Development of LRFD Specifications for Horizontally Curved Steel Girder Bridges (2006)

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A P P E N D I X A Literature Search

A-iii C O N T E N T S A-iv List of Abbreviations and Symbols A-1 A1 Introduction A-1 A1.1 General A-1 A1.2 Objective A-1 A1.3 Research Procedure A-1 A1.4 Electronic Database A-1 A1.5 Synthesis A-1 A2 Overview of Literature Search A-1 A2.1 General A-2 A2.2 Analysis of Curved Bridges A-2 A2.3 Design A-3 A3 Curved I-Girder Bridges A-3 A3.1 Analysis A-3 A3.2 Design A-3 A3.2.1 Nominal Bending Strength A-5 A3.2.2 Curvature Effects on Elastic Lateral-Torsional Buckling A-5 A3.2.3 Cross-Frame Spacing and Lateral Bracing Effects A-5 A3.2.4 Local Buckling of Curved I-Girder Flanges A-5 A3.2.5 Strength of Curved I-Girder Web Panels under Pure Shear A-6 A3.2.6 Curved I-Girder Web Panels Subjected to Bending A-6 A3.2.7 I-Girder Webs Subjected to Combined Bending and Shear A-6 A3.2.8 Lifting of Slender Curved I-Girders A-6 A3.2.9 Constructibility Limit State A-6 A4 Curved Box-Girder Bridges A-6 A4.1 Analysis A-7 A4.2 Design A-7 A5 Conclusions and Recommendations for Further Study A-8 A5.1 Analysis Methods A-8 A5.2 I-Girders A-8 A5.3 Box-Girders A-8 A5.4 Constructibility A-8 A5.5 Extreme Event Limit State A-9 References A-13 Abstracts A-37 Background Research Pertaining to Updated AASHTO LRFD Specifications for Steel Structures, Third Edition

A-iv L I S T O F A B B R E V I A T I O N S A N D S Y M B O L S a = distance between transverse stiffeners, b = width of flange, bf = width of compression flange, c = curvature parameter, C = shear strength constant, D = depth of the web panel, Dc = depth of the web panel in compression, fb = normal stress, Fb = allowable bending stress, fv = shear stress, Fv = allowable shear stress, Fy = minimum specified yield stress, fw = warping stress (flange lateral bending stress), k = elastic shear buckling coefficient, L = length of girder, Mu = ultimate vertical bending moment, R = radius of curvature, Rd = reduction factor of shear due to initial out of flatness or reduction factor of deflection, as appropriate, Rs = reduction factor of stress, t = thickness of flange, tf = thickness of flange, tw = thickness of the web panel, Vp = plastic shear capacity, Vu = ultimate shear capacity, x = subtended angle between adjacent cross frames, y = critical moment ratio,  = unbraced length of compression flange of I-girder, Ψ = reduction factor of local buckling of compression flange, and Ψw = parameter relating bend-buckling of curved I-girder web.

A1 Introduction A1.1 General The scope and intent of this literature search was to continue the exhaustive literature survey that was conducted as Phase I of the FHWA-sponsored project,“Curved Steel Bridge Research Project” (DTFH61-93-C-00136), and published as Interim Report I,“Synthesis” (1). Although the report date of the Syn- thesis is December 1994, the cut-off date of the literature survey activity is assumed to be June 1993. Therefore, the literature collected and included in this report are those published after June 1993 and up to the time this search was conducted in early 2000. References made to articles in AASHTO LRFD Bridge Design Specifications and AASHTO Standard Specifications for Highway Bridges use the article numbers existing at the time of conducting the literature search. A1.2 Objective The objective of this task was to perform a comprehensive up-to-date literature search after June 1993 on publications describing work related to the design and analysis of horizon- tally curved steel box-girders and I-girders. A brief synopsis of the information in each reference is provided for a speedy review. A1.3 Research Procedure In order to facilitate the objective, a comprehensive literature search was conducted using computerized searching, manual paging, and personal contact. The initial search resulted in the collection of approximately 100 references. Each reference was reviewed to determine its appropriateness for the design and analysis of horizontally curved steel girder bridges. References that were obviously not relevant to horizontally curved steel girders were eliminated.General references on analysis method- ology that may have applied to both steel and concrete were retained. This process resulted in over 90 references, which were placed in the electronic database. A1.4 Electronic Database The information compiled on horizontally curved steel bridge girders was stored in the Microsoft Access database management software, which runs under the Windows oper- ating system.This database program was chosen simply because the Synthesis is already in this software. Because this report is considered to be a continuation of the Synthesis, the database format was made identical to that used in the Synthesis. The information pertaining to a particular reference was stored in six different fields. The first three fields contained material about the title, author, and year of publication. The fourth field was for the source of publication (journal, con- ference, publisher, etc.), and the fifth field was for additional information, such as the order of contract number. The sixth field was for a summary or abstract of the publication. In addition to these six fields, check boxes were added to the database form for further information. These check boxes can be used to distinguish whether a reference is on box-girders or I-girders, static or dynamic, thermal or failure analysis, or other aspects of response. The information could then be used to sort and filter required publications. As an example, one can request all design work by Hall that was published after 1993, and that information can be sorted according to the year in an ascending order. The result of a filtering and/or sorting process can be on-screen (which is temporary) or can be saved as a query (which can be used at a later time).Additionally, one can also search a certain field for a particular word or name. A1.5 Synthesis Because of budget and time constraints, the collection of references could not be exhaustive, and the evaluation of the information described in each reference could not be suffi- ciently detailed. Nevertheless, the synthesis is very useful for digesting quickly what has been studied after 1993 for further understanding of the complex behavior of horizontally curved steel bridge girders. A2 Overview of Literature Search A2.1 General Ninety-six references were identified, of which 91 had abstracts. Of the 91 references with abstracts, 89 were reviewed. The remaining two references with abstracts could not be acquired. When sorted according to broad subject area, the 91 references reveal the following distribution: analysis, 45; design, 22; progress report, 6; bridge test, 3; synthesis, 3; computer program, 2; and cylindrical shells, rings, and oth- ers, 10. All of the authors of the progress reports are associ- ated with two recent multiyear national projects on steel girder curved bridges: FHWA Curved Steel Bridge Research Project (CSBRP) and NCHRP Project 12-38. A vast majority of references deal with the analysis of curved beams. A signif- icant number of these references have an academic, rather than practical, approach. Many references are for the deriva- tion of a new or modified spatial curved beam element or its derivative with six or seven kinematic degrees of freedom. Some of these so-called new curved beam elements include one or more of the following rather special effects: shear deformation, rotatory inertia terms in a consistent mass matrix, and noncoincidental centroid and shear center. These A-1

curved beam elements may be incorporated into either 1D or 2D analysis of curved girders. This literature search yielded a significantly higher proportion of references dealing with design problems (22 out of 91) than did the Synthesis results summarized in NCHRP Report 424 (2) (approximately 50 out of 540). Table A-1 shows the distribution of the 91 references based on a broad classification. Contents of some references (2, 3, 4, 5, 6, 7) are so comprehensive that it is inappropriate to classify them as either an analysis type or a design type. Therefore, the total number of references given in each cate- gory in Table A-1 will exceed 91. A2.2 Analysis of Curved Bridges Hall et al. (8, 9) presented three generic methods of curved bridge analysis, 1D line girder analysis, 2D planar grid analysis, and 3D finite element method. An early formulation of the V-load method (10) is the basis of a typical line girder analysis method applied to I-girder bridges and M/R method (11) is the 1D line girder analysis method applied to horizontally curved box girder bridges. A variety of 2D planar grid analysis meth- ods for horizontally curved bridges are available, including finite strip method, finite element method, and finite difference method. Any curved bridge analysis method that does not recognize the section depth is categorized as 2D planar grid method or 1D line girder analysis method. In 3D finite element method, the section depth is recognized and the forces in cross- frames or diaphragms are evaluated as an integral part of the analysis. A 3D skeletal space frame analysis method is not considered a 3D finite element method because of its inability to rigorously account for the plate/shell action of the girder web and the deck. Any commercially available, general-purpose finite element analysis codes—such as NASTRAN, ABAQUS, ADINA, and ANSYS—can be used to provide full 3D analyses of curved bridges. There are, however, a few special-purpose proprietary codes—such as BSDI 3D System (12). Article 4.1 of Ref. (4) requires that the analysis be performed using a rational method that accounts for the interaction of the entire superstructure. Inability to recognize the section depth in 1D and 2D analysis methods may lead to a serious compromise in the determination of member forces due to temperature variations, wind loads, skewed support, bearing orientation, centrifugal forces, and prestressing of the deck. Although it generally requires additional efforts, strategies, and time to model the structure and to pre- and post-processing the input and output data, a full 3D finite element analysis of a curved bridge automatically satisfies the requirements of Article 4.1 of Ref. (4). A2.3 Design References dealing with the design of curved box-girder bridges are fewer than references dealing with the design of curved I-girder bridges. Because of the superior torsional strength of the box-girders, closed box-shape girders are better suited for horizontally curved bridges. However, the addition cost associated with the fabrication of curved box-girders may offset the advantage afforded by the box-girders. Articles 1.8 and 2.9 of Ref. (13) require that the diaphragms or cross-frames be full-depth members designed as the primary load-carrying members and that they be attached closely to the flanges. Article 9.3 of Ref. (4) requires that eccentricity between the cross-frame members or diaphragm flanges and the girder flanges be recognized in the analysis, in the design of connection plates, and in their connection to the web and flange. In the design and construction of horizontally curved A-2 Category References Progress Report 14–19 Bridge Test 20–22 Computer Program 12, 23 Synthesis 1, 24, 25 Others 26–35 Analysis, I-Girder (Static) 6, 7, 36–60 Analysis, I-Girder (Buckling) 6, 7, 38, 61–64 Analysis, I-Girder (Dynamic) 6, 65–74 Analysis, Box-Girder (Static) 6, 75–77 Analysis, Box-Girder (Dynamic) 6, 78, 79 Design, I-Girder 2–8, 38, 80–90 Design, Box-Girder 2, 4–6, 9, 91–94 Table A-1. Distribution of references.

highway bridges in the early 1960s, these provisions were not followed closely. As a result, a large number of bridges were retrofit later, and, in some instances, inadequate cross-framing was a contributing factor for bridges to be classified as struc- turally deficient. A3 Curved I-Girder Bridges A3.1 Analysis Hall et al. (8) presented interesting results of a series of com- parative analyses made on an example I-girder bridge. It is a three-span (160 feet, 210 feet, 160 feet), four-girder (11-foot girder spacing) structure with a centerline curvature radius of 700 feet. The bridge was analyzed by a 3D finite element analysis (12), a 2D grid analysis (MSC/NASTRAN), and a 1D V-load analysis. There exists a close correlation in the dead load analysis results among all three analysis methods. Fairly close correlation existed between the 3D finite element analy- sis and the 2D grid analysis results for live load. However, the 1D V-load analysis produced significantly different vertical bending moments in the live load analysis. The discrepancy between the V-load and the finite element analysis results was up to 70%. Much of this discrepancy was probably due to the wheel load distribution factors (AASHTO Article 3.23) used to determine the primary vertical bending moments in the V-load analysis. This discrepancy in the live load analysis results will likely be improved if more accurate wheel load distribution factors are used in the V-load analysis. A3.2 Design Hall et al. (8) present a comprehensive curved I-girder bridge design example. Ref. (8) is 243 pages long and includes essen- tially all major design procedures that reflect the provisions of the “Recommended Specifications for Steel Curved-Girder Bridges” (4) (hereafter referred to as the 12-38 recommended specifications). Some of the recent research results that can be incorporated into design specifications are highlighted below. A3.2.1 Nominal Bending Strength The nominal bending strength predictor equations in the Guide Specifications appear to be quite conservative, as shown in Table A-2. Hall et al. (2) demonstrate that the lateral flange bending stress (fw) due to curvature in the McManus-Culver predictor equation in the Guide Specifications is double- counted. Therefore, the lateral flange bending stress due to curvature at the critical cross-frame location must be set equal to zero. This fact and an alternate critical stress expres- sion based on yielding of the flange due to combined lateral bending and vertical bending are implemented in the 12-38 recommended specifications. An interaction equation based on full plastification of a compact I-section has been introduced in 1988 by Nakai and Yoo (6). Schilling (87) extended this concept to flange-compact sections and noncompact sections in 1996. These interaction equations are, however, limited to doubly symmetric I-shaped sections. The ultimate strength of the section is controlled by the combined action of vertical bending and lateral flange bending. The magnitude of the lateral bending stress is affected by the radius of curvature and the cross-frame spacing. There- fore, these equations are not well suited for use as ultimate strength predictors. Since the combination of the vertical and lateral flange bend- ing moments depends on the unbraced length (i.e., cross-frame spacing) of the compression flange for a given loading and the radius of curvature, spacing of the cross-frame is an important parameter in the development of an ultimate strength predic- tor equation. Yoo et al. (94) presented yield interaction equa- tions for nominal bending strength of curved I-girders. There are 17 interaction equations encompassing composite sections that are doubly symmetric, singly symmetric, I-shaped com- pact, flange-compact, noncompact, cracked, and uncracked. It is assumed that only singly symmetric compact sections can be made hybrid. For compact-flange and noncompact sections, however, a homogeneous section is assumed because hybrid construction will not yield significantly higher moment capacities. The radius of curvature, the cross-frame spacing, the material properties, and the cross-section geometry are included as variables of the interaction equations. Although these interaction equations can be evaluated in an iterative, fast-converging manner, they are programmed to use on a PC. Comparison of selected test results from the literature versus analytically predicted vertical moment capacities is given in Table A-2. The values in the column under the heading “Interaction” were evaluated from the interaction equations. The values under the heading of “L-T Bklg”were computed by the regression formula derived by Yoo et al. (64) for slender sections, where α = 2.152, β = 2.129, γ = 0.1058, x = subtended angle between the adjacent cross-frames in radian, and y = critical vertical moment ratio = Mxcr,cv/Mxcr,st. The subscript cv stands for curved girder, and st stands for straight girder. It appears feasible that the nominal bending strength limit state of curved girders can be assessed in a manner similar to AASHTO LRFD Articles 6.10.5 and 6.10.6 because the strength predictor equa- tions are available for compact, noncompact, and slender sections. y x= −( )1 1γ β α ( ) A-3

Guide Spec. Yoo et al. Nakai (6) Fukumoto Hanshin (5) L-T Bklg. (64) Interaction (90) Testedby Specimen ID Test Results Mu (k-in.) Mu (k-in.) Ratio Mu (k-in.) Ratio Mu (k-in.) Ratio Mu (k-in.) Ratio Mu (k-in.) Ratio Mu (k-in.) Ratio L1-A 1830 1716 0.94 1865 1.02 1701 0.93 1999 1.09 1504 0.82 1107 0.61 L2-A 1830 1749 0.96 1888 1.03 1726 0.94 1993 1.09 1533 0.84 1136 0.62 GI-5 1377 * * 1279 0.93 1232 0.89 1279 0.93 989 0.72 652 0.47 C u l v e r GO-8 2120 1992 0.94 2062 0.97 2156 0.89 2213 1.04 1912 0.90 1164 0.55 M1 8098 6965 0.86 6965 0.86 7972 0.98 5657 0.70 8007 0.99 7509 0.93 M2 7754 3708 0.48 6986 0.90 7058 0.91 6508 0.84 6187 0.80 4713 0.61 M3 6131 2627 0.43 5985 0.98 6031 0.98 5356 0.87 4803 0.78 3504 0.57 M4 7203 * * 6958 0.97 5498 0.76 7972 1.11 4317 0.60 2654 0.37 M5 5902 * * 5766 0.98 4535 0.77 6790 1.15 3592 0.61 2298 0.39 M6 6287 * * 5665 0.90 4482 0.71 6649 1.06 3503 0.56 2277 0.36 M7 6547 * * 6148 0.94 4819 0.74 7176 1.10 3818 0.58 2371 0.36 M8 2935 * * 2979 1.01 1654 0.56 1451 0.49 1054 0.36 * * N a k a i M9 5548 * * 5951 1.07 4253 0.77 6934 1.25 3073 0.55 * * * The central angle for this girder exceeds the limitation of the procedure. Table A-2. Comparison of bending test results with six strength predictor equations Mu in k-in., Ratio = Analytical (DEN = 12) Mu/Test Mu.

A3.2.2 Curvature Effects on Elastic Lateral- Torsional Buckling Although the classical bifurcation type lateral-torsional buckling may not be observed in horizontally curved girders, the system eigenvalue indicates a critical elastic lateral-tor- sional buckling moment. The results of a series of theoretical and numerical analyses detailed in Ref. (64) on a number of hypothetical curved girders have been quantified in a regres- sion formula (23). Additional study results are presented in Refs. (38) and (7). A3.2.3 Cross-Frame Spacing and Lateral Bracing Effects Davidson et al. (3) presented a regression formula for the cross-frame spacing resulting from a number of three- dimensional finite element analyses of a large number of hypothetical curved bridges: where fw/fb = warping-to-bending stress ratio, L = girder span length in feet, R = radius of curvature in feet, and bf = com- pression flange width in inches. The desired cross-frame spacing can also be extracted from the V-load analysis con- cept (7) as where units for R and bf need to be consistent (either inches or feet). The placement of lateral bracing on top and bottom flanges makes the I-girders behave as pseudo-box-girders. Depending on the radius of curvature and the subtended angles of the girders, the addition of lateral bracing reduces the vertical deflection, flange bending stress, and warping stress significantly. A3.2.4 Local Buckling of Curved I-Girder Flanges Davidson and Yoo (80) presented a simple yet conserva- tive regression formula for the limiting width-to-thickness ratio of curved I-girder compression flange after a series of numerical investigations on curved I-girders of practical design proportions. b t b t where R bcv st f ⎛⎝⎜ ⎞⎠⎟ ≤ ⎛⎝⎜ ⎞⎠⎟ = − ⎛ ⎝⎜ψ ψ 1 05 4 2 . l ⎞ ⎠⎟ ≤ 1 0 4. ( ) l = ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ 5 3 3 0 5 f f R bw b f . ( ) l l= − ( )⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ − L n f f Rb L w b f 6 108 22 1 52 . ( . ) A3.2.5 Strength of Curved I-Girder Web Panels under Pure Shear Lee and Yoo (86) conducted a three-dimensional finite ele- ment investigation on a number of hypothetical curved I-girder web panels encompassing a wide variety of practical design parameters. The aspect ratio, a/D, of transversely stiffened web panels was varied from 0.5 to 2.0, and the web slenderness ratio, D/tw, was varied from 90 to 300. In practical designs of curved bridge plate girders, the value of the curvature param- eter, c = (a2)/(8Rtw), has been shown to be less than 1.0 (6). The curvature parameter varied from 0.5 to 1.0 inch. The shear strength is computed by where The constant C is to be determined by and the strength reduction factor, Rd, due to the initial out-of- flatness, D/120, permitted by the Bridge Welding Code (95) is given by R for D t F d w y = >1 0 12. ( ) 12,000 k R D t F k for d w y = + ( ) −⎛ ⎝⎜⎜ ⎞ ⎠⎟⎟0 8 0 2 6 000 6 000 . . , , 6,000 12,000k k F D t Fy w y ≤ ≤ ( )11 R for D t F d w y = <0 8 10. ( ) 6,000 k C k D t F for D t Fw y w y = × ( ) 4 5 10 9 7 2 . ( )> 7,500 k C k D t F for F D t Fw y y w y = ( ) ≤ ≤ 6 000 8 , ( ) 6,000 7,500k k C for D t Fw y = 1 0 7. ( )< 6,000 k V F Dtp y w= 0 58 6. ( ) V R V Cu d p= +( )0 6 0 4 5. . ( ) A-5

