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 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:
2
D 36,500a
a =1Ϫ8.6ϩ34
(1)
ͫ
ͩͪͬ
t R
R
F w ͙y
where 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
=1Ϫ2.9
ϩ2.2
(2)
ͫ
ͱ
ͬ
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 Specifications ,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 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)as
D 36,500a =1Ϫ4Յ192(3)ͫ
ͬt R F w ͙y D 23,000a =1Ϫ4
Յ170
(4)
ͫ
ͬ
t R
f w ͙b
The 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-
