Japanese). Nakai,H.,Kitada,T.,Ohminami,R.,and Fukumoto,K.(1984b).‘‘Ex-perimental study on shear strength of horizontally curved plate gird-ers.’’Proc.,JSCE,Tokyo,350/I-2,281–290(in Japanese).

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‘‘V-load analysis,an approximate procedure,simpli?ed and extended for

determining moments and shears in designing horizontally-curved open-framed highway bridges.’’(1984).USS highway structures design handbook,V ol.1,United States Steel Corporation,Pittsburgh. Wachowiak,T.(1967).‘‘Die Berechnung gekruemmter Stege duennwan-digen traegern auf Grund der Shalentheorie,’’PhD thesis,Politechnika Gdanska,Gdansk,Poland.

Washizu,K.(1975).Variational methods in elasticity and plasticity,2nd Ed.,Pergamon Press,Tarrytown,N.Y.

Yadlosky,J.M.(1993).‘‘Curved steel bridge research project,summary work plan(1.0).’’HDR Interim.Rep.DTFH61-92-C-00136,Federal Highway Administration,Washington,D.C.

APPENDIX II.NOTATION

The following symbols are used in this paper:

A m,

B m,

C m,

D m=constants;

a=panel length and/or distance between trans-

verse stiffeners;

b f=?ange width;

D=unsupported distance between?ange compo-

nents(AASHTO symbol);

D p=plate?exural rigidity;

F y=yield stress;

f b=calculated stress due to bending(AASHTO

symbol);

h=panel height of analytical models;

h c=panel height in compression;

k fs,k ss=?xed-simple and simple-simple boundary con-

dition constants for plate buckling,respec-

tively;

M bz,M b?=plate bending moments about z and?direc-

tions,respectively;

M cr=critical vertical bending moment of girder;

M x,M y=plate bending moments about x and y direc-

tions,respectively;

P=resultant tangential force in curved plate due

to vertical bending;

q c=lateral pressure due to curvature;

q0=transverse distributed load at plate edge under

hydrostatic loading;

R=radius of curvature at center of panel or I-

girder web thickness;

R C1,R C2,R D,

R N1,R N2

=curvature reduction factors on design D/t w by

various researchers;

r,z,?=coordinates in radial,vertical,and tangential

directions,respectively;

t=panel plate thickness;

t f=?ange thickness;

t w=web thickness of I-shaped girder;

w=transverse displacement component;

x,y,z=rectangular coordinates of?at panel;

Z h,Z m,Z n,Z s=curvature parameters used by various research-

ers;

?=constant used in predicting maximum‘‘bulg-

ing’’displacement of curved panel;

?=constant used in predicting maximum stress of

curved panel;

?x,?y=summation constants;

?max=maximum transverse‘‘bulging’’displacement

of curved panel;

?=Poisson’s ratio;

?bz,?b?=plate bending stress;

?m=membrane stress at top of panel;

?=shear stress;and

?c v,?st=refers to curved and straight(?at)values,re-

spectively.

212/JOURNAL OF BRIDGE ENGINEERING/AUGUST1999

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外文翻译原文

204/JOURNAL OF BRIDGE ENGINEERING/AUGUST1999

JOURNAL OF BRIDGE ENGINEERING /AUGUST 1999/205 ends.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 ?ange resultant force.The maximum web bending stress was shown to occur at 0.20h from the compression ?ange for the simple support stiffness condition and 0.24h for the ?xed 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 ?xed condition cases it was determined that there was no signi?cant change in the membrane stresses (from free to ?xed)but that there was a signi?cant 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 ?anges based on curvature and web slenderness for a panel aspect ratio of 1.0and a web-to-?ange 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 de?ections and web bending stresses that would occur for girders with a curvature corre-sponding to the initial imperfection out-of-?atness 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: 2 D 36,500a a =1?8.6?34 (1) ? ??? t R R F w ?y where D =unsupported distance between ?anges;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 ?ange/web boundaries were modeled as ?xed,and the boundaries at the transverse stiffeners were modeled as ?xed and simple.Longitudinal stiffeners were modeled with moments of inertias as multiples of the AASHO (Standard 1969)values for straight https://www.wendangku.net/doc/1a12530297.html,ing analytical results obtained for the slenderness required to limit the plate bending stresses in the curved panel to those of a ?at panel with the maximum allowed out-of-?atness (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 =1?2.9 ?2.2 (2) ? ? ? t R f R w ?b where the calculated bending stress,f b ,is in psi.It was further concluded 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 Speci?cations ,and (2)is used in the ‘‘Allowable Stress Design’’portion for girders stiffened with one longitudinal stiffener.This work was continued 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 ?anges 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 ?at panel and increases with an increase in curvature. In 1980,Daniels et al.summarized the Lehigh University ?ve-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)as D 36,500a =1?4?192(3)? ?t R F w ?y D 23,000a =1?4 ?170 (4) ? ? t R f w ?b The latter equation is currently used in the ‘‘Allowable Stress Design’’portion of the Guide Speci?cations 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 ?nite-difference method.Simple support boundary condi-tions were assumed along the curved boundaries (top and bot-tom at the ?ange locations)and both simple and ?xed 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,de?ection,membrane stress,bend-ing stress,and torsional stress were obtained,but no equations for design use were presented.Signi?cant 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 ?nite-ele-ment method to demonstrate the elastic ?nite displacement be-havior of curved I-girder webs under bending using models with and without ?ange rigidities.Rotation was not allowed (?xed condition)about the vertical axis at the ends of the panel (transverse stiffener locations).Again,the nonlinear distribu-

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