A3.2.6 Curved I-Girder Web Panels Subjected to Bending Davidson et al. (84, 85) examined the limiting values of the web slenderness ratio of curved I-girders based on the control of excessive bulging and the amplified web stresses. Based on a lateral pressure analogy developed in this study and a series of nonlinear finite element analyses, formulations for the reduction in allowable web slenderness due to curvature effects were presented based upon both a limit on allowable bulging transverse displacement and based upon maximum allowable stress. where where where the subscripts in reduction factors shown in Equations 14 and 16 stand for deflection and stress, respectively. A3.2.7 I-Girder Webs Subjected to Combined Bending and Shear Davidson et al. (82, 83) discovered from the results of a large number of displacement finite element analyses that the vertical moment capacities of curved I-girders are reduced somewhat less than 5% when the girders are simultaneously subjected to shearing forces of 60% of their shear capacities. An interaction graph showing these small adjustments is given in Refs. (7), (38), (82), and (83). Effects of longitudinal stiff- eners on curved I-girder webs are presented in Refs. (7), (38), and (83). and D t D R D t w c w c c w ψ = + ⎛⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ 1 0 161 0 128. . ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ 2 2 0 5 1 1 5 D R D R c c. ( . 17) Rs w = 1 16 ψ ( ) D t D t R w cv w st s ⎛ ⎝⎜ ⎞ ⎠⎟ ≤ ⎛ ⎝⎜ ⎞ ⎠⎟ ( )15 R R D d = ≤0 185 1 0 14. . ( ) D t D t R w cv w st d ⎛ ⎝⎜ ⎞ ⎠⎟ ≤ ⎛ ⎝⎜ ⎞ ⎠⎟ ( )13 A3.2.8 Lifting of Slender Curved I-Girders Davidson (38) and Yoo (7) present the results of a series of finite element analyses to determine the critical lifting points under three variations of lifting schemes of a long slender curved girder during handling, transporting, and erection. The stresses that are induced by the self-weight of the girder dur- ing lifting are likely to be substantially less than the specified minimum yield stress of the girder. However, the excessive displacements and rotations during lifting can be problematic. A3.2.9 Constructibility Limit State The past performance record of horizontally curved highway bridges has been excellent. Most problems associated with curved bridges occurred during construction. The constructi- bility issue of curved bridges, along with the deck casting sequence, has become onerous.Article 2.5 of Ref. (4) mandates that the constructibility limit state be checked at each critical stage during construction. Hall et al. (8) demonstrate how this constructibility limit state check is made. A4 Curved Box-Girder Bridges A4.1 Analysis Razaqpur and Li (75, 76) present a new thin-walled, multi- cell box-girder finite element that can model extension, flexure, torsion, warping, distortion, and shear lag effects. The element is one-dimensional. The element is based on the generalized Vlasov’s thin-walled beam theory and the finite element tech- nique. Interaction between the longitudinal and transverse deformations of the box-girder is accounted for by using a series of parameters. The cross-sections need to be prismatic, and, because the deck is represented by a sector plate, the supports need to be in the radial direction. Yabuki et al. (77) present an incremental nonlinear analysis technique on an essentially one-dimensional curved box-girder model, includ- ing local buckling and initial stress effects on the ultimate strength of the girder. After comparing analytically predicted values with the test results of two scaled model specimens, Yabuki et al. concluded that the degree of correlation was sat- isfactory. Galdos et al. (78) present a methodology for deter- mining impact factor for curved box-girder bridges. Their recommendation has been implemented in the recommended specifications. Generally, impact factors for box-girders are higher than for I-girders. Using the finite element method, Sennah and Kennedy (79) present the results of a parametric study conducted on 120 simply supported, curved, composite, multicell bridges for the natural frequencies and mode shapes. The analytically obtained values were compared with results from tests on four 1/12 scaled model bridges. A-6

Hall et al. (9) presented interesting results of a series of comparative analyses made on an example box-girder bridge. The bridge is a three-span (160 feet, 210 feet, 160 feet), two- tub-girder (22 feet, 6 inches girder spacing) structure with a centerline radius of curvature of 700 feet. The bridge was ana- lyzed by a 3D finite element analysis (12), a 2D grid analysis (MSC/NASTRAN), and a 1D M/R analysis. There was a close correlation of the vertical bending moments obtained from all three analysis methods in all cases considered. Good cor- relation of these values was expected because the geometry and the relatively high torsional rigidity of the box-girders in this particular example minimize the increase in the vertical bending moments due to curvature. As the spacing between centers of the two box-girders exceeds 14 feet, the wheel load distribution was determined according to the computation of simple beam reactions (footnote “f” of AASHTO Table 3.23.1). Therefore, the total number of truck wheel loads was identical in all three analysis methods. The concrete deck was modeled as a continuum in the 3D finite element model. In both the grid analysis method and the M/R method, girders are represented as one-dimensional elements and the rigid concrete deck cannot be represented as a continuum. The 3D finite element analysis also properly recognizes the physical location of the two bearings at each support point. Two bear- ings at each support point cannot be physically represented in the grid model, or in the M/R method, due to the limitations imposed by one-dimensional modeling of the girders.Although the vertical bending moments compared well among the three analysis methods, the difference in torsional moments and shears between the 3D finite element analysis and the other analysis methods is more significant. The grid and M/R meth- ods tended to underestimate the torques in each girder at the supports and overestimate the shear in the diaphragms between the boxes at each support. A4.2 Design Hall et al. (9) presented a comprehensive curved box-girder (tub-girder) design example. Ref. (9) is 152 pages long and includes essentially all major design procedures reflecting the provisions of the “Recommended Specifications for Steel Curved-Girder Bridges” (4). Cheung and Foo (91) presented the results of a parametric study, using the finite strip method, on three simply sup- ported concrete-steel composite box-girder bridge models, including single-, double-, and triple box-cross sections. Internal forces were evaluated using a 20-term Fourier series. The models do not include diaphragms. Vertical bending moment ratios were presented for a variety of loading cases and other design parameters, such as span lengths, subtended angles, and span-to-depth ratios. Sennah and Kennedy (92, 93) presented the results of a parametric study, using the finite element method, on 120 simply supported curved box-girder bridge models. The ana- lytically obtained results are compared with test results on four 1/12 scaled model bridges. The parametric study and tests were conducted on concrete-steel composite box-girder bridges. A series of regression equations were given for verti- cal bending moments, deflections, and maximum axial forces in bracings. The Hanshin guidelines (5) include the following inter- action equation for the ultimate strength of a box-girder as where fb = total normal stress, including stresses due to vertical bending, lateral flange bending, and cross-sectional distortion; fv = shear stress; Fb = allowable bending stress; and Fv = allowable shear stress. Although both (fb/Fb) and (fv/Fv) are not permitted to be greater than 1.0, the sum of the square of these ratios may be taken as high as 1.2. The Hanshin guidelines do not use composite girders, but do use the allowable stress design method. The practice of curved box-girder design in North America heavily relies on composite tub-girder construction.Top flanges of tub-girders are designed according to the provisions of open cross-section, I-girder designs for factored dead loads and construction loads prior to hardening of the deck. Top flanges of horizontally curved tub-girders are also braced laterally to resist torsional shear flow. However, these lateral bracing mem- bers are not subjected to any additional loading after the deck is hardened.Yoo et al. (94) presented a simplified design method using top flange lateral bracing members by transforming a lattice wall into a pseudo-box wall. A5 Conclusions and Recommendations for Further Study This task resulted in 91 references published from June 1993 after the “Synthesis”(1) pertaining to the analysis, design, and experimental investigations of horizontally curved I-girder and box-girder bridges. These references were reviewed briefly and are placed in an electronic database that is easy to access, query, and update as additional research is completed. Some recommendations deserving immediate attention are sum- marized below. f F f F b b v v ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝⎜ ⎞ ⎠⎟ ≤ 2 2 1 2 18. ( ) A-7

A5.1 Analysis Methods Horizontally curved girders are one of the least understood structural elements in common use today. Designers, owners, and contractors will undoubtedly benefit from research lead- ing to improved fundamental understanding of curved girder bridge behavior. New analysis tools are available, including finite element computer programs that permit detailed analy- ses of both structural elements and entire structures. These analysis tools have not been adequately applied to horizontally curved girder bridges. The Guide Specifications require that the entire superstructure be considered in the analysis.However, approximate methods that do not have well-defined limitations are permitted. Even the refined methods have limitations and assumptions that remain unexplored. There is a great need to quantify the reliability of various analysis methods and to explore how and when each method might be best applied. It is well known that all V-load analyses are not the same. It is less well known that all 3D finite element analyses do not provide the same results. Application of the methods needs to be studied and compared with field measurements. The fail- ure mechanism of a curved bridge needs to be defined. Tests of full-scale bridges to failure are needed. Any refined analy- sis methods need to be correlated with the experimentally obtained data. A5.2 I-Girders Multiple I-girders attached to a common composite re- inforced concrete deck should be investigated because the shear center of the bridge superstructure shifts above the deck after the deck has hardened. Tests on web stiffening should be con- ducted to evaluate the shear strength, the bend-buckling strength, and the effectiveness of various details. Tests of shear connectors that are subjected to large lateral forces near cross- frames in fatigue are needed. Evaluation of the effectiveness of bottom flange bracing at various stages of bridge construction and service is needed. A5.3 Box-Girders There is a need to investigate an interaction equation for the ultimate strength of box-girders under the combined action of flexure and torsion. The effectiveness of internal cross-bracing in reducing cross-section distortion needs to be investigated. Although Ref. (96) is widely used to check the distortion- induced stresses in curved box-girders, the adequacy of its use for other-than-straight box-girders is not well defined. Design rules for internal diaphragms need to be advanced. Design rules of longitudinal stiffeners in curved compression flanges are urgently needed. A5.4 Constructibility During lifting and when the girders are temporarily un- braced, lateral deflections and twists are large. Research is needed to define which conditions limit the applicability of small deflection theory. Research is also needed to assess the lateral deflection and twist limitations during construc- tion to ensure that stresses and deflections will not exceed those permitted. A5.5 Extreme Event Limit State Design rules of horizontally curved girder bridges under extreme event limit state are urgently needed. No simple equation is available to determine the natural frequency of horizontally curved bridges. A-8

A-9 1. Zureick,A.,Naqib,R., and Yadlosky, J. M.(1994).“Curved steel bridge research project.” Interim Report I, “Synthesis,” FHWA Contract No. DT FH61-93-C-00136, FHWA, McLean, VA, December. 2. Hall, D. H., Grubb, M. A., and Yoo, C. H. (1999). NCHRP Report 424: Improved Design Specifications for Horizontally Curved Steel Girder Highway Bridges. Transportation Research Board, National Research Council, Washington, D.C. 3. Davidson J. S., Keller, M. A., and Yoo, C. H. (1996). “Cross-frame spacing and parametric effects in horizontally curved I-girder bridges.” Journal of Structural Engineering, ASCE, Vol. 122, No. 9, September. 4. Hall, D. H., and Yoo, C. H. (1998). “Recommended specifications for steel curved-girder bridges.” Appendix D of the final report submitted to National Cooperative Highway Research Program, Project 12-38, Transportation Research Board, National Research Council, Washington, D.C., December. 5. Hanshin Express Public Corporation and Steel Structure Study Committee (1988). “Guidelines for the Design of Horizontally Curved Girder Bridges (Draft).” Hanshin Expressway Public Cor- poration, October. 6. Nakai, H., and Yoo, C. H. (1988). Analysis and Design of Horizontally Curved Steel Bridges. McGraw-Hill Book Company, New York. 7. Yoo, C. H. (1996). “Progress Report on FHWA-CSBRP-Task D.” FHWA Contract No. DTFH61-92-C-00136, Auburn University, Department of Civil Engineering Interim Report submitted to HDR Engineering, Inc., Pittsburgh Office, Pittsburgh, PA, August. 8. Hall, D. H., Grubb, M. A., and Yoo, C. H. (1999).“Design example, horizontally curved steel I girder bridge.” Appendix E of the final report submitted to National Cooperative Highway Research Pro- gram, Project 12-38, Transportation Research Board, National Research Council, Washington, D.C., May. 9. Hall, D. H., Lawin, A. R., and Yoo, C. H. (1999). “Design example, horizontally curved steel box girder bridge.”Appendix F of the final report submitted to National Cooperative Highway Research Pro- gram, Project 12-38, Transportation Research Board, National Research Council, Washington, D.C., July. 10. Richardson, Gordon and Associates, “Analysis and Design of Hor- izontally Curved Steel Bridge Girders,”United States Steel Structural Report, ADUCO 91063, May 1963. 11. Tung and Foundation, “Approximate Torsional Analysis of Curved Box Girders by the M/R Method,” Engineering Journal, AISC, Vol. 7, No. 3, July 1970. 12. Hall, D. H. (1994). “BSDI 3D System.” Internal Document, Bridge Software Development International, Ltd., Coopersburg, PA. 13. AASHTO (1993). Guide Specifications for Horizontally Curved High- way Bridges. American Association of State Highway and Trans- portation Officials, Inc., Washington, D.C. 14. Duwadi, S. R., Grubb, M. A., Yoo, C. H., and Hartmann, J. (2000). “Federal highway administration’s horizontally curved steel bridge research project.” Proceedings of the 5th International Bridge Con- ference, Tampa, FL, April 3–5. 15. Duwadi, S. R., Hall, D. H.,Yadlosky, J. M.,Yoo, C. H., and Zureick,A. (1995).“FHWA-CSBRP I-girder component testing.”Proceedings of the Structures Congress 13, ASCE, Vol. 2, pp. 1695–1698, Boston, MA, April 24–28. 16. Duwadi, S. R., Yadlosky, J. M., and Yoo, C. H. (1994).“Horizontally curved steel bridge research-update 2.” Proceedings of the Structures Congress 12, ASCE,Vol. 2, pp. 1071–1076, Atlanta, GA, April 24–28. 17. Hall,D. H.(1997).“Proposed curved girder provisions for AASHTO.” Building to Last Structures Congress—Proceedings, Vol. 1, ASCE, New York, NY, pp. 151–154. 18. Yoo, C. H. (1998). “Recommended specifications for horizontally curved steel-girder highway bridges.” Proceedings of the 1998 Pacific Rim Steel Structures Conference, Seoul, Korea, October 13–16. pp. 37–40. 19. Yoo, C. H., Hall, D. H., and Sabol, S. A. (1995). “Improved design specifications for horizontally curved steel-girder highway bridges.” Proceedings of the Structures Congress 13, ASCE,Vol. 2, pp. 1699–1702, Boston, MA, April 2–5. 20. Galambos, T. V., Hajjar, J. F., Leon, R. T., Huang,W. H., Pulver, B. E., and Rudie, B. J. (1996). “Stresses in steel curved girder bridges.” Minnesota Department of Transportation,Office of Research Admin- istration, Transportation Building, 395 John Ireland Boulevard, St. Paul, Minnesota 55155-1899. 21. Richardson, J. A., and Douglas, B. M. (1993). “Results from field testing a curved box-girder bridge using simulated earthquake loads.” Earthquake Engineering and Structural Dynamics, Vol. 22, No. 10. 22. Utah State University. (2000). “Bridge Test Report.” 23. Arman EPST. (1994).“A finite element program for automatic static and dynamic analysis and design of horizontally curved I-girder bridges: CIG4BR (Curved I-Girder 4 degrees of freedom bridge).” AKMAN Engineering Production Software Technologies, 9350 Washington Blvd., Lanham, Maryland 20706-3119, USA. 24. Kitada, T., Nakai, H., and Murayama, Y. (1993). “State-of-the art on research, design and construction of horizontally curved bridges in Japan.” Proceedings of the SSRC Annual Technical Session, Milwaukee, WI. References

25. Zureick,A., and Naqib, R. (1999).“Horizontally curved steel I-girders state-of-the-art analysis methods.” Journal of Bridge Engineering, ASCE, Vol. 4, No. 1, February. 26. Bouabdallah, M. S., and Batoz, J. L. (1996).“Formulation and eval- uation of a finite element model for the linear analysis of stiffened composite cylindrical panels.”Finite Elements in Analysis and Design, Vol. 21, No. 4, April. 27. Bozhevolnaya, E., and Kildegaard, A. (1997). “Experimental study of a uniformly loaded curved sandwich beam.”Composite Structures, Vol. 40, No. 2, December. 28. Cleghorn, W. L., Tabarrok, B., and Lee, T. W. (1993). “Vibration of rings with unsymmetrical cross-sections: A finite element approach.” Journal of Sound and Vibration, Vol. 168, No. 1, November. 29. Dhondt, G., and Kohl, M. (1999). “Effect of the geometry and the load level on the dynamic failure of rotating disks.” International Journal of Solids and Structures, Vol. 36, No. 6, February. 30. Grubb, M. A., Yadlosky, J. M., and Herrmann, A. W. (1993). “Behavior of horizontally curved steel highway bridges.” Structural Engineering in Natural Hazards Mitigation, Proceedings of the Symposium on Structural Engineering in Natural Hazards Miti- gation. ASCE, New York, NY. 31. Meyerpiening, H. R., and Anderegg, R. (1995).“Buckling and Post- buckling investigations of imperfect curved stringer-stiffened com- posite shell. A. Experimental investigation and effective width evaluation.” Thin-Walled Structures, Vol. 23, No. 1–4. 32. Moas, E., Boitnott, R. L., and Hayden, G. O. (1994).“Analytical and experimental investigation of the response of the curved, composite frame/skin specimens.” Journal of the American Helicopter Society, Vol. 39, No. 3, July. 33. Shanmugam, N. E., Thevendran, V., Richard Liew, J. Y., and Tan, L. O. (1995). “Experimental study on steel beams curved in plan.” Journal of Structural Engineering, ASCE, Vol. 121, No. 2, February. 34. Shim,V. P. W., and Quah, S. E. (1998).“Solution of impact-induced flexural waves in a circular ring by the method of characteristics.” Journal of Applied Mechanics-Transactions of the ASME, Vol. 65, No. 3, September. 35. Stiftinger, M. A., Skrnajakl, I. C., and Rammerstorfer, F. G. (1995). “Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shell, part B. Computational investiga- tions.” Thin-Walled Structures, Vol. 23, No. 1–4. 36. Choi, J. K., and Lim, J. K. (1993). “Simple curved shear beam ele- ments.”Communications in Numerical Methods in Engineering, Vol. 9, No. 8, August. 37. Choi, J. K., and Lim, J. K. (1995). “General curved beam elements based on the assumed strain fields.”Computers and Structures,Vol.55, No. 3, May. 38. Davidson, J. S. (1996). “Nominal bending and shear strength of horizontally curved steel I-girder bridges.” Dissertation submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the degree of Doctor of Philosophy. 39. Dorfi, H. R., and Busby, H. R. (1994). “Effective curved composite beam finite element based on the hybrid-mixed formulation.”Com- puters and Structures, Vol. 53, No. 1, October. 40. Fu, C. C., and Hsu, Y. T. (1995). “Development of an improved curvilinear thin-walled Vlasov element.” Computers and Structures, Vol. 54, No. 1, January. 41. Hall, D. H. (1996). “Curved girders are special.” Engineering Struc- tures, Vol. 18, No. 10, October. 42. Ibrahimbegovic, A. (1995). “On finite element implementation of geometrically nonlinear Reissner’s beam theory: Three-dimensional curved beam elements.” Computer Methods in Applied Mechanics and Engineering, Vol. 122, No. 1–2, April. 43. Kang, Y. J., and Yoo, C. H. (1994). “Thin-walled curved beams. II: analytical solutions for buckling of arches.” Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. 44. Kim, J. G., and Kim, Y. Y. (1998).“New higher-order hybrid-mixed curved beam element.” International Journal for Numerical Methods in Engineering, Vol. 43, No. 5, November. 45. Koziey, B. L., and Mirza, F. A. (1994). “Consistent curved beam element.” Computers and Structures, Vol. 51, No. 6, June. 46. Lee, S. S., Koo, J. S., and Choi, J. M. (1996).“Development of a new curved beam element with shear effect.” Engineering Computations (Swansea, Wales), Vol. 13, No. 2–4. 47. Lim, C. W., Wang, C. M., and Kitipornchai, S. (1997).“Timoshenko curved beam bending solutions in terms of Euler-Bernoulli solu- tions.” Archive of Applied Mechanics, Vol. 67, No. 3, February. 48. Lin, S. M. (1998). “Exact solutions for extensible circular curved Timoshenko beams with nonhomogeneous elastic boundary con- ditions.” Acta Mechanica, Vol. 130, No. 1–2. 49. Litewka, P., and Rakowski, J. (1997). “Efficient curved beam finite element.”International Journal for Numerical Methods in Engineering, Vol. 40, No. 14, July. 50. Mentrasti, L. (1996). “Shear-torsion of large curvature beams— Part II.Applications to thin bodies.”International Journal of Mechan- ical Sciences, Vol. 38, No. 7, July. 51. Orth, F. J., and Surana, K. S. (1994). “P-version two-dimensional beam element for geometrically nonlinear analysis.”Computers and Structures, Vol. 50, No. 3, February. 52. Paavola, J., and Salonen, E. M. (1999).“Strain and stress analysis of a curved tapered beam model.” Computers and Structures, Vol. 72, No. 4–5. 53. Pi, Y. L., and Trahair, N. S. (1996). “Nonlinear elastic behaviour of I-beams curved in plan.” Research Report—University of Sydney, School of Civil and Mining Engineering, No. 734, December, pp. 1–45. 54. Raveendranath, P., Singh, G., and Pradhan, B. (1999). “Two-noded locking-free shear flexible curved beam element.” International Journal for Numerical Methods in Engineering, Vol. 44, No. 2, January. 55. Pak, R.Y. S., and Stauffer, E. J. (1994).“Nonlinear finite deformation analysis of beams and columns.” Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. 56. Ryu, H. S., and Sin, H. C. (1996).“Curved beam elements based on strain fields.”Communications in Numerical Methods in Engineering, Vol. 12, No. 11, November. 57. Sengupta, D., and Dasgupta, S. (1997). “Static and dynamic appli- cations of a five noded horizontally curved beam element with shear deformation.” International Journal for Numerical Methods in Engineering, Vol. 40, No. 10, May. 58. Simpson, M. D., and Birkemoe, P. C. (1997). “Nonlinear finite element modeling of curved girder experiments.”Structures—Design, Concrete and Reinforced Concrete Structures, Bridges Proceedings, Annual Conference, Canadian Society for Civil Engineering,Vol. 7, Canadian Society for Civil Engineering, Montreal, Quebec, Canada. 59. Thevendran,V.,Chen,S.,Shanmugam,N. E.,and Liew, J. Y. R. (1999). “Nonlinear analysis of steel-concrete composite beams curved in plan.” Finite Elements in Analysis and Design, Vol. 32, No. 3. 60. Yang, S. Y., and Sin, H. C. (1995).“Curvature-based beam elements for the analysis of Timoshenko and shear-deformable curved beams.” Journal of Sound and Vibration, Vol. 187, No. 4, November. 61. Hu, N., Hu, B., Yan, B., Fukunaga, H., and Sekine, H. (1999). “Two kinds of CO-type elements for buckling analysis of thin-walled A-10

curved beams.” Computer Methods in Applied Mechanics and Engi- neering, Vol. 171, No. 1. 62. Kang, Y. J., and Yoo, C. H. (1994). “Thin-walled curved beams. I: Formulation of nonlinear equations.” Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. 63. Kang, Y. J., Yoo, C. H., Yu., C., and Lee, H. E. (1993). “Lateral buckling behavior of thin-walled arches.” Proceedings of the Fourth East Asia-Pacific Conference on Structural Engineering and Construction, pp. 401–406, Seoul, Korea, September 20–22. 64. Yoo, C. H., Kang, Y. J., and Davidson, J. S. (1996). “Buckling analy- sis of curved beams by finite-element discretization.” Journal of Engineering Mechanics, ASCE, Vol. 122, No. 8, August. 65. Gendy, A. S., and Saleeb, A. F. (1994).“Vibration analysis of coupled extensional/flexural/torsional modes of curved beams with arbitrary thin-walled sections.” Journal of Sound and Vibration, Vol., 174, No. 2, July. 66. Howson, W. P., and Jemah, A. K. (1999).“Exact out-of-plane natural frequencies of curved Timoshenko beams.” Journal of Engineering Mechanics, ASCE, Vol. 125, No. 1, January. 67. Howson, W. P., Jemah, A. K., and Zhou, J. Q. (1995). “Exact natu- ral frequencies for out-of-plane motion of plane structures com- posed of curved beam members.”Computers and Structures, Vol. 55, No. 6, June. 68. Hsiao, K. M., and Yang, R. T. (1995). “Co-rotational formulation for nonlinear dynamic analysis of curved Euler beam.” Computers and Structures, Vol. 54, No. 6, March. 69. Huang, D.,Wang, T. L., and Shahawy, M. (1995).“Dynamic behavior of horizontally curved I-girder bridges.” Computers and Structures, Vol. 57, No. 4, November. 70. Jiang, J., and Olson, M. D. (1994).“Vibration analysis of orthogonally stiffened cylindrical shells using super finite elements.” Journal of Sound and Vibration, Vol. 173, No. 1, May. 71. Kang, K. J., Bert, C. W., and Striz, A. G (1996). “Vibration analysis of horizontally curved beams with warping using DQM.” Journal of Structural Engineering, ASCE, Vol. 122, No. 6, June. 72. Kawakami, M., Sakiyama, T., Matsuda, H., and Morita, C. (1995). “In-plane and out-of-plane free vibrations of curved beams with variable sections.” Journal of Sound and Vibration, Vol. 187, No. 3, November. 73. Wang, R. T., and Sang,Y. L. (1999).“Out-of-plane vibration of multi- span curved beam due to moving loads.”Structural Engineering and Mechanics, Vol. 7, No. 4. 74. Yildirim,V. (1999).“In-plane free vibration of symmetric cross-ply laminated circular bars.” Journal of Engineering Mechanics, ASCE, Vol. 125, No. 6, June. 75. Razaqpur, A. G., and Li, H. G. (1994). “Refined analysis of curved thin-walled multicell box girders.”Computers and Structures, Vol. 53, No. 1. 76. Razaqpur, A. G., and Li, H. G. (1997).“Analysis of curved multicell box girder assemblages.” Structural Engineering and Mechanics, Vol. 5, No. 1. 77. Yabuki, T., Arizumi, Y., and Vinnakota, S. (1995).“Strength of thin- walled box girders curved in plan.” Journal of Structural Engineering, ASCE, Vol. 121, No. 5. 78. Galdos, N. H., Schelling, D. R., and Sahin, M. A. (1993). “Method- ology for impact factor of horizontally curved box bridges.” Journal of Structural Engineering, ASCE, Vol. 119, No. 6. 79. Sennah, K. M., and Kennedy, J. B. (1997).“Dynamic characteristics of simply supported curved composite multi-cell bridges.”Canadian Journal of Civil Engineering, No.4, August. 80. Davidson, J. S., and Yoo, C. H. (1996).“Local Buckling of Curved I-Girder Flanges.” Journal of Structural Engineering, ASCE,Vol. 122, No. 8, August. 81. Davidson, J. S., and Yoo, C. H. (2000). “Evaluation of strength for- mulations for horizontally curved flexural members.” Journal of Bridge Engineering, ASCE, Vol. 5, No. 3, August. 82. Davidson, J. S., Ballance, S. R., and Yoo, C. H. (2000). “Behavior of curved I-girder webs subjected to combined bending and shear.” Journal of Bridge Engineering, ASCE, Vol. 5, No. 2, May. 83. Davidson, J. S., Ballance, S. R., and Yoo, C. H. (2000). “Effects of longitudinal stiffeners on curved I-girder webs.” Journal of Bridge Engineering, ASCE, Vol. 5, No. 2, May. 84. Davidson, J. S., Ballance, S. R., and Yoo, C. H. (1999). “Finite dis- placement behavior of curved I-girder webs subjected to bending.” Journal of Bridge Engineering, ASCE, Vol.4, No. 3, August. 85. Davidson, J. S., Ballance, S. R., and Yoo, C. H. (1999). “Analytical model of curved I-girder web panels subjected to bending.” Journal of Bridge Engineering, ASCE, Vol. 4, No. 3, August. 86. Lee, S. C., and Yoo, C. H. (1999). “Strength of curved I-girder web panels under pure shear.” Journal of Structural Engineering, ASCE, Vol. 125, No. 8. 87. Schilling, C. G. (1996). “Yield-interaction relationships for curved I-girders.”Journal of Bridge Engineering, ASCE,Vol.1,No.1,February. 88. Weaver, DL. (1996). “Steel girder bridges.” Construction Specifier, Vol. 49, No. 5, May. 89. Yoo, C. H. (1993). “Some Considerations in the design and con- struction of horizontally curved highway bridges.” Proceedings of the Fourth East Asia-Pacific Conference on Structural Engineering and Construction, p. 55–60, Seoul, Korea, September 20–22. 90. Yoo, C. H., and Davidson, J. S. (1997).“Yield interaction equations for nominal bending strength of curved I-girders.” Journal of Bridge Engineering, ASCE, Vol. 2, No. 2, May. 91. Cheung, M. S., and Foo, S. H. C. (1995). “Design of horizontally curved composite box-girder bridges—A simplified approach.” Canadian Journal of Civil Engineering, Vol. 22, No. 1. 92. Sennah, K., and Kennedy, J. B. (1998).“Shear distribution in simply- supported curved composite cellular bridges.” Journal of Bridge Engineering, ASCE, Vol. 3, No. 2. 93. Sennah, K., and Kennedy, J. B. (1999). “Simply supported curved cellular bridges: Simplified design method.” Journal of Bridge Engi- neering, ASCE, Vol. 4, No. 2. 94. Yoo, C. H., Davidson, J. S., and Zhang, J. (1998). “Top flange diag- onal bracing of horizontally curved box girders.” Proceedings of the Engineering Mechanics Conference: A Force for the 21st Century, La Jolla, CA, May 18–20. 95. ANSI/AASHTO/AWS D1.5-96 (1996), Bridge Welding Code. 96. Heins and Hall (1981). Designer’s Guide to Steel Box Girder Bridges, Bethlehem Steel Corporation (Booklet No. 3500). A-11

A-13 AUTHOR: Zureick,A., Naqib, R., and Yadlosky, J. M. (1994) TITLE: Curved steel bridge research project INFO: Interim Report I, “Synthesis,” FHWA Contract No. DT FH61-93-C-00136, FHWA, McLean, VA, December. ABSTRACT: A comprehensive literature search on horizon- tally curved girder bridges up to 1993. AUTHOR: Hall, D. H., Grubb, M. A., and Yoo, C. H. (1999) TITLE: NCHRP Report 424: Improved Design Speci- fications for Horizontally Curved Steel Girder Highway Bridges INFO: Transportation Research Board, National Research Council, Washington, D.C. ABSTRACT: The primary objective of NCHRP Project 12-38 was to develop revised Guide Specifi- cations for Horizontally Curved Highway Bridges, based on current practice and technol- ogy, that could be recommended to AASHTO for possible adoption. The revised Guide Spec- ifications were to be applicable to the design, fabrication, and erection of horizontally curved steel I-girder and box-girder bridges. The research team was to complete ten tasks in this project. Chapter 1 provides a detailed descrip- tion of each task. Six appendixes were also pre- pared and submitted: Appendix A: I-Girder Curvature Study; Appendix B: Curved Girder Design and Construction, Current Practice; Appendix C: A Unified Approach for Designing and Constructing Horizontally Curved Girder Bridges, Proposed AASHTO Specifications (Highlights of Major Changes); Appendix D: Recommended Specifications for Steel Curved- Girder Bridges and Commentary; Appendix E: Design Example, Horizontally Curved Steel I-Girder Bridge; and Appendix F: Design Exam- ple, Horizontally Curved Steel Box-Girder Bridge. This report summarizes the develop- ment of the revised Guide Specifications and the Appendixes. Appendixes D through F are not published in Report 424. AUTHOR: Davidson J. S., Keller, M. A., and Yoo, C. H. (1996) TITLE: Cross-frame spacing and parametric effects in horizontally curved I-girder bridges INFO: Journal of Structural Engineering, ASCE, Vol. 122, No. 9, September. ABSTRACT: The finite element method was used to create detailed models of horizontally curved steel I-girder bridges connected by cross-frames. The effects of a number of parameters on the behavior of the curved girder system were established and compared to the effect of these parameters in straight girder systems. The parameters that most significantly affect the behavior of the systems were determined to be the degree of curvature, span length, and flange width. An equation was developed based on a nonlinear statistical regression to provide a preliminary design limit for the cross-frame spacing interval. AUTHOR: Hall, D. H., and Yoo, C. H. (1998) TITLE: Recommended specifications for steel curved- girder bridges INFO: Appendix D of the final report submitted to National Cooperative Highway Research Pro- Abstracts

gram, Project 12-38, Transportation Research Board, National Research Council, Washing- ton, D.C., December. ABSTRACT: Recommended specifications and commentary. AUTHOR: Hanshin Express Public Corporation and Steel Structure Study Committee (1988) TITLE: Guidelines for the Design of Horizontally Curved Girder Bridges (Draft) INFO: Hanshin Expressway Public Corporation, October. ABSTRACT: This is the only semi-official design guide of horizontally curved girder bridges in the world today other than AASHTO Guide Specifica- tions. The document has not been officially adopted by the Japanese specification-issuing body. The guide includes an I-girder design example and a box-girder design example. AUTHOR: Nakai, H., and Yoo, C. H. (1988) TITLE: Analysis and Design of Horizontally Curved Steel Bridges INFO: McGraw-Hill Book Company, New York. ABSTRACT: The one and only comprehensive technical sequence book on the subject in the world today. The book presents fundamentals of thin-walled curved girder behavior and intro- duces Japanese curved bridge design and con- struction practice. The book explains the background research for most provisions of the Hanshin Guidelines. AUTHOR: Yoo, C. H. (1996) TITLE: Progress Report on FHWA-CSBRP-Task D INFO: FHWA Contract No. DTFH61-92-C-00136, Auburn University, Department of Civil Engi- neering Interim Report submitted to HDR Engineering, Inc., Pittsburgh Office, Pitts- burgh, PA, August. ABSTRACT: Task D of CSBRP is to develop analytical pre- dictor equations for the nominal bending strength and shear strength of I-girders. Included in the interim report, in addition to the nominal bending strength and shear strength of I-girders, are the design of trans- verse stiffeners and longitudinal stiffeners of I-girders; a guide for an optimum cross-frame spacing; and an independent analysis (NASTRAN) of the component test frame, the local buckling of curved I-girder compression flanges, and the lifting and transporting of slender curved I-girders. AUTHOR: Hall, D. H., Grubb, M. A., and Yoo, C. H. (1999) TITLE: Design example, horizontally curved steel I girder bridge INFO: Appendix E of the final report submitted to National Cooperative Highway Research Pro- gram, Project 12-38, Transportation Research Board, National Research Council, Washing- ton, D.C., May. ABSTRACT: I-girder bridge design example highlighting the design procedure following the revised provisions. AUTHOR: Hall, D. H., Lawin, A. R., and Yoo, C. H. (1999) TITLE: Design example, horizontally curved steel box girder bridge INFO: Appendix F of the final report submitted to National Cooperative Highway Research Pro- gram, Project 12-38, Transportation Research Board, National Research Council, Washing- ton, D.C., July. ABSTRACT: Box-girder bridge design example highlighting the design procedure following the revised provisions. AUTHOR: Hall, D. H. (1994) TITLE: BSDI 3D System INFO: Internal Document, Bridge Software Develop- ment International, Ltd., Coopersburg, PA. ABSTRACT: A special-purpose 3D finite element program package tailored to analyze highway bridge superstructures under dead load and any design live loads, including impact and cen- trifugal load. Temperature and wind load analyses can readily be conducted. The eccen- tric load effects resulting from bearing forces on skewed bridges and curved bridges can be examined routinely. AUTHOR: Duwadi, S. R., Grubb, M. A., Yoo, C. H. and Hartmann, J. (2000) A-14

TITLE: Federal highway administration’s horizontally curved steel bridge research project INFO: Proceedings of the 5th International Bridge Conference, Tampa, FL, April 3–5. ABSTRACT: Since 1992, the Federal Highway Administra- tion (FHWA) has had a major concentrated research project in the area of horizontally curved steel bridges, the objective of which is to conduct research to better define the funda- mental behavior of such bridges. This project involves theoretical work leading to the devel- opment of refined predictor equations, and verification of those equations through linear and non-linear analysis and experimental test- ing of I-girder components. The overall exper- imental program involves testing of a series of full-scale bending and shear curved steel I-girder components, and subsequently, a full- size bridge. The objectives of this paper are to summarize the development and refinement of predictor equations and to describe the work leading to the first series of experimental tests, which involve testing of full-scale bend- ing components. The research team consists of HDR Engineering, Inc.; Auburn University; Georgia Institute of Technology; BSDI, Ltd.; and FHWA structures and support staff. AUTHOR: Duwadi, S. R., Hall, D. H., Yadlosky, J. M., Yoo, C. H., and Zureick, A. (1995) TITLE: FHWA-CSBRP I-girder component testing INFO: Proceedings of the Structures Congress 13, ASCE, Vol. 2, Boston, MA, April 24–28. ABSTRACT: This paper is an update on the continuing research ongoing under the FHWA Curved Steel Bridge Research Program (CSBRP). An overview of this comprehensive program was presented at the 1994 ASCE Structures Con- gress (Duwadi et al. 1994). A brief highlight of all the tasks involved is presented. The major focus of this paper is on the proposed experi- mental testing of the I-beam components at the FHWA laboratory at the Turner Fairbank Highway Research Center (FHWA 1993). AUTHOR: Duwadi, S. R., Yadlosky, J. M., and Yoo, C. H. (1994) TITLE: Horizontally curved steel bridge research— update 2 A-15 INFO: Proceedings of the Structures Congress 12, ASCE, Vol. 2, Atlanta, GA, April 24–28. ABSTRACT: The AASHTO Guide Specifications for Hori- zontally Curved Highway Bridges was initially issued in 1980 and is based on research per- formed by the CURT project in the early 1970s. They cover both I-girders and box-girders. AISI later financed the development of related provisions for load factor design of curved girders. These were later adopted by AASHTO as part of the same curved girder guide speci- fications. More than 12 years of design and construction experience with the guide speci- fications has uncovered a number of major deficiencies. As a result of this, there exists an urgent need to conduct a series of research studies so that improved recommended guide specifications can be presented to AASHTO for consideration for adoption. To address these needs, the Federal Highway Administration and the AASHTO-sponsored National Coop- erative Highway Research Program are jointly administrating a coordinated program of research intended to provide a revised recom- mended ASD and LRFD design specification and eventually a recommended LRFD-based design specification. This paper presents an overview of these two major research projects. AUTHOR: Hall, D. H. (1997) TITLE: Proposed curved girder provisions for AASHTO INFO: Building to Last Structures Congress— Proceedings, Vol. 1, ASCE, New York, NY. ABSTRACT: This paper discusses the draft proposed speci- fication for design of horizontally curved steel highway bridges. A draft of the provisions has been submitted to AASHTO for consideration of adoption. An overview of the provisions is provided, including highlights of modifica- tions to the existing AASHTO Guide Specifica- tions for horizontally curved girders. AUTHOR: Yoo, C. H. (1998) TITLE: Recommended specifications for horizontally curved steel-girder highway bridges INFO: Proceedings of the 1998 Pacific Rim Steel Structures Conference, Seoul, Korea, October 13–16.

ABSTRACT: An overview of the Recommended Specifica- tions for Horizontally Curved Steel-Girder Highway Bridges is presented. The research project was funded by NCHRP Project 12-38. AUTHOR: Yoo, C. H., Hall, D. H., and Sabol, S. A. (1995) TITLE: Improved design specifications for horizon- tally curved steel-girder highway bridges INFO: Proceedings of the Structures Congress 13, ASCE, Vol. 2, Boston, MA, April 2–5. ABSTRACT: A research program focused on the develop- ment of improved design specifications for horizontally curved steel girder highway bridges has been underway for the past two years. The objective of the research is to pre- pare improved guide specifications and com- mentary based on current technology and available information. A thorough literature search and evaluation and survey of current practice throughout the world have been reflected in the development of the specifica- tions and commentary. AUTHOR: Galambos, T. V., Hajjar, J. F., Leon, R. T., Huang, W. H., Pulver, B. E., and Rudie, B. J. (1996) TITLE: Stresses in steel curved girder bridges INFO: Minnesota Department of Transportation, Office of Research Administration, Trans- portation Building, 395 John Ireland Boule- vard, St. Paul, Minnesota 55155-1899. ABSTRACT: A steel curved I-girder bridge system may be more susceptible to instability during con- struction than bridges constructed of straight I-girders. The primary goal of this project is to study the behavior of the steel superstructure of curved steel I-girder bridge systems during all phases of construction and to ascertain whether the linear elastic analysis software used by Mn/DOT during the design process repre- sents well the actual stresses in the bridge. Sixty vibrating wire strain gages were applied to a two-span, four-girder bridge, and the resulting stresses and deflections were compared with computational results for the full construction sequence of the bridge. The computational results from the Mn/DOT analysis software were first shown to compare well with results from a program developed specifically for this project (called the “UM program”) since the A-16 latter permits more detailed specification of actual loading conditions on the bridge during construction. The UM program, in turn, corre- lated well with the field measurements, espe- cially for the primary flexural stresses. Warping stresses induced in the girders, and the stresses in the crossframes, were more erratic, but showed reasonable correlation. It is concluded that Mn/DOT’s analysis software captures the behavior well for these types of curved girder bridge systems and that the stresses in these bridges may be relatively low if their design is controlled largely by stiffness. AUTHOR: Richardson, J. A., and Douglas, B. M. (1993) TITLE: Results from field testing a curved box-girder bridge using simulated earthquake loads INFO: Earthquake Engineering and Structural Dynamics, Vol. 22, No. 10. ABSTRACT: This paper presents the results of a unique field test on a curved highway overpass. In the test, large horizontal loads were applied to the superstructure of the bridge and quickly released, causing the bridge to vibrate. The resulting large-amplitude vibrations were intended to be similar to the vibrations caused by earthquakes (horizontal accelerations of up to 25 percent of gravity were measured on the bridge deck). Well-defined lateral, longitudi- nal, vertical and torsional vibration modes were identified from the test data. The vibra- tion modes were used to verify an analytical model of the bridge’s dynamic response. For this paper, the model was verified using only the fundamental vibration mode, which was primarily a horizontal vibration mode. Using a system identification procedure, the dynamic response model was adjusted until its fre- quency and mode shape matched the meas- ured frequency and mode shape. Parameters in the verified model were compared with the same parameters calculated from information in the structural drawings. Because the funda- mental mode represents a horizontal mode, the bridge parameters identified in this paper were parameters that strongly influence the horizontal response of the bridge. AUTHOR: Utah State University (2000) TITLE: Bridge Test Report

INFO: Utah DOT Report UT-03.02 ABSTRACT: A series of static live load tests were conducted on NB I-15 to WB I-215 Connector Ramp, which was to be replaced due to its low struc- tural evaluation rating (structurally deficient). The bridge is a three-span continuous struc- ture (41 feet 6 inches, 69 feet 3 inches, and 41 feet 6 inches) with five I-girders spaced at 8 feet 10 inches on a radius of 480 feet. The girders are noncomposite and prismatic. There are no cross frames. Only small channel sec- tion diaphragms are connected to connection plates/transverse stiffeners. Utah State Univer- sity is the principal investigator with the instrumentation support provided by Bridge Diagnostic Inc. under the general oversight by FHWA/Utah DOT. AUTHOR: Arman EPST. (1994) TITLE: A finite element program for automatic static and dynamic analysis and design of horizon- tally curved I-girder bridges: CIG4BR (Curved I-Girder 4 Degrees of Freedom Bridge) INFO: AKMAN Engineering Production Software Technologies, 9350 Washington Blvd., Lan- ham, Maryland 20706-3119, USA. ABSTRACT: This is a user documentation for CIG4BR (Curved I-Girder 4 Degrees of Freedom Bridge). The main program was developed based on two-dimensional modeling of the bridge superstructure, including warping and the effect of cross frames and diaphragms. There are four worked out examples included. AUTHOR: Kitada, T., Nakai, H., and Murayama, Y. (1993) TITLE: State-of-the art on research, design and con- struction of horizontally curved bridges in Japan INFO: Proceedings of the SSRC Annual Technical Session, Milwaukee, WI. ABSTRACT: This paper emphasizes the necessity of research study on the ultimate strength con- cerning the horizontally curved plate and box- girders through a survey of references published in Japan since 1977. Then the out- line of a latest draft design method for curved plate girders is presented. Next, the applicable ranges of parameters on buckling stability of A-17 the curved box-girders are shown on the basis of a questionnaire for about 260 curved box- girder bridges. Finally, a special curved contin- uous spiral girder bridge under construction is also introduced together with a few numerical results based on the elasto-plastic and finite displacement analysis for this bridge under seismic load. AUTHOR: Zureick, A., and Naqib, R. (1999) TITLE: Horizontally curved steel I-girders state-of- the-art analysis methods INFO: Journal of Bridge Engineering, ASCE, Vol. 4, No. 1, February. ABSTRACT: Current AASHTO specifications pertaining to the analysis and design of horizontally curved bridges are based upon research work con- ducted prior to 1978. Since then, a significant amount of work has been conducted to further advance analysis methods and to better under- stand the behavior of these complex structural systems. Unfortunately, the results of these var- ious research efforts are scattered and, in some cases, unevaluated. This paper complements and updates survey articles published in 1968 and 1978 and presents highlights of the analyt- ical work conducted on horizontally curved steel I-girder bridges. AUTHOR: Bouabdallah, M. S., and Batoz, J. L. (1996) TITLE: Formulation and evaluation of a finite element model for the linear analysis of stiffened com- posite cylindrical panels INFO: Finite Elements in Analysis and Design, Vol. 21, No. 4, April. ABSTRACT: A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-order shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilib- rium equations are satisfied. The elements are

free of membrane and shear locking with cor- rect satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed show- ing the validity of the models. AUTHOR: Bozhevolnaya, E., and Kildegaard, A. (1997) TITLE: Experimental study of a uniformly loaded curved sandwich beam INFO: Composite Structures,Vol. 40, No. 2, December. ABSTRACT: A sandwich curved beam subjected to a uniform loading is experimentally investigated. Load- deflection and thrust-deflection dependencies are shown to be nonlinear, while load- deformation dependencies for sandwich faces are found to be linear. A technical solution for implementation of the simple support is real- ized. An actual stiffness of the beam tie is meas- ured experimentally and estimated theoretically. AUTHOR: Cleghorn, W. L., Tabarrok, B., and Lee, T. W. (1993) TITLE: Vibration of rings with unsymmetrical cross- sections: A finite element approach INFO: Journal of Sound and Vibration, Vol. 168, No. 1, November. ABSTRACT: A finite element model is developed for free, coupled in-plane and out-of-plane, vibration of a curved beam with an arbitrary unsym- metrical cross-section. Solutions of the governing differential equations of static equi- librium are used as shape functions for deriv- ing the element stiffness and mass matrices. The performance of the element developed is assessed by comparing results obtained with those found experimentally, and comparing those obtained using a commercially available finite element package. AUTHOR: Dhondt, G., and Kohl, M. (1999) TITLE: Effect of the geometry and the load level on the dynamic failure of rotating disks INFO: International Journal of Solids and Structures, Vol. 36, No. 6, February. ABSTRACT: The simplified and the higher-order curved beam theory published previously [Interna- tional Journal of Solids and Structures 30(1), A-18 137–149 (1993) and International Journal of Solids and Structures 31(14), 1949–1965 (1994)] are applied to rotating disks with rec- tangular cross-section subject to an instanta- neous radial failure. The dependence of the angular size of the debris on the inner to outer radius ratio and on the load level is examined and compared with semi-empirical and exper- imental results in the literature. Finite element calculations confirm the results obtained with the higher-order beam theory for moderate to high inner to outer radius ratios. It is shown that agreement with the experimental results can be further improved by a proper choice of the boundary conditions. AUTHOR: Grubb, M. A., Yadlosky, J. M., and Herrmann, A. W. (1993) TITLE: Behavior of horizontally curved steel highway bridges INFO: Structural Engineering in Natural Hazards Mitigation, Proceedings of the Symposium on Structural Engineering in Natural Hazards Mitigation. ASCE, New York, NY. ABSTRACT: Horizontally curved highway bridges represent up to 25 percent of the market for new steel bridge construction each year. However, unan- swered questions remain concerning the funda- mental behavior of horizontally curved steel girders that may not be adequately addressed in current design specifications. While there has been some isolated research, a coordinated large-scale effort to study curved girder behav- ior has not been undertaken in over 20 years. The FHWA and the NCHRP are launching sig- nificant research programs on curved steel bridges in late 1992 and early 1993, respectively. However, additional research is recommended to supplement these programs to ensure cover- age of areas these programs cannot address. One particular area of interest that may not be addressed is field instrumentation and testing of actual in-service curved steel bridges. Field measurements can provide valuable data to complement and verify previous research. The entire curved-girder research effort must be coordinated from the start to avoid needless repetition of effort. All projects must be over- seen by technically competent individuals who are willing to devote the time needed to success- fully overcome all barriers to success.

AUTHOR: Meyerpiening, H. R., and Anderegg, R. (1995) TITLE: Buckling and post-buckling investigations of imperfect curved stringer-stiffened composite shell. A. Experimental investigation and effec- tive width evaluation INFO: Thin-Walled Structures, Vol. 23, No. 1–4. ABSTRACT: A box-like stringer-stiffened thin-walled CFRP structure was subjected to load cases well beyond the limits of local buckling. The devel- opment of the deflection pattern was recorded via optical means and analyzed numerically. In addition, the structure was modeled and ana- lyzed using the MARC FE program in the non- linear deflection range. The geometrical imperfections of the test structure were recorded by mechanical scanning and optical methods and introduced into the mathemati- cal model. For the perfect (“ideal”) and the geometrically imperfect (“real”) structural model, the results of the FE analyses were com- pared and used to judge the effect of geomet- ric imperfections on the post-buckling behavior of the structure. The effective axial stiffness for the various post-buckling states was evaluated and related to analytical esti- mates of effective width values for orthotropic sheet-like panels. AUTHOR: Moas, E., Boitnott, R. L., and Hayden, G. O. (1994) TITLE: Analytical and experimental investigation of the response of the curved, composite frame/skin specimens INFO: Journal of the American Helicopter Society, Vol. 39, No. 3, July. ABSTRACT: Six-foot-diameter, semicircular graphite/epoxy specimens representative of generic aircraft frames were loaded quasi-statically to determine their load response and failure mechanisms for large deflections that occur in airplane crashes. These frame/skin specimens consisted of a cylin- drical skin section co-cured with a semicircular I-frame. The skin provided the necessary lateral stiffness to keep deformations in the plane of the frame in order to realistically represent defor- mations as they occur in actual fuselage struc- tures.Various frame laminate stacking sequences and geometries were evaluated by statically loading the specimen until multiple failures A-19 occurred. Two analytical methods were com- pared for modeling the frame/skin specimens: a two-dimensional shell finite element analysis and a one-dimensional, closed-form, curved beam solution derived using an energy method. Reduced flange effectiveness was included in the beam analysis to account for the curling phenomenon that occurs in thin flanges of curved beams. Good correlation was obtained between experimental results and the analytical predictions of the linear response of the frames prior to the initial failure. The spec- imens were found to be useful for evaluating composite frame designs. AUTHOR: Shanmugam, N. E., Thevendran, V., Richard Liew, J. Y., and Tan, L. O. (1995) TITLE: Experimental study on steel beams curved in plan INFO: Journal of Structural Engineering, ASCE, Vol. 121, No. 2, February. ABSTRACT: This paper is concerned with the ultimate load behavior of I-beams curved in plan. Results obtained from experiments on two sets of I-beams (one comprising rolled sections and the other built-up sections) are presented. The test results for deformations and ultimate strength are found to be in good agreement with the corresponding values predicted by using the elasto-plastic finite element analysis. The effects of residual stresses and radius of curvature to span-length ratio (R/L) on ulti- mate strength are considered. Each beam was subjected to a concentrated load applied at an intermediate point where the beam was later- ally restrained. Test results indicate that the load-carrying capacity decreases with the decrease in the R/L ratio. AUTHOR: Shim, V. P. W., and Quah, S. E. (1998) TITLE: Solution of impact-induced flexural waves in a circular ring by the method of characteristics INFO: Journal of Applied Mechanics-Transactions of the ASME, Vol. 65, No. 3, September. ABSTRACT: A study of elastic wave propagation in a curved beam (circular ring) is presented. The govern- ing equations of motion are formulated in two forms based on Timoshenko beam theory. Solutions are obtained using the method of

characteristics, whereby a numerical scheme employing higher-order interpolation is pro- posed for the finite difference equations. Results obtained are verified by experiments; it is found that use of the higher-order numeri- cal scheme improves correlation with experi- mental results. Comparison of the relative accuracy between the two mathematical for- mulations—one in terms of generalized forces and velocities and the other in terms of gener- alized displacements—shows the former to be mathematically simpler and to yield more accurate results. AUTHOR: Stiftinger, M. A., Skrnajakl, I. C., and Ram- merstorfer, F. G. (1995) TITLE: Buckling and postbuckling investigations of imperfect curved stringer-stiffened composite shell, part B. Computational investigations INFO: Thin-Walled Structures, Vol. 23, No. 1–4. ABSTRACT: Computational investigations of the buckling and post-buckling behavior of a stringer- stiffened composite wing torsion box employ- ing the finite element method are presented. Perfect and imperfect configurations—consid- ering geometrical imperfections as well as ini- tial stresses—are discussed. Two different load cases are investigated: pure axial loading and axial loading with a superimposed constant torsion moment. The buckling behavior is determined by tracing the load-deflection curves using Riks’ path-following method. Additional investigations, such as accompany- ing eigenvalue analyses and first-ply failure cal- culations, are performed for the first load case. Special attention is put on the modeling of the stiffened regions, where eccentrically placed layers have to be taken into account due to the bonding of the stiffeners to the inner surface of the box. The results show interesting phenom- ena, such as additional buckling points in the post-buckling region, which, however, can hardly be detected by simply considering the load-axial displacement path. AUTHOR: Choi, J. K., and Lim, J. K. (1993) TITLE: Simple curved shear beam elements INFO: Communications in Numerical Methods in Engineering, Vol. 9, No. 8, August. A-20 ABSTRACT: We have developed two curved beam elements with two nodes, CSCC and CSLC, based on Timoshenko’s beam theory and the curvilinear co-ordinate system. These curved beam ele- ments have been modified from the conven- tional strain element, which has been applied only to Euler beam analysis. Therefore, they do not have any spurious constraints and locking characteristics. They also show rapid and sta- ble convergence on the wide ranges of beam length to height ratio for linear and non-linear analyses. AUTHOR: Choi, J. K., and Lim, J. K. (1995) TITLE: General curved beam elements based on the assumed strain fields INFO: Computers and Structures, Vol. 55, No. 3, May. ABSTRACT: Simple two-noded and three-noded general curved beam elements have been formulated on the basis of assumed strain fields and Tim- oshenko’s beam theory. The two-noded ele- ment is formulated from constant strain fields and the three-noded one is from linear strain fields. These curved beam elements, designed in a local curvilinear coordinate system, are transformed into a global Cartesian coordinate system in order to analyze effectively the gen- eral curved beam structures located arbitrarily in space. Since strain functions are assumed independently, these elements are free from any locking phenomena. Through various numerical tests, it is shown that the suggested general curved beam elements give better con- vergent characteristics than the modified isoparametric curved beam elements that have been shown in existing studies. These elements are also easy to formulate due to their consis- tent form of strain fields. AUTHOR: Davidson J. S. (1996) TITLE: Nominal bending and shear strength of hori- zontally curved steel I-girder bridges INFO: Dissertation submitted to the Graduate Fac- ulty of Auburn University in partial fulfillment of the requirements for the degree of Doctor of Philosophy. ABSTRACT: The currently used AASHTO Guide Specifi- cations for Horizontally Curved Highway Bridges is primarily based on research per-

formed as part of the CURT project during the early 1970s. Since that time, numerous prob- lems with the Guide Specifications have been revealed. The Guide Specifications in its origi- nal form is disjointed and difficult to use. There is significant discontinuity in the com- pressive strength formulation between com- pact and noncompact sections, and the strength predicted by the formulations does not approach that predicted by the formula- tions for straight girders as the radius of the curved girder approaches infinity. For these reasons, among others, it has never been adopted as an integral part of the AASHTO Standard Specifications for Highway Bridges. The present research addresses many of the strength-related issues in the design of hori- zontally curved I-girder bridges, including 1) the overall lateral-torsional buckling and large displacement behavior including the design spacing of cross-frames and diaphragms; 2) the local buckling behavior of the curved com- pression flanges; 3) the buckling and finite dis- placement behavior of the curved web panels under bending, shear, and combined bending and shear; and 4) construction issues involving the large displacement behavior of single long slender I-girder during lifting and transport- ing. Theoretical and analytical results are pre- sented on the behavior of such systems along with design recommendations. Also, in all applicable areas, an in-depth review and com- parison of existing Japanese research results is presented. AUTHOR: Dorfi, H. R., and Busby, H. R. (1994) TITLE: Effective curved composite beam finite ele- ment based on the hybrid-mixed formulation INFO: Computers and Structures, Vol. 53, No. 1, October. ABSTRACT: A laminated curved-beam finite element with six displacement degrees of freedom and three stress parameters is derived and evaluated. Both thermal and hygrothermal effects are included. The element is based on the Hellinger-Reissner principle and the hybrid- mixed formulation. The Timoshenko beam theory and classical lamination theory are employed in the finite element description. Within an element linear displacement inter- polation is used; the generalized stresses are A-21 interpolated by either stress functions based on the equilibrium equations (P1) or constant stress approximation (P2). The beam element stiffness is obtained explicitly and numerical results show very good displacement predic- tion compared with analytical solutions. Gen- eralized stresses are predicted accurately at the mid-point of the finite element only for con- stant stress interpolation. The P1-type element yields more accurate displacement and stress prediction. AUTHOR: Fu, C. C., and Hsu, Y. T. (1995) TITLE: Development of an improved curvilinear thin- walled Vlasov element INFO: Computers and Structures, Vol. 54, No. 1, January. ABSTRACT: The purpose of this study is to develop a more exact horizontally curved beam finite element in a cylindrical coordinate system. The varia- tion method is used to formulate the stiffness matrix, and the results of the solution are com- pared with another study using a closed form solution. The stiffness matrix for a curved beam element can be used in analyzing the behavior of horizontally curved bridges for either open or closed sections. AUTHOR: Hall, D. H. (1996) TITLE: Curved girders are special INFO: Engineering Structures,Vol. 18, No. 10, October. ABSTRACT: Horizontally curved I-girders have been used in highway bridges for over 30 years. Their structural behavior is known to be quite dif- ferent from straight girders because of the always present nonuniform torsion. Early studies of these members were based on a strength of materials approach. Modifications of these results were made to account for dis- tortion and amplification effects. More recent investigations in Japan have involved inelastic finite element studies. From these studies, var- ious modifications of straight I-girder bending strength equations have been presented. Bend- ing tests of curved beams are compared with these equations. It is suggested that current research may lead to the thought that curved girders are the general case, and straight gird- ers may be considered a special case.

AUTHOR: Ibrahimbegovic, A. (1995) TITLE: On finite element implementation of geomet- rically nonlinear Reissner’s beam theory: Three-dimensional curved beam elements INFO: Computer Methods in Applied Mechanics and Engineering, Vol. 122, No. 1–2, April. ABSTRACT: Finite element implementation of the three- dimensional finite-strain beam theory of Reissner is considered in this work. In contrast with some earlier works on the subject, dis- cussed here are the beam elements whose ref- erence axes are arbitrary space-curved lines. We have shown that an improved representa- tion of curved reference geometry significantly increases the accuracy of the results. However, it also makes the choice of non-locking finite element interpolations somewhat more deli- cate. A hierarchical displacement interpolation proposed here is proved to be capable of elim- inating both shear and membrane locking phenomena. A very satisfying non-locking performance is demonstrated for a set of prob- lems in nonlinear elasto-statics. AUTHOR: Kang, Y. J., and Yoo, C. H. (1994) TITLE: Thin-walled curved beams. II: analytical solu- tions for buckling of arches INFO: Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. ABSTRACT: In recent years, after Yoo queried the validity of Vlasov’s equations regarding the lateral buck- ling of a circular arch, papers have been pub- lished with a certain degree of disagreement among various researchers. Based on a com- prehensive and consistent formulation of curved beam theory presented in the previous paper, closed-form solutions thus obtained are used for comparison of the present theoretical study to others. In the present paper, major potential sources of discrepancies are traced, and the rigor and validity of the present for- mulation is thereby demonstrated. Also, the study on the buckling modes of arches sub- jected to the condition of uniform bending gives an insight for better understanding of lat- eral buckling characteristics of arches. AUTHOR: Kim, J. G., and Kim, Y. Y. (1998) TITLE: New higher-order hybrid-mixed curved beam element A-22 INFO: International Journal for Numerical Methods in Engineering, Vol. 43, No. 5, November. ABSTRACT: The purpose of this work is to show the suc- cessful use of nodeless degrees of freedom in developing a highly accurate, locking free hybrid-mixed CO curved beam element. In the performance evaluation process of the present field-consistent higher-order ele- ment, the effect of field consistency and the role of higher-order interpolation on both displacement-type and hybrid-mixed-type elements are carefully examined. Several benchmark tests confirm the superior behav- ior of the present element. AUTHOR: Koziey, B. L., and Mirza, F. A. (1994) TITLE: Consistent curved beam element INFO: Computers and Structures,Vol. 51, No. 6, June. ABSTRACT: Curved beam finite elements with shear defor- mation have required the use of reduced inte- gration to provide improved results for thin beams and arches due to the presence of a spu- rious shear strain mode. It has been found that the spurious shear strain mode results from an inconsistency in the displacement fields used in the formulation of these elements. A new curved beam element has been formulated. By providing a cubic polynomial for approxima- tion of displacements, and a quadratic polyno- mial for approximation of rotations a consistent formulation is ensured thereby eliminating the spurious mode. A rotational degree of freedom which varies quadratically through the thickness of the element is included. This allows for a parabolic variation of the shear strain and hence eliminates the need for use of the shear correction factor kappa as required by the Timoshenko beam theory. This rotational degree of freedom also provides a cubic variation of displacements through the depth of the element. Thus, the normal to the centroidal axis is neither straight nor normal after shearing and bending allow- ing for warping of the cross section. Material nonlinearities are also incorporated, along with the modified Newton-Raphson method for nonlinear analysis. Comparisons are made with the available elasticity solutions and those predicted by the quadratic isoparametric beam element. The results indicate that the consis-

tent beam element provides excellent predic- tions of the displacements, stresses and plastic zones for both thin and thick beams and arches. AUTHOR: Lee, S. S., Koo, J. S., and Choi, J. M. (1996) TITLE: Development of a new curved beam element with shear effect INFO: Engineering Computations (Swansea, Wales), Vol. 13, No. 2–4, pp. 9–25. ABSTRACT: Two-noded curved beam elements, CMLC and 1MLC, are developed on the basis of Timo- shenko’s beam theory and curvilinear coordi- nates. These elements are developed by the separation of the radial displacement into the bending and the shear deflection and the pro- jection of the shear deflection into bending deflection. In the CMLC element, field- consistent membrane strain interpolation is adapted for removing the membrane locking. The CMLC element shows the rapid and sta- ble convergence on the wide range of radius, thickness and length of the curved beam. The field-consistent membrane strain and the sep- aration of radial displacement produce the most efficient linear element possible. AUTHOR: Lim, C. W., Wang, C. M., and Kitipornchai, S. (1997) TITLE: Timoshenko curved beam bending solutions in terms of Euler-Bernoulli solutions INFO: Archive of Applied Mechanics, Vol. 67, No. 3, February, pp. 179–190. ABSTRACT: This paper presents the exact relationships between the deflections and stress resultants of Timoshenko curved beams and that of the cor- responding Euler-Bernoulli curved beams. The curved beams considered are of rectangular cross sections and constant radius of curva- ture. They may have any combinations of clas- sical boundary conditions, and are subjected to any loading distribution that acts normal to the curved beam centerline. These relation- ships allow engineering designers to directly obtain the bending solutions of Timoshenko curved beams from the familiar Euler- Bernoulli solutions without having to perform the more complicated shear deformation analysis. A-23 AUTHOR: Lin, S. M. (1998) TITLE: Exact solutions for extensible circular curved Timoshenko beams with nonhomogeneous elastic boundary conditions INFO: Acta Mechanica, Vol. 130, No. 1–2, pp. 67–79. ABSTRACT: A generalized Green function of nth-order ordinary differential equation with forcing function composed of the delta function and its derivatives is obtained. The generalized Green function can be easily and effectively applied to both the boundary value problems and the initial value problems. The generalized Green function is expressed in terms of n lin- early independent normalized homogeneous solutions. It is the generalization of those given by Pan and Hohenstein, and Kanwal. Accord- ingly, the exact solution for static analysis of an extensible circular curved Timoshenko beam with general nonhomogeneous elastic bound- ary conditions, subjected to any transverse, tangential and moment loads is obtained. The three coupled governing differential equations are uncoupled into one complete sixth-order ordinary differential characteristic equation in the tangential displacement. The explicit rela- tions between the angle of rotation due to bending, the transverse displacement and the tangential displacement are obtained. AUTHOR: Litewka, P., and Rakowski, J. (1997) TITLE: Efficient curved beam finite element INFO: International Journal for Numerical Methods in Engineering, Vol. 40, No. 14, July, pp. 2629–2652. ABSTRACT: The plane two-node curved beam finite ele- ment with six degrees of freedom is consid- ered. Knowing the set of 18 exact shape functions their approximation is derived using the expansion of the trigonometric functions in the power series. Unlike the ones commonly used in the FEM analysis the functions sug- gested by the authors have the coefficients dependent on the geometrical and physical properties of the element. From the strain energy formula the stiffness matrix of the ele- ment is determined. It is very simple and can be split into components responsible for bend- ing, shear and axial forces influences on the displacements. The proposed element is totally

free of the shear and membrane locking effects. It can be referred to the shear-flexible (param- eter d) and compressible (parameter e) sys- tems. Neglecting d or e yields the finite elements in all necessary combinations, i.e., curved Euler-Bernoulli beam or curved Timo- shenko beam with or without the membrane effect. Applying the elaborated element in the calculations a very good convergence to the analytical results can be obtained even with a very coarse mesh without the commonly adopted corrections as reduced or selective integration or introduction of the stabilization matrices, additional constraints, etc., for the small depth-length ratio. AUTHOR: Mentrasti, L. (1996) TITLE: Shear-torsion of large curvature beams—Part II. Applications to thin bodies INFO: International Journal of Mechanical Sciences, Vol. 38, No. 7, July, pp. 723–733. ABSTRACT: The shear-torsion state of stress in a curved beam, whose cross section is a thin rectangle with sides not parallel to the plane of the beam, is determined in closed form. The maximum value of the shear stress is attained at the con- cave boundary of the beam. The shear-torsion moment of inertia Jst in multi-connected thin- walled cross sections is evaluated. Several examples of cross sections are discussed. AUTHOR: Orth, F. J., and Surana, K. S. (1994) TITLE: P-version two-dimensional beam element for geometrically nonlinear analysis INFO: Computers and Structures, Vol. 50, No. 3, Feb- ruary, pp. 383–392. ABSTRACT: This paper presents a p-version geometrically nonlinear formulation (GNL) based on the total Lagrangian approach for a three-node two-dimensional curved beam element. The hierarchical element approximation functions and the corresponding nodal variables are derived directly from the Lagrange family of interpolation functions. The resulting element displacement approximation is hierarchical and can be of arbitrary and different polyno- mial orders in the longitudinal and the trans- verse directions of the beam element and ensures CO continuity. The element geometry is described by the coordinates of the nodes A-24 located on the axis of the beam (middle sur- face) and the nodal vectors describing the top and bottom surfaces of the element. The ele- ment properties are established using the prin- ciple of virtual work and the hierarchical element displacement approximation. In for- mulating the properties of the element com- plete two-dimensional stresses and strains are considered; hence, the element is equally effec- tive for very slender as well as extremely deep beams. Incremental equations of equilibrium are derived and solved using the standard Newton-Raphson method. The total load is divided into increments, and for each incre- ment of load, equilibrium iterations are per- formed until each component of the residuals is within a present tolerance. Numerical exam- ples are presented to show the accuracy, effi- ciency and advantages of the present formulation. The results obtained from the present formulation are compared with those reported in the literature. The formulation presented here removes virtually all of the drawbacks present in existing GNL beam finite element formulations and has many additional benefits. First, the currently available GNL beam formulations are based on fixed order of approximation for the displacements and thus are not hierarchical and have no provision for changing the order of approximation for the displacements u and v. Secondly, the element displacement approximations in the existing formulations are either based on a linearized displacement field for which a true Lagrangian formulation is not possible and the incremen- tal load step size is severely limited or are based on nonlinear nodal rotation function approach in which case even though the description of the displacement field for the element is exact additional complications arise due to the non- cummutative nature of the nonlinear nodal rotation functions. The p-version displacement approximation used here does not involve the traditional nodal rotations that have been used in the existing beam formulations, thus the dif- ficulties associated with their use are not pres- ent in this formulation. AUTHOR: Paavola, J., and Salonen, E. M. (1999) TITLE: Strain and stress analysis of a curved tapered beam model

INFO: Computers and Structures, Vol. 72, No. 4–5, pp. 565–577. ABSTRACT: The strain and stress resultant expressions for a tapered curved beam model are derived. The model is a one-dimensional version of the finite element–based shell theory model of Irons and it may be considered as a generaliza- tion of the well-known Timoshenko beam model. To gain insight into the properties of the model, the expressions needed are devel- oped analytically without use of finite ele- ments. An effort is made to clarify a statement given previously, which is needed in the appli- cation of the theory and apt to lead to some confusion in its interpretation. Small displace- ment theory is applied. The stress resultant expressions are derived using the principle of virtual work and in addition directly from the stresses acting on the beam cross-section. The stresses obtained by the model in the isotropic elastic case are compared in two simple exam- ple problems with those by the Timoshenko model and with the exact values. AUTHOR: Pi, Y. L., and Trahair, N. S. (1996) TITLE: Nonlinear elastic behaviour of I-beams curved in plan INFO: Research Report—University of Sydney, School of Civil and Mining Engineering, No. 734, December, pp. 1–45. ABSTRACT: The vertical deflections perpendicular to the plane of a horizontal beam curved in plan are coupled with its twist rotations, and its axial deflections are coupled with its horizontal radial deflections. Because of the first of these couplings, a horizontally curved beam sub- jected to vertical loading has both primary bending and torsion actions. In the nonlinear range, second-order couplings between the vertical and horizontal deflections and the twist rotations are developed, and the nonlin- ear behavior of the curved beam becomes more complicated. This paper studies the lin- ear, neutral, and nonlinear equilibrium of elas- tic horizontally curved I-beams under vertical loading, and develops a curved finite element model for their analysis. It is found that when the initial curvature of a curved beam is small, the primary coupling is also small and bending is the major action. In this case, the nonlinear A-25 behavior is similar to the elastic flexural- torsional buckling of a straight beam. How- ever, if the initial curvature of the curved beam is not small, the primary coupling becomes sig- nificant and both torsion and bending are major actions. In this case nonlinear behavior develops very early and is quite different from rite flexural-torsional buckling behavior of a straight beam. AUTHOR: Raveendranath, P., Singh, G., and Pradhan, B. (1999) TITLE: Two-noded locking-free shear flexible curved beam element INFO: International Journal for Numerical Methods in Engineering, Vol. 44, No. 2, January, pp. 265–280. ABSTRACT: A new two-noded shear flexible curved beam element which is impervious to membrane and shear locking is proposed herein. The ele- ment with three degrees of freedom at each node is based on curvilinear deep shell theory. Starting with a cubic polynomial representa- tion for radial displacement (w), the displace- ment field for tangential displacement (u) and section rotation (theta) are determined by employing force-moment and moment-shear equilibrium equations. This results in a poly- nomial displacement field whose coefficients are coupled by generalized degrees of freedom and material and geometric properties of the element. The procedure facilitates quadratic polynomial representation for both u and theta for curved element configurations, which reduces to linear and quadratic polynomials for u and theta, respectively, for straight ele- ment configuration. These coupled poly- nomial coefficients do not give rise to any spurious constraints even in the extreme thin regimes, in which case, the present element exhibits excellent convergence to the classical thin beam solutions. This simple C degree ele- ment is validated for beams having straight/ curved geometries over a wide range of slen- derness ratios. The results indicate that per- formance of the element is much superior to other elements of the same class. AUTHOR: Pak, R. Y. S., and Stauffer, E. J. (1994) TITLE: Nonlinear finite deformation analysis of beams and columns

INFO: Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. ABSTRACT: A method for solving the finite-displacement problem of a curved elastic beam with axial, shear, and flexural deformation subject to dis- tributed and point loads is presented. Within the context of the kinematic assumptions of the Timoshenko theory, a Lagrangian formulation of the problem is enveloped. In terms of three cross-sectional stress resultants, three Euler equations of equilibrium for the beam are derived with the aid of a variational principle for finite deformation. Upon linearization to small strains and the adoption of a linear elastic constitutive relation between the stress and strain tensors, it is shown that the problem is reducible to a single second-order nonlinear ordinary differential equation. Subject to appro- priate boundary conditions, the resulting two- point boundary-value problem is solved by a finite-element method. By virtue of a continua- tion algorithm, accurate solutions of the system of nonlinear equations can be obtained for a variety of bifurcation and buckling problems. Comprehensive results are presented for two cantilever beams as illustrations. AUTHOR: Ryu, H. S., and Sin, H. C. (1996) TITLE: Curved beam elements based on strain fields INFO: Communications in Numerical Methods in Engineering, Vol. 12, No. 11, November. ABSTRACT: Two curved beam elements with two nodes and three nodes are designed based on strain fields. At the element level, curvature and membrane strain fields are approximated independently and shear strain fields are incor- porated into the formulation by the equilib- rium equations. The displacement fields are obtained by integrating the assumed strain fields. Two examples are given to verify the for- mulations and demonstrate the numerical per- formance of the two curved beam elements. Analysis results obtained reveal that the ele- ments describe the curved beam behavior cor- rectly and show exceptional accuracy throughout a wide slenderness range. AUTHOR: Sengupta, D., and Dasgupta, S. (1997) TITLE: Static and dynamic applications of a five noded horizontally curved beam element with shear deformation A-26 INFO: International Journal for Numerical Methods in Engineering, Vol. 40, No. 10, May. ABSTRACT: A five-noded thirteen DOF horizontally curved beam element with or without an elas- tic base is presented. One set of fourth-degree Lagrangian polynomials in natural co-ordinates is used for interpolation of beam geometry and vertical displacement while the angles of trans- verse rotation and twist are interpolated by another set of third-degree polynomials. For elastic subgrade, the reactive forces offered at any point are assumed to be proportional to the corresponding displacements at that point. The effect of shear deformation is accounted for in the stiffness matrix. In mass matrix eval- uation, for dynamic problems, translational as well as rotary inertias have been considered and studied separately. For numerical integra- tion of the stiffness matrix, a four-point Gaussian scheme has been found to be ade- quate. Numerical results for a number of sam- ple problems and their comparison with analytical solutions have been presented for circular as well as for non-circular curved beams. Displacements, bending moment and torque for static loading with or without elas- tic foundation, as well as natural frequencies and mode shapes, are computed for different cases. Examples include the problem of a can- tilever beam of spiral geometry with different parametric values of the spiral and the agree- ment with the analytical results establishes the efficacy of the element. The performance of the element has been found to be excellent in both static and dynamic conditions. Sufficient details are presented so that the formulation may be readily used. It is hoped that the large number of numerical illustrations will eluci- date the validity and the range of applicability of the element and will also serve as a bench- mark for future researchers. AUTHOR: Simpson, M. D., and Birkemoe, P. C. (1997) TITLE: Nonlinear finite element modeling of curved girder experiments INFO: Structures—Design, Concrete and Reinforced Concrete Structures, Bridges Proceedings, Annual Conference—Canadian Society for Civil Engineering, Vol. 7, Canadian Society for Civil Engineering, Montreal, Quebec, Canada.

ABSTRACT: Research involving the ultimate strength char- acteristics of curved steel girder components began in North America in the early 1970s under the direction of the Consortium of Uni- versity Research Teams (CURT). Both analyti- cal and experimental programs attempted to isolate the parameters that affected the flexural and shear strength of curved steel girders. Much of the understanding gained from the CURT project is still reflected in the current Guide Specification for Horizontally Curved Highway Bridges. Additional experimental and analytical studies were performed in Japan during the 1980s to further advance the state- of-the-art understanding of curved girder behavior. Although some researchers have used analytical techniques to investigate curved girder behavior, few have employed the flexibility and generality of the finite element method. Thus, the purpose of this paper is to demonstrate the use of the finite element method as applied to the analysis of curved steel girders. In particular, a select number of ultimate strength experiments are modeled using inelastic, large deformation finite ele- ment solutions. Residual stresses are also included, and their effect on behavior is discussed. AUTHOR: Thevendran, V., Chen, S., Shanmugam, N. E., and Liew, J. Y. R. (1999) TITLE: Nonlinear analysis of steel-concrete composite beams curved in plan INFO: Finite Elements in Analysis and Design, Vol. 32, No. 3, pp. 125–139. ABSTRACT: This paper deals with the behavior of struc- tural steel-concrete composite beams curved in plan. The finite element package ABAQUS has been used to study the nonlinear behavior and ultimate load-carrying capacity of such beams. A three-dimensional finite element model has been adopted. Shell elements have been used to simulate the behavior of concrete slab and steel girder, and rigid beam elements to simulate the behavior of shear studs. The proposed finite element model has been vali- dated by comparing the computed values with available experimental results. An acceptable correlation has been observed between the computed and experimental results obtained for beams of realistic proportion. A-27 AUTHOR: Yang, S. Y., and Sin, H. C. (1995) TITLE: Curvature-based beam elements for the analy- sis of Timoshenko and shear-deformable curved beams INFO: Journal of Sound and Vibration, Vol. 187, No. 4, November. ABSTRACT: Curved beam elements with six degrees of free- dom and two-, three-, four- and five-node Timoshenko straight beam elements with four, five, six and seven degrees of freedom, respec- tively, are proposed to eliminate stiffness lock- ing when applied to the dynamic analysis of the beams. The elements are based on curva- ture assumptions so that they can represent the bending energy accurately, and the shear strain energy is incorporated into the formulation by way of the equilibrium equation. Eigen- value problems of Timoshenko and shear- deformable curved beams are analyzed by using the elements. The results of the eigen- analysis show that the curvature-based beam elements are free of locking and are efficient. AUTHOR: Hu, N., Hu, B., Yan, B., Fukunaga, H., and Sekine, H. (1999) TITLE: Two kinds of CO-type elements for buckling analysis of thin-walled curved beams INFO: Computer Methods in Applied Mechanics and Engineering, Vol. 171, No. 1. ABSTRACT: This paper deals with the spatial buckling analysis of curved beams. First, a second-order expansion for the finite rigid-rotations in nonlinear strain expressions is derived and employed to produce the geometric stiffness matrix. This second-order accurate geometric stiffness matrix can ensure that all significant instability modes can be predicted. Further- more, Timoshenko’s and Vlasov’s beam theo- ries are combined to develop two kinds of the CO-type finite element formulations for arbi- trary cross-section thin-walled curved beams, which include the isoparametric curved beam element and the strain curved beam element. These two kinds of elements include both shear and warping deformations caused by bending moments and bimoments. In numer- ical examples, the effect of the second-order terms in the nonlinear strains on the buckling load is investigated. Furthermore, efficiencies

of the proposed two kinds of elements are studied in the buckling analysis of curved beam structures. AUTHOR: Kang, Y. J., and Yoo, C. H. (1994) TITLE: Thin-walled curved beams. I: Formulation of nonlinear equations INFO: Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, October. ABSTRACT: An extensive investigation on the buckling and large displacement behavior of thin- walled circular beams has been conducted theoretically. Equilibrium equations govern- ing the linear, the bifurcation buckling, and the large displacement behavior have been derived using the principle of minimum total potential energy. An explicit and clear approximation of the curvature effect is made in the derivation process. The paper con- cludes with a series of fundamental nonlinear equations that describe the elastic behavior of thin-walled curved beams. A companion paper examines closed-form solutions for arch-buckling problems based on the formu- lations presented in this paper and demon- strates the rigor and the validity of the present formulation. AUTHOR: Kang, Y. J., Yoo, C. H., Yu, C., and Lee, H. E. (1993) TITLE: Lateral buckling behavior of thin-walled arches INFO: Proceedings of the Fourth East Asia-Pacific Conference on Structural Engineering and Construction, Seoul, Korea, September 20–22. ABSTRACT: Abstract could not be located. AUTHOR: Yoo, C. H., Kang, Y. J., and Davidson, J. S. (1996) TITLE: Buckling analysis of curved beams by finite- element discretization INFO: Journal of Engineering Mechanics, ASCE, Vol. 122, No. 8, August. ABSTRACT: Recently, an extensive theoretical investiga- tion on the buckling and large-displacement behavior of thin-walled circular beams was reported in a two-paper series. Equilibrium equations governing the linear, bifurcation A-28 buckling, and large-displacement behaviors were derived using the principle of minimum total potential energy. This paper first pre- sents the transformation process for finite- element stiffness relationships for a spatial curved beam element with a total of 14 de- grees of freedom. It then presents numerical data demonstrating the applicability of the method for the lateral buckling of arches and the lateral-torsional buckling of horizontally curved beams. A numerical comparison between the present formulations and those presented by others is made, along with a comparison to results obtained using three- dimensional finite-element models. Based on results from the lateral bifurcation buckling of horizontally curved beams, a regression equation is formulated representing the reduction in critical moment due to the sim- ple addition of curvature. A comparison of results using this regression equation with results from ultimate strength experimental testing of horizontally curved girders by others resulted in an unexpected excellent correlation. AUTHOR: Gendy, A. S., and Saleeb, A. F. (1994) TITLE: Vibration analysis of coupled extensional/ flexural/torsional modes of curved beams with arbitrary thin-walled sections INFO: Journal of Sound and Vibration, Vol. 174, No. 2, July. ABSTRACT: A three-dimensional, two-field, variational formulation is employed to derive the differ- ential equations governing the dynamics of stretching, shearing, bending and twisting, as well as warping modes of deformations in a spatially curved beam with arbitrary cross- section. Correspondingly, the finite element discretization was developed for free vibra- tion analysis based on a Timoshenko-Vlasov thin-walled theory, including the effects of flexural-torsional coupling, shear deforma- tions due to flexure as well as torsional warp- ing, and rotary inertia. Attention was given to the significant curvature effects on the results in cases involving unsymmetrical cross-sections of the thin-walled type. Several numerical examples are given to demonstrate the high accuracy and effectiveness of the element developed.

AUTHOR: Howson, W. P., and Jemah, A. K. (1999) TITLE: Exact out-of-plane natural frequencies of curved Timoshenko beams INFO: Journal of Engineering Mechanics, ASCE, Vol. 125, No. 1, January. ABSTRACT: A powerful and efficient method is presented for finding exact out-of-plane natural frequen- cies of plane structures composed of curved Timoshenko beams. Initially, exact dynamic stiffness is derived from the governing differ- ential equations of motion in a form that can be used directly in the stiffness method of analysis. This enables any appropriate struc- ture to be modeled according to standard techniques, which, in this case, yield a tran- scendental eigenvalue problem. Then, it is shown how any desired natural frequency may be obtained with certainty by employing a modification to a well-established algorithm, which ensures that no natural frequencies can be missed and avoids the usual approximations associated with traditional finite elements. Finally, comparisons are made with published results, and an example shows how the natural frequencies of a continuous curved beam are altered when the effects of shear deflection and rotary inertia are considered. AUTHOR: Howson, W. P., Jemah, A. K., and Zhou, J. Q. (1995) TITLE: Exact natural frequencies for out-of-plane motion of plane structures composed of curved beam members INFO: Computers and Structures,Vol. 55, No. 6, June. ABSTRACT: Exact finite elements form the basis of a new and convenient procedure for converging with certainty upon any required natural frequency of out-of-plane motion of any plane structure composed of slender elastic curved members. Solution of the inherent transcendental eigen- value problem is achieved through a variation on the powerful Wittrick-Williams algorithm. Two illustrative examples are included. AUTHOR: Hsiao, K. M., and Yang, R. T. (1995) TITLE: Co-rotational formulation for nonlinear dynamic analysis of curved Euler beam INFO: Computers and Structures, Vol. 54, No. 6, March. A-29 ABSTRACT: A co-rotational finite element formulation for the dynamic analysis of a planar curved Euler beam is presented. The Euler-Bernoulli hypothesis and the initial curvature are prop- erly considered for the kinematics of a curved beam. Both the deformational nodal forces and the inertial nodal forces of the beam element are systematically derived by consistent lin- earization of the fully geometrically nonlinear beam theory in element coordinates, which are constructed at the current configuration of the corresponding beam element. An incremental- iterative method based on the Newmark direct integration method and the Newton-Raphson method is employed here for the solution of the nonlinear dynamic equilibrium equations. Numerical examples are presented to demon- strate the effectiveness of the proposed element and to investigate the effect of the initial curva- ture on the dynamic response of the curved beam structures. AUTHOR: Huang, D., Wang, T. L., and Shahawy, M. (1995) TITLE: Dynamic behavior of horizontally curved I-girder bridges INFO: Computers and Structures, Vol. 57, No. 4, November. ABSTRACT: The purpose of this paper is to investigate the dynamic behavior of horizontally curved I-girder bridges due to one or two trucks (side by side) moving across rough bridge decks. The bridge is modeled as a planar grillage beam system composed of horizontally curved beam elements and straight beam elements. Warping torsion is taken into consideration in the analysis. The analytical vehicle is simulated as a nonlinear vehicle model with 11 inde- pendent degrees of freedom according to the HS2O-44 truck design loading contained in the American Association of State Highway and Transportation Officials (AASHTO) spec- ifications. The study used four different classes of road surface roughness generated from power spectral density for very good, good, average, and poor roads. The analytical results are very significant and show that the dynamic behavior of curved I-girder bridges is quite dif- ferent from that of straight girder bridges. The impact factors of bending and shear for inside

girders of curved I-girder bridges are signifi- cantly smaller than those for outside girders. AUTHOR: Jiang, J., and Olson, M. D. (1994) TITLE: Vibration analysis of orthogonally stiffened cylindrical shells using super finite elements INFO: Journal of Sound and Vibration, Vol. 173, No. 1, May. ABSTRACT: An efficient numerical technique for the free vibration analysis of orthogonally stiffened cylindrical shell structures is presented. The new technique is developed within the frame- work of a super finite element method and involves formulation of special cylindrical shell and curved beam elements. The displacement functions used in the present formulation combine the usual polynomials with some analytical functions carefully chosen to pro- vide a good approximation of basic vibration modes in conjunction with coarse grid model- ing strategy. Detailed results of vibration appli- cation of these elements are presented. AUTHOR: Kang, K. J., Bert, C. W., and Striz, A. G. (1996) TITLE: Vibration analysis of horizontally curved beams with warping using DQM INFO: Journal of Structural Engineering, ASCE, Vol. 122, No. 6, June. ABSTRACT: The differential quadrature method (DQM) is applied to computation of the eigenvalues of small-amplitude free vibration for horizon- tally curved beams, including a warping con- tribution. Natural frequencies are calculated for single-span, curved, wide-flange uniform beams having a range of nondimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curva- ture, included angle of the curve, polar mass moment of inertia, and various end condi- tions. Results are compared with existing exact and numerical solutions by other methods for cases in which they are available. The DQM provides accuracy even when only a limited number of grid points are used. In addition, results are given for a cantilever that has one clamped end and one free end; no previous results are known for this case. Finally, para- metric results are presented in dimensionless form. A-30 AUTHOR: Kawakami, M., Sakiyama, T., Matsuda, H., and Morita, C. (1995) TITLE: In-plane and out-of-plane free vibrations of curved beams with variable sections INFO: Journal of Sound and Vibration, Vol. 187, No. 3, November. ABSTRACT: Abstract could not be located. AUTHOR: Wang, R. T., and Sang, Y. L. (1999) TITLE: Out-of-plane vibration of multi-span curved beam due to moving loads INFO: Structural Engineering and Mechanics, Vol. 7, No. 4. ABSTRACT: This paper presents an analytic method of examining the out-of-plane vibration of con- tinuous curved beam on periodical supports. The orthogonality of two distinct sets of mode shape functions is derived. The forced vibra- tion of beam due to moving loads is examined. Two types of moving loads, which are concen- trated load and uniformly distributed load, are considered. The response characteristics of beams induced by these loads are investigated as well. AUTHOR: Yildirim, V. (1999) TITLE: In-plane free vibration of symmetric cross-ply laminated circular bars INFO: Journal of Engineering Mechanics, ASCE, Vol. 125, No. 6, June. ABSTRACT: A parametric study is performed to investigate influences of the opening angles, the slender- ness ratios, the material types, the boundary conditions, and the thickness-to-width ratios of the cross section on the in-plane natural fre- quencies of symmetric cross-ply laminated cir- cular composite beams. Governing equations are obtained based on the classical beam the- ory. The transfer matrix method is successfully applied to calculate exact natural frequencies with the help of an effective numerical algo- rithm, which was previously used for isotropic materials. The effects of the shear deformation, the axial deformation, and the rotary inertia are included in the formulation based on the first-order shear deformation theory. The physical system is considered a continuous sys- tem. To verify the present theory, two examples

are worked out for straight beams. An agree- ment is presented with the reported results. AUTHOR: Razaqpur, A. G. and Li, H. G. (1994) TITLE: Refined analysis of curved thin-walled multi- cell box girders INFO: Computers and Structures, Vol. 53, No. 1. ABSTRACT: The generalized Vlasov’s thin-walled beam theory was combined with the finite element technique to develop a new curved thin-walled multicell box girder finite element that can model extension, flexure, torsion, torsional warping, distortion, distortional warping and shear lag effects. For multicell box girders, sev- eral distortional modes were introduced to describe the complete distortional behavior of the cross-section. Interaction between the lon- gitudinal and transverse deformations of the box girder was also accounted for. The element is one-dimensional. It has three nodes and employs the conventional polynomial shape functions. For modeling flexure, Timoshenko beam theory was used to take account of shear deformations. A computer program was devel- oped based on the proposed element for the analysis of single and multicell curved box girder bridges. Numerical examples are pre- sented to demonstrate the accuracy and effi- ciency of the proposed element. Compared with the standard finite element method, the proposed method needs substantially less computer time, memory and input data. The output is also in a form that can be used in design without further manipulation. AUTHOR: Razaqpur, A. G., and Li, H. G. (1997) TITLE: Analysis of curved multicell box girder assem- blages INFO: Structural Engineering and Mechanics, Vol. 5, No. 1. ABSTRACT: A method of analysis is proposed for curved multicell box girder grillages. The method can be used to analyze box girder grillages com- prising straight and/or curved segments. Each segment can be modeled by a number of beam elements. Each element has three nodes, and the nodal degrees of freedom (DOFs) consist of the six DOFs for a conventional beam plus DOFs to account for torsional warping, distor- tion, distortional warping, and shear lag. This A-31 element is an extension of a straight element that was developed earlier. For a more realistic analysis of the intersection regions of non- collinear box girder segments, the concept of a rigid connector is introduced, and the compat- ibility requirements between adjoining ele- ments in those regions are discussed. The results of the analysis showed good agreement with the shell finite element results, but the proposed method of analysis needs a fraction of the time and effort that the shell finite ele- ment analysis needs. AUTHOR: Yabuki, T.,Arizumi,Y., and Vinnakota, S. (1995) TITLE: Strength of thin-walled box girders curved in plan INFO: Journal of Structural Engineering, ASCE, Vol. 121, No. 5. ABSTRACT: This paper presents a numerical method for predicting the influence of local buckling in component plates and distortional phenome- non on the ultimate strength of thin-walled, welded steel box girders curved in plan. A modified stress-strain curve allowing for local buckling of plate components is proposed. The effect of distortion is considered by incorpo- rating additional strain in the reference stage of the incremental process in the nonlinear flex- ure analysis. Theoretical predictions obtained using the proposed method are compared with experimental test results. Nonlinear behavior of the test girders is illustrated, and reasonable agreement between tests and theory is observed. AUTHOR: Galdos, N. H., Schelling, D. R., and Sahin, M. A. (1993) TITLE: Methodology for impact factor of horizontally curved box bridges INFO: Journal of Structural Engineering, ASCE, Vol. 119, No. 6. ABSTRACT: This paper presents a method for determining the dynamic impact factor of horizontally curved steel box-girder bridges under vehicle loadings.The two-dimensional planar grid anal- ogy is used to model box bridges. The vehicle is idealized as a pair of concentrated forces,with no mass traveling on circumferential paths with constant velocity. Analysis of multigirder and continuous span bridges indicates a tendency for

frequency clustering. Static solutions for these bridges are compared with mode superposition and direct integration. The bridge behavior was studied under several truck loading paths. Static and dynamic bridge behavior was observed under these loadings. A rational methodology for determining the impact factor is developed, and alternate impact-factor criteria are proposed to replace the current American Association of State Highway and Transportation Officials guide specification. AUTHOR: Sennah, K. M., and Kennedy, J. B. (1997) TITLE: Dynamic characteristics of simply supported curved composite multi-cell bridges INFO: Canadian Journal of Civil Engineering, No. 4, August. ABSTRACT: The use of horizontally curved composite bridges in the interchange facilities of modern highway systems has become increasingly pop- ular for economic as well as aesthetic consider- ations. Cellular steel composite sections with a concrete deck are quite suitable in resisting tor- sional and warping effects induced by highway curvature. However, this type of structure has inherently created design problems for the designer in estimating the level of dynamic response when subjected to moving vehicles, wind, or seismic conditions. This paper pres- ents a summary of an extensive parametric study, using the finite-element method, in which 120 simply supported, curved compos- ite bridge prototypes are analyzed to evaluate their natural frequencies and mode shapes. The parameters considered in the study are end-diaphragm thickness, cross-bracing sys- tem, aspect ratio, span-to-depth ratio, degree of curvature, number of lanes, and number of cells. Results from tests on four 1/12 linear- scale simply supported composite three-cell bridge models of different curvatures are used to substantiate the analytical modeling. The stipulation made in bridge codes for treating a curved bridge as a straight one is examined. Based on the data generated from the para- metric study, expressions for the first flexural frequency and hence the dynamic load allowance are deduced. Recommendations for enhancing the torsional resistance to dynamic forces on curved bridges are presented. Two design examples are illustrated. A-32 AUTHOR: Davidson, J. S., and Yoo, C. H. (1996) TITLE: Local buckling of curved I-girder flanges INFO: Journal of Structural Engineering, ASCE, Vol. 122, No. 8, August. ABSTRACT: Two analytical approaches are used to investi- gate the effect of curvature on the elastic local buckling behavior of horizontally curved I-girder compression flanges. First, a governing differential equation for the isolated curved flange plate elements under various loading and boundary conditions is derived in polar coordinates, and the equation is solved using the finite-difference technique. Numerical results for various curvatures and aspect ratios are generated. Second, the finite-element method using MSCINASTRAN is used where the entire cross section of the curved I-girder is modeled. A comparison of the results from the two approaches is made and differences are explained. The effect of curvature on the elas- tic buckling of the flange plates is observed and a simple but conservative formulation repre- senting this effect is derived for design use. AUTHOR: Davidson, J. S., and Yoo, C. H. (2000) TITLE: Evaluation of strength formulations for hori- zontally curved flexural members INFO: Journal of Bridge Engineering, ASCE, Vol. 5, No. 3, August. ABSTRACT: In 1992 the Federal Highway Administration (FHWA), along with 13 states, began a project to study the behavior of horizontally curved steel bridges. The project is referred to as the Curved Steel Bridge Research Project (CSBRP) and involves studying the behavior of curved mem- bers through theoretical, analytical, and experi- mental research. In this paper, detailed finite element models representing a curved three- girder test frame that is planned under the CSBRP experimental phase are used to evaluate the effects of curvature on the bending strength of curved I-girders. Linear-elastic static, buck- ling, and combined material and geometric nonlinear analyses are conducted using models that represent the test frame and component test specimens that will be inserted into it. The results are compared with various predictor equations developed from analytical work by the writers and with related work by other

researchers, including Japanese research that is not readily available in the U.S. It is demon- strated that the predictor equations developed by the writers are accurate in representing the behavior of the system. Limitations and needed improvements are described as well. AUTHOR: Davidson, J. S., Balance, S. R., and Yoo, C. H. (2000) TITLE: Behavior of curved I-girder webs subjected to combined bending and shear INFO: Journal of Bridge Engineering, ASCE, Vol. 5, No. 2, May. ABSTRACT: Two previous papers by the authors described the buckling and finite displacement behavior of curved I-girder web panels subjected to pure bending. The papers presented a theoretically pure analytical model and presented equations that describe the reduction in strength due to curvature. This paper describes the buckling and finite displacement behavior of curved web panels under combined bending and shear. Unlike in straight girder web panels, the addi- tion of shear in curved panels is shown to increase the transverse “bulging” displacement of the web prior to buckling. The accompanying decrease in moment carrying capacity is ana- lyzed in a manner similar to that used for the combined bending and shear nominal strength interaction for straight girder design. Prelimi- nary recommendations are made towards forming design criteria for curved webs. AUTHOR: Davidson, J. S., Ballance, S. R., and Yoo, C. H. (2000) TITLE: Effects of longitudinal stiffeners on curved I-girder webs INFO: Journal of Bridge Engineering, ASCE, Vol. 5, No. 2, May. ABSTRACT: Two previous papers by the authors described the buckling and finite displacement behavior of curved I-girder web panels without longitu- dinal stiffeners subjected to pure bending. The papers presented a theoretically pure analytical model and presented equations that describe the reduction in strength due to curvature. This paper describes the optimum location and strength effects of one and two longitudi- nal stiffeners attached to curved I-shaped plate A-33 girders. Using lateral load and lateral pressure analogies similar to that described in the ear- lier papers, strength reduction equations are formulated for curved plate girders with longi- tudinal stiffeners. A comprehensive compari- son is made between the equations developed in this investigation and design equations used in Japanese and American design guides. The applicability and superiority of the method is demonstrated. AUTHOR: Davidson, J. S., Ballance, S. R., and Yoo, C. H. (1999) TITLE: Finite displacement behavior of curved I-girder webs subjected to bending INFO: Journal of Bridge Engineering, ASCE, Vol. 4, No. 3, August. ABSTRACT: Curvature greatly complicates the behavior of curved plate girders used in bridges. The out- of-plane “bulging” displacement of the curved web results in an increase in stress, which must be considered in the design of plate girders with significant curvature. This paper presents results from geometric nonlinear finite- element analyses used to evaluate the finite dis- placement behavior of such panels and to for- mulate deflection amplification factors that can be applied to analytical models to get con- servative values for predicting the maximum displacements and stresses of the curved panel. Equations are developed that represent the reduction in nominal strength of the curved web due to the effects of curvature. The appli- cability of the method is demonstrated, and a comprehensive comparison is made between the equations developed in this investigation and design equations used in Japanese and American design guides. AUTHOR: Davidson, J. S., Ballance, S. R., and Yoo, C. H. (1999) TITLE: Analytical model of curved I-girder web pan- els subjected to bending INFO: Journal of Bridge Engineering, ASCE, Vol. 4, No. 3, August. ABSTRACT: Curvature greatly complicates the behavior of curved plate girders used in bridges. The out- of-plane “bulging” displacement of the curved web results in an increase in stress, which must

be considered in the design of plate girders with significant curvature. The currently used Guide Specifications for Horizontally Curved Bridges provides web slenderness reduction equations that account for curvature effects, but these equations were based on a regression of limited data from experimental and unit- strip analyses conducted in the early 1970s. This paper presents a theoretically pure ana- lytical model that can be used to predict the transverse displacement and induced plate bending stresses of curved I-shaped plate girder web panels subjected to bending. Several boundary conditions are demonstrated and compared, and the finite-element method is used to verify the closed-form solutions. The effects of curvature on the elastic buckling behavior of curved web panels is also pre- sented. Furthermore, a comprehensive litera- ture review is presented, including numerous Japanese publications not readily available to American researchers. AUTHOR: Lee, S. C., and Yoo, C. H. (1999) TITLE: Strength of curved I-girder web panels under pure shear INFO: Journal of Structural Engineering, ASCE, Vol. 125, No. 8. ABSTRACT: The bifurcation buckling and the ultimate strength of curved web panels subjected to pure shear are investigated by the finite- element method. To evaluate the ultimate strength of curved web panels subjected to pure shear to the point of failure, both geo- metric and material nonlinearities are con- sidered. From the nonlinear analysis it is shown that curved web panels are capable of developing considerable post-buckling strength after the bifurcation point. Results of the present study are compared with shear strength of straight girder web panels sub- jected to pure shear. Comparisons indicate that the straight girder equations can effec- tively predict the shear strength of curved web panels subjected to pure shear. The deflection curves that are due to a unit-generalized dis- placement at nodal coordinate and the exact element stiffness matrix are derived based on the solution for the general system. A finite element method can be developed based on the results for the dynamic analysis. Mean- A-34 while, the stiffness locking phenomena accompanied in some other curved beam ele- ment methods do not exist in the proposed method. AUTHOR: Schilling, C. G. (1996) TITLE: Yield-interaction relationships for curved I-girders INFO: Journal of Bridge Engineering, ASCE, Vol. 1, No. 1, February. ABSTRACT: Yield-interaction relationships are developed for compact, compact-flange, and noncom- pact sections of curved girders under com- bined vertical and lateral (warping torsion) moments. These relationships define the bending strength of such compact, compact- flange, and noncompact sections if suitable web-slenderness, compression-flange slen- derness, and compression-flange bracing lim- its are satisfied. The most convenient form of these relationships is equations defining reduced flange widths as a function of lateral moment. The correct vertical-bending strength can be calculated from a reduced steel section consisting of the actual web and reduced widths of flanges. Compact flange sections have a compact compression flange and a noncompact web. For straight girders, there is no advantage in using a compact compression flange with a noncompact web. For curved girders, however, the compact flange permits a larger lateral moment to be carried in combination with a given vertical moment. AUTHOR: Weaver, D. L. (1996) TITLE: Steel girder bridges INFO: Construction Specifier, Vol. 49, No. 5, May. ABSTRACT: The majority of steel bridges constructed today consist of straight or curved girder superstruc- tures with maximum spans of 60 m or less. Although the basic concept of steel girder bridges has remained unchanged since the 1920s, advances in materials and design prac- tice have allowed them to remain one of the most cost-effective and popular types of bridge superstructure systems. Some of the refinements in steel girder bridge design and construction are presented.

AUTHOR: Yoo, C. H. (1993) TITLE: Some considerations in the design and con- struction of horizontally curved highway bridges INFO: Proceedings of the Fourth East Asia-Pacific Conference on Structural Engineering and Construction, Seoul, Korea, September 20–22 (Invited paper). AUTHOR: Horizontally curved bridges are much more complex than bridges with girders having ini- tial curvature in the horizontal plane. Presented in the paper are some of the major design con- siderations that are normally not considered in the straight bridge design. Likewise, erection of horizontally curved girders requires careful precautions that are normally extended in the erection of straight girders. A simple span curved I-girder is unstable. Depending upon the severity of curvature, the site condition, and the availability of heavy equipment, an erection scheme can be devised taking the maximum advantage of the situation presented. AUTHOR: Yoo, C. H., and Davidson, J. S. (1997) TITLE: Yield interaction equations for nominal bend- ing strength of curved I-girders INFO: Journal of Bridge Engineering, ASCE, Vol. 2, No. 2, May. ABSTRACT: Horizontally curved I-girders are subjected to combined vertical bending and torsion under gravity loading alone. The torsional behavior of open I-shaped girders is commonly and conveniently equated to self-equilibrating lat- eral bending moments in the flanges. The interaction of vertical bending and this lateral flange bending effectively reduces the vertical moment carrying capacity of the section. Yield interaction equations for predicting the nom- inal bending strength of horizontally curved steel I-girders subjected to vertical moment and torsion are derived. Singly symmetric composite and noncomposite I-shaped cross sections in both the positive and negative moment zones are considered. Strength crite- ria considered are: (1) complete plastification of the cross section for compact sections; (2) partial yield penetration for compact-flange sections; and (3) initial yielding at the extreme flange tip for noncompact sections. A total of A-35 17 interaction cases are ultimately considered. These strength criteria are based purely on the static equilibrium of the cross section with no secondary amplification considered. The limi- tations and applicability of the derived equa- tions toward design use are demonstrated and analyzed. AUTHOR: Cheung, M. S., and Foo, S. H. C. (1995) TITLE: Design of horizontally curved composite box- girder bridges—A simplified approach INFO: Canadian Journal of Civil Engineering,Vol. 22, No. 1. ABSTRACT: Because of their excellent torsional capacity, box girders are used extensively in modern bridge construction having curved alignments. Applications of most design codes have been limited to bridges where the radius of curva- ture is much greater than the span length and cross-sectional dimensions. To meet the prac- tical requirements arising during the design process, simple design methods are needed for curved bridges. This paper presents the results of a parametric study on the relative behavior of curved and straight box-girder bridges and on the development of a simplified design method for the combined longitudinal moment of curved bridges. The combined moment includes the effects of flexure, torsion, and distortion. Three simply supported concrete-steel composite bridge models— including single-cell, twin-cell, and three-cell box girders—that were subjected to loadings as specified in the Ontario Highway Bridge Design Code were analyzed using the finite strip method. The parameters considered in the study include types of cross section; types, locations, and magnitudes of loads; span lengths; and radius of curvature. Preliminary analysis of the results suggests that the behav- ior of horizontally curved box-girder bridges is dependent on a variety of parameters, but most importantly on the span-to-radius ratio. Empirical relationships for combined longitu- dinal moment between curved and straight box-girder bridges are also proposed. AUTHOR: Sennah, K., and Kennedy, J. B. (1998) TITLE: Shear distribution in simply-supported curved composite cellular bridges

INFO: Journal of Bridge Engineering, ASCE, Vol. 3, No. 2. ABSTRACT: Composite steel-concrete multicell box girder bridges are quite often used in modern bridge superstructures with curved alignments. They provide excellent torsional resistance as well as elegant appearance. While current design practices in North America recommend few analytical methods for the design of curved multicell box bridges, practical requirements in the design process require a need for a sim- plified design method. This paper summarizes the results from an extensive parametric study, using a finite-element model, in which 120 simply-supported curved bridge prototypes are analyzed to evaluate the shear distribution in the webs due to truck loading as well as dead load. Results from tests on four 1/12 linear- scale simply-supported curved composite con- crete deck-steel three-cell bridge models are used to substantiate the analytical modeling. The parameters considered in the study are: cross-bracing system, aspect ratio, number of lanes, number of cells, and degree of curvature. Based on the data generated from the para- metric study, expressions for the shear distri- bution factors for truck loadings as well as dead load are deduced. An illustrative design example is presented. AUTHOR: Sennah, K., and Kennedy, J. B. (1999) TITLE: Simply supported curved cellular bridges: Sim- plified design method INFO: Journal of Bridge Engineering, ASCE, Vol. 4, No. 2. ABSTRACT: The use of curved composite bridges in inter- changes of modern highway systems has become increasingly popular for economic and aesthetic considerations. Bridges with a concrete deck composite with a steel multicell section can adequately resist torsional and warping effects induced by high curvature. Although current design practices in North America recommend few analytical methods for the design of curved multicell box girder bridges, economical requirements in the design process point to a need for a simplified design method. This paper summarizes the A-36 results from an extensive parametric study, using the finite-element method, in which simply supported curved composite multicell bridge prototypes are analyzed to evaluate the moment and deflection distributions between girders, as well as the axial forces expected in the bracing system, due to truck loading as well as dead load. Results from tests on four, 1/12 linear-scale, simply supported curved composite concrete deck-steel multicell bridge models are used to substantiate and verify the analytical modeling. The parame- ters considered in the study are cross-bracing system, aspect ratio, number of lanes, number of cells, and degree of curvature. Based on the data generated from the parametric study, expressions for moment and deflection distri- bution factors are deduced. Expressions for the maximum axial force in bracing members are also derived. An illustrative design exam- ple is presented. AUTHOR: Yoo, C. H., Davidson, J. S., and Zhang, J. (1998) TITLE: Top flange diagonal bracing of horizontally curved box girders INFO: Proceedings of the Engineering Mechanics Conference: A Force for the 21st Century, La Jolla, CA, May 18–20. ABSTRACT: Steel/concrete composite box girders with inclined webs, otherwise known as tub girders, have been successfully designed and built in the U.S. As modern advancement in welding has permitted box girder fabrication with rela- tive ease, it is expected that the use of compos- ite box girders will grow in urban interchanges. The superior torsional stiffness of box girders, however, cannot be realized until the compos- ite concrete deck has been hardened. For non- composite dead loads, top flange diagonal bracing in horizontally curved girders acts as primary load carrying members. This paper presents a rational design guide for a diagonal tub flange bracing member highlighting sev- eral important design considerations. Sup- porting data used for computation were generated by a three-dimensional finite ele- ment analysis on a hypothetical horizontally curved box girder bridge.

A-37 Aydemir, M., White, D. W., and Jung, S. K. (2004). “Shear Strength and Moment Shear Interaction in HPS Hybrid I-Girders,”Structural Engineering, Mechanics and Materials Report No. 25, School of Civil and Environmental Engineer- ing, Georgia Institute of Technology, Atlanta, GA, 203 pp. This research examines the maximum strength behavior of transversely stiffened prismatic straight hybrid I-girders having Grade 50 steel webs and Grade 70 steel flanges. A parametric suite of test specimens is designed, and full nonlinear analyses of these girders are conducted using shell finite elements. The primary focus of this work is the evaluation of the postbuckling shear strength as well as the moment-shear interaction char- acteristics of these types of girders. The members considered include 18 different cross-sections and are subjected to various ratios of the maximum moment-to-shear within the critical test panel. The key cross-section parameters varied within the study are the web slenderness D/tw, the panel aspect ratio do/D, and the ratio of the depth of the web in compression to the total web depth Dc/D.A total of 147 combinations of geometry and loading conditions are studied. Barth,K. E.,White,D. W.,Righman,J. E.,and Yang,L.(2005). “Evaluation of Web Compactness Limits for Singly and Doubly Symmetric Steel I-Girders,”Journal of Constructional Steel Research, 61(10), 1411–1454. This paper presents the results of research aimed at evaluat- ing the web compactness limit for steel I-girders. Specifically, the paper tests the implications of a new web compactness limit equation provided in AASHTO (2003) versus the web compactness limit in the 2001 AASHTO LRFD Specifications. In both Specifications, these limits are required for the nominal moment capacity to equal the plastic moment capacity of the girder, provided other requirements are also satisfied. The 2001 AASHTO LRFD web compactness limit is the same as the limit in the AISC LRFD (1999) Specifications, except the AASHTO provisions place an additional restriction on the web slenderness if the flange slenderness exceeds 75% of the corresponding flange compactness limit, via an interaction equation involving the flange and the web slenderness values. The origin of the AASHTO (2003) and (2001) web compact- ness limits is presented along with a performance evaluation of these equations. Specifically, resulting moment capacities from a comprehensive suite of finite element analyses are compared to the capacities that the respective limits are intended to pro- vide.Results indicate there are some limitations in the AASHTO (2001) and AISC (1999) web compactness limits (particularly for singly symmetric cross-sections) that are removed by the AASHTO (2003) web compactness provisions. Barth,K. E.,Hartnagel,B. A.,White,D. W.,and Barker,M. G. (2004).“Recommended Procedures for Simplified Inelastic Design of Steel I-Girder Bridges,”Journal of Bridge Engineer- ing, ASCE, 9(3), 230–242. This paper presents summary recommendations pertaining to new AASHTO procedures for simplified inelastic design of steel I-girder bridges. First, key developments are summarized that lead to the proposed inelastic design approach. The paper then outlines a set of equations that provide an improved char- acterization of the inelastic moment-rotation response for a wide range of I-beams and plate girders. Effective plastic moment predictions based on these equations are combined with the recently proposed design method, resulting in greater accuracy and simplicity of the proposed approach. The ease of use of the resulting procedure is illustrated by a design example. Barth, K. E., and White, D. W. (1997). “Finite Element Evaluation of Pier Moment-Rotation Characteristics in Continuous-Span Steel I-Girders.” Engineering Structures, 20(8), 761–778. This paper summarizes the implementation and execution of a reasonably comprehensive set of finite element parametric Background Research Pertaining to Updated AASHTO LRFD Specifications for Steel Structures, Third Edition Prepared by Donald W. White, Georgia Tech

studies to fill in knowledge gaps in the available experimen- tal data pertaining to the hogging moment-plastic rotation behavior of steel and composite steel-concrete bridge girders. The paper highlights key requirements for proper finite ele- ment modeling of the behavior, outlines the design of the finite element parametric studies, and presents the important “numerical test”results.Based on the finite element predictions, simple moment-rotation relationships are developed for use in design and rating of bridge structures. Beshah, F. (2005). “Moment-Shear Test Series,” Volume 3, Curved Steel Bridge Research Project, Federal Highway Administration, Curved Steel Bridge Research Project, Federal Highway Administration, December. As part of the Federal Highway Administration’s curved steel bridge research project (CSBRP), to further perform experi- mental and analytical study in the fundamental behavior of horizontally curved steel bridges, static load tests on four full- scale curved I-girders were performed under high moment and low shear loading. Each of the four girders had 7.74 m center- line length measured along arc length, 63.63 m radius of curvature, 4.77 m unbraced length, and 8 by 1,219 mm web plate. All girders were fabricated from AASHTO M270 Grade 345 steel. The varied dimensional parameters include the width to thickness ratio of the flange, and the transverse stiffener spacing. Strain gages, potentiometers, load-cells, and tilt meters were placed on all test girders to study their responses. This report, Volume 3 of the final report, presents detailed descriptions and results from these static load tests. Measured data are compared with nonlinear finite element analysis prediction. Furthermore, the measured data are also used to evaluate design specifications in current bridge codes. Beshah, F. (2006).“Girder Pair Test,”Volume 5, Curved Steel Bridge Research Project, Federal Highway Administration, May. As part of the Federal Highway Administration’s curved steel bridge research project (CSBRP), to further perform experimental and analytical study on the fundamental behav- ior of horizontally curved steel bridges, a static load test on a full-scale girder pair was performed. The test evaluated the effect of lateral bracing. This report,Volume 5 of the final report, presents a detailed description and results from this static load test. Measured data are presented and compared with nonlinear finite element analysis prediction. The measured data are also used to evaluate design specifications in current highway bridge codes. Analyt- ical studies are also included to investigate the effect of lateral brace size. Beshah, F. (2006).“Testing of Composite Bridge,”Volume 9, Curved Steel Bridge Research Project, Federal Highway Administration, November. As part of the Federal Highway Administration’s curved steel bridge research project (CSBRP), to further perform experi- mental and analytical study on the fundamental behavior of horizontally curved steel bridges, static load tests on a full- scale composite curved steel I-girder bridge were conducted. The tests determined the behavior under a number of loading stages up to failure. The bridge was a simple span consisting of three I-girders spaced at 2,667 mm. A 203-mm-thick con- crete deck, consisting of radial and longitudinal reinforce- ment, was composite with the steel girders. The bridge was instrumented with strain gages, potentiometers, load-cells, linear voltage displacements, transducers, and tilt meters. This volume,Volume 9 of the final report, presents detailed descriptions and results from these tests. Measured data are presented, and comparison to nonlinear finite element analysis prediction is presented in Volume 6 of the final report. Chang, C.-J., and White, D. W. (2006).“Construction Simu- lation of Curved Steel I-Girder Bridges,” Volume 7, Curved Steel Bridge Research Project, Federal Highway Adminis- tration, May. This study addresses the development of a prototype software system for analysis of horizontally curved steel I-girder bridges, with emphasis on construction simulation. The approach adopted in this research is based on the use of open-walled section beam theory for the I-girders and a beam grillage representation for the composite bridge slab. An approximate approach is targeted for capturing the influ- ence of girder web distortions on composite I-girder responses. Also, recommendations are provided for the use of grillage idealizations in analyzing curved I-girder bridge structural sys- tems.A key focus of the research is on simulating steel erection and staged slab casting processes. The resulting capabilities allow engineers to check deflections, reactions and/or stresses at different stages of the steel erection or concrete casting and determine required crane capacities, tie-down, jacking or come- along forces, incremental displacements due to removal of temporary supports, etc. Also, the capabilities can be used to determine the influence of different steel detailing methods on the bridge geometry, such as the web-plumbness under the steel or total dead load, as well as the implications of geomet- ric tolerances on the structural performance. The fundamen- tal requirements necessary to ensure accuracy of the analysis results are addressed. A-38

Chang, C.-J.,White, D. W., Beshah, F., and Wright,W. (2005). “Design Analysis of Curved I-Girder Bridge Systems—An Assessment of Modeling Strategies,” Annual Proceedings, Structural Stability Research Council, 349–369. This paper investigates the qualities and limitations of a number of modeling strategies for design analysis of curved I-girder bridge systems, ranging from modified line-girder analyses to finite element approaches. A representative full- scale composite curved I-girder bridge, tested at the FHWA Turner-Fairbank Highway Research Center (TFHRC), is uti- lized for assessment of the different approaches.The predictions of key lateral and vertical displacements, reactions, cross-frame forces, and major-axis and flange lateral bending stresses are evaluated with respect to the test bridge. The paper highlights a number of subtle but critical blunders that can occur in the context of the different methods. Engineers need to be aware of these pitfalls in order to avoid potential problems due to inaccurate analysis predictions. The paper closes with a dis- cussion of the efficacy of the various approaches. Jung, S.-K., and White, D. W. (2006). “Inelastic Behavior of Horizontally Curved Composite I-Girder Bridge Structural Systems,” Volume 6, Curved Steel Bridge Research Project, Federal Highway Administration, May. This study addresses the design and analysis of a full-scale horizontally curved composite test bridge selected to examine the system and component responses of a representative curved composite structure under all loading stages: non-composite dead load, composite service live load, and ultimate loading. The curved composite test bridge is designed at or above a number of maximum limits in AASHTO (2003 and 2004) design provisions. Refined three-dimensional finite element analysis (FEA) models are developed and applied in this research for both the elastic design-analysis as well as linear elastic, geometric nonlinear and full nonlinear FEA simula- tion of the test bridge system. The full nonlinear FEA model involves the simulation of staged construction, followed con- secutively by applied loads on the composite structure in a single continuous process. The composite test bridge was built and tested at the FHWA Turner-Fairbank Highway Research Center for all loading stages. This study provides the synthesis and assessment of the experimental results and corresponding full nonlinear FEA predictions for the test bridge. Further- more, parametric FEA studies are performed to investigate the responses for a wider range of bridges, with various attributes including skewed supports, integral abutments and different cross-frame detailing methods. Based on the system and com- ponent responses of the test bridge as well as the additional FEA parametric studies, this research investigates the impli- cations of the use of enhanced flexural resistance equations that account for flange lateral bending effects from any source, specified in AASHTO (2004) for straight bridge I-girders, in the context of curved I-girder bridges. Jung, S.-K., and White, D. W. (2006).“Shear Strength of Hor- izontally Curved Steel I-Girders—Finite Element Studies,” Journal of Constructional Steel Research, 62(4), 329–342. This paper presents the results of finite element analysis (FEA) studies of four curved steel I-girder shear components tested experimentally in previous research, as well as para- metric extensions of these tests. These studies focus on the influence of horizontal curvature on the maximum strength of transversely stiffened members with web slenderness D/tw approximately equal to the largest value permitted in AASHTO (2004),and with panel aspect ratios of do/D = 1.5 and 3.0. These ratios are larger than previously considered in experimental tests of curved I-girders with similar or larger slenderness. The girders studied have subtended angles between their bracing locations of Lb/R = 0.05 and 0.10, and web panel do/R values ranging from 0.0287 to 0.10. The FEA models incorporate the measured material stress-strain relationships and section dimensions from the physical tests,detailed modeling of the test boundary conditions, residual stresses due to flame cutting and welding, and initial geometric imperfections in the form of buckling mode shapes.The load transfer mechanisms of the test girders are investigated via elastic buckling and full nonlinear analyses. The parametric studies are performed to investigate the effects of residual stresses and geometric imperfections, the behavior of equivalent straight girders, and the influence of reduced flange size on the peak shear capacity and moment- shear interaction. Jung, S.-K., and White, D. W. (2006). “Strength Behavior of Horizontally Curved Composite I-Girder Bridge Structural Systems,”Annual Proceedings, Structural Stability Research Council, to appear, 20 pp. This paper discusses the strength behavior of a representative horizontally curved composite (steel and concrete) I-girder bridge system that was designed near the limits of the AASHTO (2004) bridge design specifications and tested at the Federal Highway Administration (FHWA) laboratory for its ultimate loading capacity. Particular focus is placed on the force transfer mechanisms within the bridge system as a whole as the test bridge approaches the strength limit.The results clearly indicate that there is a high degree of interaction among the bridge girders. However, despite the complexities associated with the interconnectedness of the bridge girders, all the bridge component and system responses are accurately predicted by A-39

linear elastic analysis up to the strength limit based on the pro- visions of Article 6.10.7.1 in AASHTO (2004) Specifications. Article 6.10.7.1 uses the composite section plastic moment, reduced by the flange lateral bending effects due to torsion, as the base flexural resistance for the strength limit states.AASHTO (2004) presently does not allow the use of Article 6.10.7.1 for the strength limit state checks in horizontally curved I-girder bridges, although these provisions are applicable to general straight I-girder bridges. It limits the base resistance for the strength checks to the member yield moment My, rather than Mp, for these bridge types (via Article 6.10.7.2). The findings from this and other research support the potential liberalization of the AASHTO strength design provisions for horizontally curved I-girder bridges. Jung, S.-K., White, D. W., Beshah, F., and Wright, W. (2005). “Ultimate Strength of Horizontally-Curved Composite I-Girder Bridge Structural Systems,” Annual Proceedings, Structural Stability Research Council, 327–347. This paper provides an overview of a full-scale composite curved I-girder bridge tested at the FHWA Turner-Fairbank Highway Research Center. The paper presents a summary of the broad research plan for the composite bridge test. This is followed by a detailed discussion of key aspects related to the strength design and behavior of the composite test bridge structural system: (1) a brief introduction to the unified flex- ural resistance equations for curved and straight I-girder design recently implemented in AASHTO (2004), (2) current design restrictions in AASHTO (2004) and why, (3) potential liberal- ization of these restrictions, (4) refined finite element analysis (FEA) modeling, (5) design attributes of the test bridge per- taining to AASHTO (2004), and (6) preliminary results from refined full nonlinear FEA and from physical testing of the test bridge system. Kim,Y. D., Jung, S. K., and White, D. W. (2006).“Transverse Stiffener Requirements in Straight and Horizontally Curved Steel I-Girders,” Journal of Bridge Engineering, ASCE, to appear. A number of prior research studies have demonstrated that transverse stiffeners in straight I-girders are loaded predom- inantly by bending induced by their restraint of web lateral deflections at the shear strength limit state, not by in-plane tension field forces. This is at odds with present specification approaches for the design of these components. Furthermore, recent studies have confirmed that curved I-girders are capable of developing substantial shear postbuckling resistance due to tension field action and have demonstrated that the AASHTO LRFD equations for the tension field resistance in straight I-girders may be applied to curved I-girders within specific limits. However, the corresponding demands on transverse stiffeners in curved I-girders are still largely unknown. In this paper, the behavior of one- and two-sided transverse stiffeners in straight and horizontally curved steel I-girders is investigated by refined full nonlinear finite element analysis. New recom- mendations are developed for design of transverse stiffeners in straight and curved I-girders based on the combined solutions from this research and from prior research studies. White, D. W., and Chang, C.-J. (2006). “Improved Flexural Stability Design of I-Section Members in AISC (2005)—A Case Study Comparison to AISC (1989) ASD,” Engineering Journal, AISC, to appear. The AISC (2005) provisions for the flexural stability design of steel I-section members have been updated relative to pre- vious specifications to simplify their logic, organization and application, while also improving their accuracy and generality. This paper gives a brief overview of the updated provisions and compares and contrasts their flexural resistance calcula- tions with the corresponding calculations from the previous AISC (1989) ASD specification. The relative simplicity and accuracy of the AISC (2005) equations is highlighted. White, D. W., and Grubb, M. A. (2005).“Unified Resistance Equations for Design of Curved and Tangent Steel Bridge I-Girders,”Proceedings of the 2005 TRB Bridge Engineering Conference, Transportation Research Board, Washington, DC, July, 121–128. The provisions of the 2004 AASHTO Load and Resistance Factor Design (LRFD) Specifications for steel I- and box-girder bridge design have been updated relative to previous specifi- cations to simplify their logic, organization and application while also improving their accuracy and generality. These pro- visions provide a unified approach for the flexural design of both tangent and curved I- and box-girder bridges. Updated resist- ance equations are a key component of this unified approach. This paper provides an overview of the updated resistance equations for I-section members. The primary focus is on the handling of coupled major-axis bending, minor-axis bending and torsion from any source in both straight and horizontally curved I-section members. White, D. W. (2004).“Unified Flexural Resistance Equations for Stability Design of Steel I-Section Members—Overview,” Structural Engineering, Mechanics and Materials Report No. 24a, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 42 pp. The AASHTO (2004) and AISC (2005) provisions for the flexural design of steel I-section members have been revised A-40

in their entirety relative to previous specifications to simplify their logic, organization and application, while also improving their accuracy and generality. This report provides a technical overview of these developments with respect to the stability limit states. The updated AISC and AASHTO flexural resist- ances are, with minor exceptions, fundamentally the same. However, the organization and format in AASHTO (2004) emphasizes the streamlined design of various I-section member types common for bridge spans of 30 m (100 ft) or larger, while AISC (2005) emphasizes the streamlined design of compact doubly-symmetric I-section members. This report presents the fundamental logic and the associated calculations of these new provisions as a single set of flowcharts for design of all types of steel I-section members, with the exception of com- posite members in positive bending. That is, all types of com- posite I-section members in negative bending and all types of noncomposite I-section members are addressed, including hybrid members with different yield strengths for the tension flange, compression flange and/or web, longitudinally stiffened members, members with channel caps on the compression flange or cover plates on one or both flanges, and members with section transitions and/or variable web depth. The flex- ural resistance equations are presented in a unified format applicable for all of the above types of I-section members. In each of the sections of the report, the presentations start with an emphasis on the “how to” of the resistance calculations. This is followed by more in-depth discussions of the back- ground to the calculation procedures. Various improvements relative to the prior AISC and AASHTO specifications are highlighted. A paper version of the above report is under review for poten- tial publication in the Journal of Structural Engineering, ASCE. White, D. W., and Jung, S.-K. (2004).“Unified Flexural Resis- tance Equations for Stability Design of I-Shaped Members— Uniform Bending Tests,”Structural Engineering, Mechanics and Materials Report No. 28, School of Civil and Environ- mental Engineering,Georgia Institute of Technology,Atlanta, GA, 128 pp. The AASHTO (2004) and AISC (2005) provisions for flex- ural design of steel I-section members have been revised in their entirety relative to previous specifications to simplify their logic, organization and application, while also improv- ing their accuracy and generality. This report evaluates the lateral-torsional and flange local buckling (LTB and FLB) resistance predictions from these and previous specifications versus uniform bending experimental test results. A total of 154 rolled and 123 welded I-section member LTB tests, and 11 rolled and 36 welded I-section member FLB tests, are con- sidered.Reliability indices are estimated for Load and Resistance Factor Design (LRFD) of buildings based on the test statistics combined with established statistics for material and fabrica- tion bias factors and the ASCE 7 load model. The notional reliability for LTB is found to be reasonably constant and con- sistent with the targeted level in the first LRFD specification of 1986. The unified resistance equations, combined with a simple design-oriented procedure for calculation of elastic LTB K factors, are shown to capture the test results accurately through- out the inelastic and elastic LTB ranges, leading to substantial liberalization of the calculated resistances in certain cases. The mean resistances for inelastic LTB and FLB are captured accurately by a linear equation in the corresponding slender- ness parameters. The reliability for FLB is found to be slightly higher than that for LTB. A paper version of the above report is under review for poten- tial publication in the Journal of Structural Engineering, ASCE. White, D. W., and Kim,Y. D. (2004).“Unified Flexural Resis- tance Equations for Stability Design of I-Shaped Members— Moment Gradient Tests,”Structural Engineering, Mechanics and Materials Report No. 29, School of Civil and Environ- mental Engineering,Georgia Institute of Technology,Atlanta, GA, 149 pp. The AASHTO (2004) and AISC (2005) provisions for flex- ural design of steel I-section members have been revised in their entirety relative to previous specifications to simplify their logic, organization and application, while also improving their accuracy and generality. This report evaluates the lateral- torsional and flange local buckling (LTB and FLB) resistance predictions from these specifications versus moment gradient experimental test results. Uniform bending experimental tests are addressed in a companion report. Two types of moment gradient tests are considered: (1) tests in which the moment varies linearly within the critical unbraced length and (2) tests in which the member is subjected to concentrated transverse load at a specified height relative to the depth of the cross- section, resulting in a multi-linear moment diagram within the critical unbraced length. A total of 27 welded and 10 rolled I-section member FLB tests of Type (1),73 rolled and 93 welded member LTB tests of Type (1), and 129 rolled and 111 welded member LTB tests of Type (2) are considered. Reliability indices are estimated for Load and Resistance Factor Design (LRFD) of buildings based on the statistics from these tests combined with established statistics for material and fabrication bias factors and the ASCE 7 load model. The report demonstrates that in certain cases, the reliability index is substantially larger for moment gradient loading compared to that for uniform bending. However, for other cases, the notional reliability for members subjected to moment gradient is essentially the same as that estimated in the companion report for uniform moment. A paper version of the above report is under review for poten- tial publication in the Journal of Structural Engineering, ASCE. A-41

White, D. W., and Barker, M. (2004). “Shear Resistance of Transversely-Stiffened Steel I-Girders,” Structural Engi- neering, Mechanics and Materials Report No. 26, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 105 pp. This report evaluates the accuracy and ease of use of 12 of the most promising models for the shear resistance of trans- versely stiffened steel I-girders. Several models that are well established in civil engineering practice as well as a number of other recently proposed models are considered.Since the model developed in Basler’s seminal research is the method of choice in current American practice, the report focuses on the merits and limitations of the alternative models relative to Basler’s. Statistical analyses are conducted on the predictions by the various models using an updated data set from 129 experi- mental shear tests, including 30 hybrid and 11 horizontally curved I-girders. The results support the conclusion that the form of Basler’s model implemented in AASHTO (2004) and AISC (2005) gives the best combination of accuracy and sim- plicity for calculation of the shear resistance of transversely stiffened I-girders. A paper version of the above report is under review for potential publication in the Journal of Structural Engineering, ASCE. White, D. W., Barker, M., and Azizinamini,A. (2004).“Shear Strength and Moment-Shear Interaction in Transversely- Stiffened Steel I-Girders,”Structural Engineering, Mechanics and Materials Report No. 27, School of Civil and Environ- mental Engineering,Georgia Institute of Technology,Atlanta, GA, 82 pp. With the advent of HPS495W steel, hybrid I-girder designs have become advantageous in bridge design. One limit on hybrid I-girder designs,which decreases their beneficial aspects, is that the use of tension field action is not permitted in deter- mining the shear resistance. This is a significant penalty for hybrid I-girders. Also, the checking of moment-shear strength interaction is a significant complicating factor in the design and capacity rating of I-girders that use tension field action. The requirements for the shear design of hybrid I-girders and the equations for moment-shear strength interaction in AISC (1999) and AASHTO (1998) were developed originally without the benefit of a large body of experimental tests and refined finite element solutions. This report presents the results from the collection and analysis of the data from a total of 186 high-shear low-moment, high-shear high-moment, and high-moment high-shear experimental I-girder tests. Refer- ences to corroborating refined finite element studies are pro- vided. Particular emphasis is placed on the extent to which web shear postbuckling (tension-field action) strength is devel- oped in hybrid I-girders, as well as on the interaction between the flexural and shear resistances in hybrid and non-hybrid I-section members. The results of the study indicate that within certain constraints that address the effects of small flange size, Basler’s shear resistance model can be used with the unified flexural resistance provisions in AASHTO (2004) and AISC (2005) without the need for consideration of M-V strength interaction. Also, the report shows that a form of the Cardiff model can also be used with the unified flexural resistance provisions without the need to consider M-V strength inter- action. These conclusions apply to both non-hybrid and hybrid I-girder designs. A paper version of the above report is under review for potential publication in the Journal of Structural Engineering, ASCE. White, D. W., and Jung, S.-K. (2003). “Simplified Lateral- Torsional Buckling Equations for I- and Channel-Section Members,”Structural Engineering,Mechanics and Materials Report No. 24a, School of Civil and Environmental Engineer- ing, Georgia Institute of Technology, Atlanta, GA, 23 pp. This report presents a recommended simplified form of the fundamental beam-theory LTB equations for doubly sym- metric members, specialized to the elastic LTB resistance of I- and channel-section members. This recommended form is exact for doubly symmetric I-section members and has the advantage of improved accuracy relative to the corresponding AASHTO (1998) equations when applied to singly symmetric I-shapes, including composite I-section members in negative bending. Also, it avoids significantly conservative errors that occur within the double-formula approach for a wide range of rolled wide-flange shapes. Furthermore, the physical sig- nificance of each of the terms within the simplified equations is easy to understand. These equations are utilized as the base elastic LTB expressions within the AASHTO (2004) and AISC (2005) specifications. White, D. W., and Jung, S.-K. (2003). “Simplified Lateral- Torsional Buckling Equations for Singly-Symmetric I-Section Members,”Structural Engineering,Mechanics and Materials Report No. 24b, School of Civil and Environmental Engineer- ing, Georgia Institute of Technology, Atlanta, GA, 29 pp. In the companion report, the authors develop and discuss the advantages of a recommended set of equations for the elastic lateral-torsional buckling (LTB) resistance of I- and channel-section members. These equations are utilized as the base elastic LTB expressions within the AASHTO (2004) and AISC (2005) specifications. The companion report focuses on the characteristics of the recommended equations pertaining to doubly symmetric I-section members and channels.This report extends these developments to singly symmetric I-section members, including composite I-sections in negative bending A-42

and channel-capped I-sections. Two approaches are high- lighted for calculation of the elastic LTB resistance of these general member types: 1. Ad hoc application of the doubly symmetric equations recommended in a companion report,which is similar to the approach taken in AASHTO (1998) using an alternative set of LTB equations for doubly symmetric I-section members. 2. A simplified form of the rigorous equations obtained from open-walled section beam theory. A key advantage of the first approach is that it leads to a single set of equations for all I-section members and channels. Also, these equations are simpler to apply than the rigorous beam theory equations for general I-shapes.The recommended doubly symmetric equations give an improved approximation of the rigorous beam theory solution for singly symmetric I-section members compared to the AASHTO (1998) equa- tions.Also, these equations can be applied as a conservative but typically adequate approximation of the elastic LTB resistance for composite I-section members in negative bending. The main disadvantage of the first approach is that it is not rigor- ous. Therefore, the behavior of the equations must be studied parametrically to ensure that they predict the physical strengths adequately for all practical singly symmetric geometries. Any ranges of parameters that produce unacceptable error must be disallowed. The key advantage of the second approach is that an exact or a highly accurate approximation of the beam theory LTB resistance is obtained. The primary disadvantage of this approach is that the equations are more complex. Also, their extension to the handling of composite I-girders in neg- ative bending is not as straightforward. White,D. W.,and Jung,S.-K.(2003).“Effect of Web Distortion on the Buckling Strength of Noncomposite Discretely-Braced I-Beams,”Report No. 24c, School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 34 pp. The influence of the web distortional flexibility typically is not addressed explicitly in standards for flexural design of steel I-section members. This is due in part to the fact that there are no simple closed-form solutions that account for these effects, whereas closed-form solutions are well established for elastic lateral-torsional buckling using open-walled section beam theory. Open-walled section beam theory is of course based on the assumption that the cross-section profile does not deform. Also, strength predictions are generally good for a wide range of experimental tests without explicit accounting for web distortion in the lateral-torsional buckling calcula- tions. Nevertheless, it is clear from prior research studies that the web distortional flexibility can lead to a substantial reduc- tion relative to the beam theory lateral-torsional buckling resistance for I-sections with torsionally stiff flanges and rel- atively thin webs. AASHTO (2004) and AISC (2005) give spe- cific limits on the cross-section geometry and yield strengths intended to control the unconservative errors associated with the neglect of web distortion effects. This report evaluates the effectiveness of these limits. Potential future directions for improved calculation of I-section member flexural resistances are suggested for cases where the influence of web distortion is significant. A paper version of the above report is under review for potential publication in the Journal of Engineering Structures. White, D. W. (2002). “LRFD Article 6.10 Draft # 3, Sample Case Studies,”Report to AISC-AASHTO Task Committee for Updating of AASHTO LRFD Article 6.10, October, 84 pp. The goal of this report is to provide engineers with an over- all conceptual understanding of how the nominal strengths defined by the proposed resistance equations in Draft # 2 of AASHTO Article 6.10 relate to the nominal strengths defined within the current AASHTO LRFD (2001) Specifications, the AISC LRFD (1999) Specifications, the AASHTO Guide Spec- ifications for Curved Steel Bridge Design (2002), and several additional references and analytical solutions of important note. This is achieved by focusing on specific example case studies. The report addresses the following specific limit states: 1. Generic influence of flange and web slenderness in straight I-girders, 2. Lateral-torsional buckling (LTB) for various key categories of I-shaped member geometries in which the flanges are compact, 3. Strengths as a function of the unbraced length for girders where the maximum resistance is influenced by compression flange local buckling (FLB), 4. Strength and ductility of composite I-girders in positive bending, and 5. Strength under combined major-axis flexure (vertical bending) and flange lateral bending, using members from the second and third of the above groups. A total of 13 different cases are considered pertaining to straight I-girders with zero lateral bending. Three specific examples and results from a focused parametric study are presented for composite I-girders in positive bending. Four cases are considered with respect to strength under combined vertical and flange lateral bending. Except where noted other- wise, all of the studies are focused on homogeneous members with Fy = 50 ksi. A-43

White, D. W., Zureick,A. H., Phoawanich, N., and Jung, S.-K. (2001). “Development of Unified Equations for Design of Curved and Straight Steel Bridge I-Girders,” Final Report to American Iron and Steel Institute Transportation and Infrastructure Committee, Professional Services Industries, Inc., and Federal Highway Administration, October, 551 pp. This research examines analytically and computationally the maximum strength behavior of curved and straight steel I-girders subjected to uniform vertical bending, high shear and low vertical bending moment, and high-shear and high- vertical bending moment, combined with lateral bending due to torsion and/or applied design loads. The theoretical and practical background, qualities, and limitations of existing design predictor equations are outlined. Key existing equations and new predictor equations developed as part of this research are evaluated based on the results of a reasonably comprehen- sive finite element parametric study, as well as the data from and finite element analyses of prior experimental tests. Based on consideration of the maximum strength predictions as well as pre- and post-peak load-deflection results, a unified set of strength equations is recommended that can be applied to both curved and straight I-girders for all loading conditions, including lateral bending and torsion. White, D. W., and Barth, K. E. (1998).“Strength and Ductility of Compact-Flange I Girders in Negative Bending,” Journal of Constructional Steel Research, 45(3), 241–280. This paper reviews available experimental and finite element test data pertaining to the negative moment-plastic rotation behavior of continuous-span steel I-girders with compact or ultra-compact flanges. Current American specification formu- las for the pier-section strength of these types of members and a moment-plastic rotation model recently developed by the authors are examined against the experimental and finite element test results. Several weaknesses in current specifica- tion provisions are observed. The new M-θp model avoids these weaknesses and provides a lower-bound approximation of the complete moment-plastic rotation response at the pier section. White,D. W.,Ramirez,J. A.,and Barth,K. E.(1997).“Moment Rotation Relationships for Unified Autostress Designs of Continuous-Span Bridge Beams and Girders.”Final Report, Joint Transportation Research Program, West Lafayette, IN, 117 pp. This report summarizes the development and trial appli- cation of simplified moment-rotation relationships for inelastic design of continuous-span beam and girder bridges. The research described within involves the execution of a reason- ably comprehensive set of finite element parametric studies to fill in knowledge gaps in the available experimental data pertaining to the hogging moment-plastic rotation behavior of steel and composite steel-concrete bridge girders. Based on these studies, relationships have been developed for the moment-plastic rotation behavior at the pier sections in these types of bridges. The moment-rotation model is validated against available experimental data, several focused new exper- imental tests, and current American specification strength formulas. This study concludes with a detailed trial inelastic design of a three-span continuous plate-girder bridge using suggested new inelastic design procedures. The characteristics of the calculations and the resulting proportions are compared with those of an elastic design of the same bridge by current AASHTO LRFD procedures. The elastic design is a modified version of a three-span continuous plate-girder bridge example recently published by the American Iron and Steel Institute for a Highway Structures Design Handbook. Wright, D. W., and Beshah, F. (2006). “Construction of Test Bridge,” Volume 8, Curved Steel Bridge Research Project, Federal Highway Administration, November. As part of the Federal Highway Administration’s curved steel bridge research project (CSBRP), to further perform experimental and analytical study in the fundamental behavior of horizontally curved steel bridges, response of the super- structure of full-scale simple-span composite curved steel I-girder bridge during all phases of construction sequence was monitored. The bridge was a simple span consisting of three I-girders spaced at 2,667 mm. A 203-mm-thick concrete deck, consisting of radial and longitudinal reinforcement, was composite with the steel girders. The bridge was instru- mented with strain gages, potentiometers, load-cells, linear voltage displacements transducers, and tilt meters. This volume, Volume 8 of the final report, presents the response of the superstructure during all phases of the con- struction sequence. Zureick, A. H., White, D. W., Phoawanich, N., and Park, J. (2002).“Shear Strength of Horizontally Curved Steel I Gird- ers—Experimental Tests,” Volume 4, Curved Steel Bridge Research Project, Federal Highway Administration, March, 157 pp. This report presents the results of four full-scale curved steel I-girder component tests conducted to examine their shear behavior and to determine their maximum shear strengths. The girders were made of AASHTO M270 Grade 345 steel A-44

and had a nominal web depth and web thickness of 1,219 mm (48 in) and 8 mm (5/16 in), respectively. The resulting nominal web slenderness ratio D/tw was 154. Two of the girders, labeled as S1 and S1-S, had a nominal radius R = 63,630 mm (208.75 ft) and a transverse stiffener spacing such that the ratio do/D was 3 for S1 and 1.5 for S1-S (producing do/R = 0.0575 and 0.0287). The other two test components, labeled as S2 and S2-S, were identical to S1 and S1-S except that their radii were 36,580 mm (120 ft), resulting in do/R = 0.10 and 0.050. All of the girders were braced against radial deflections at intervals of 3,658 mm (12 ft) along the girder arc. Therefore, the ratio Lb/R was equal to 0.0575 for S1 and S1-S and 0.10 for S2 and S2-S, where Lb is the distance between the bracing systems along the girder arc. All the test girders were instrumented to determine their maximum shear resistance as well as the mechanisms associated with the development of their shear strengths. Of particular interest was the extent to which the curved webs were capable of developing postbuckling strength, and the influence of the horizontal curvature and panel aspect ratio on the development of this strength. A-45