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Live Load Distribution Factors for Concrete Box Girder Bridges Shin Tai Song1 Y H Chai2 and Susan E Hida3 Abstract The current American Association of State Highway and Transportation Offi cials AASHTO Load and Resistance Factor Design LRFD Specifi cations impose fairly strict limits on the use of its live load distribution factor for design of highway bridges These limits include requirements for a prismatic cross section a large span length to width ratio and a small plan curvature Refi ned analyses using 3D models are required for bridges outside of these limits These limits place severe restrictions on the routine design of bridges in California as box girder bridges outside of these limits are frequently constructed This paper presents the results of a study investigating the live load distribution characteristics of box girder bridges and the limits imposed by the LRFD specifi cations Distribution factors determined from a set of bridges with parameters outside of the LRFD limits are compared with the distribution factors suggested by the LRFD specifi cations For the range of parameters investigated results indicated that the current LRFD distribution factor formulas generally provide a conservative estimate of the design bending moment and shear force DOI 10 1061 ASCE 1084 0702 2003 8 5 273 CE Database subject headings Live loads Load distribution Bridges girder Bridges concrete Box girders California Introduction Concrete multicell box girder bridges are one of the most com mon types of highway bridges built in California Voids are intro duced in the superstructure of these structures to reduce its self weight while maintaining a relatively large fl exural and torsional stiffness as well as strength These structures are not only very effi cient in terms of their ability to resist external loads they may also hide unsightly utilities in the interior and enhance the aes thetics of the surrounding environment Under live load condi tions a vehicle traveling on a bridge will fi rst transfer its loads including vehicular contents and any forces associated with dy namic effects to the bridge deck The deck then acts like a con nection element distributing the load to different girders The dis tribution of traffi c loads to different girders as commonly characterized by a live load distribution factor is conceptually an estimate of the portion of the traffi c load distributed to each girder The live load distribution factor is extensively used to streamline the design of bridges in the United States For most bridges including multicell box girder bridges dis tribution of traffi c loads to the different girders is not uniform The girder closest to the traffi c load is expected to resist the largest portion of the load The live load distribution factor g is commonly defi ned Barker and Puckett 1997 AASHTO 1998 as g Fdistributed Fbeamline 1 where Fdistributedcorresponds to the largest bending moment or shear force distributed to the girder for all load combinations while Fbeamlinecorresponds to the maximum bending moment or shear force determined from a simple beam line analysis of one lane of traffi c assuming that the bridge superstructure can be idealized as a continuous beam For routine design of bridges the distribution factor g is used in conjunction with results from the simple beam line analysis to estimate the design bending moment or shear force in the girder i e Bending Moment or Shear Force in the Girder5Fbeamline3g 2 Guidelines for estimating the live load distribution to different girders are currently available in various bridge design codes For example the 1998 AASHTO LRFD Bridge Design Specifi cations AASHTO 1998 referred to herein as the LRFD specifi cations provide a set of distribution factor formulas for estimating the bending moment and shear force in the interior and exterior gird ers These formulas which are primarily based on a study by Zokaie et al 1991 express the distribution factor in terms of a set of simple geometric parameters such as span length girder spacing overall depth of the girder number of cells etc The current set of distribution factor formulas in the LRFD specifi ca tions are intended for lane loads instead of wheel loads and dif ferent formulas are provided for single lane loading and multiple lanes loading Note that a multiple presence factor m as reproduced in Table 1 AASHTO 1998 is also embedded in the formulas to refl ect the reduced likelihood of all traffi c lanes being loaded simultaneously Thus the distribution factor formulas rec ommended by the LFRD specifi cations strictly correspond to mgformula 1Graduate Research Assistant Dept of Civil and Environmental Engineering Univ ofCalifornia Davis CA95616 E mail ssong ucdavis edu 2Associate Professor Dept of Civil and Environmental Engineering Univ of California Davis CA 95616 E mail yhchai ucdavis edu 3Senior BridgeEngineer CaliforniaDept ofTransportation Sacramento CA 95816 E mail Susan Hida dot ca gov Note Discussion open until February 1 2004 Separate discussions must be submitted for individual papers To extend the closing date by one month a written request must be fi led with the ASCE Managing Editor The manuscript for this paper was submitted for review and pos sible publication on November 6 2001 approved on September 23 2002 This paper is part of the Journal of Bridge Engineering Vol 8 No 5 September 1 2003 ASCE ISSN 1084 0702 2003 5 273 280 18 00 JOURNAL OF BRIDGE ENGINEERING ASCE SEPTEMBER OCTOBER 2003 273 Objectives Although the objective of the distribution factor formulas is to simplify the design process fairly strict restrictions are imposed by the LRFD specifi cations on the use of these formulas Limits are placed on the bridge curvature the span length to width ratio and the number of bridge girders and requirements for prismatic cross sections with parallel girders are imposed More specifi cally the LRFD specifi cations distribution factor formulas may only be used for design of bridges with 1 constant deck width 2 number of girders not less than four 3 parallel beams with approximately equal stiffness 4 roadway overhand less than 910 mm 5 superstructure span length exceeding 2 5 times its width and 6 angular change of less than 12 in plan for a torsionally stiff closed section AASHTO 1998 Box girder bridges outside of these limits are frequently con structed in California consequently such constraints place severe restrictions on the routine design of bridges Design outside of these limits requires refi ned 3D models such as grillage or fi nite element models which are currently not a part of the standard bridge design process and often require special skills and analysis tools Among the aforementioned restrictions restrictions 1 3 5 and 6 are particularly restrictive A study was recently conducted on the limits imposed by constraint numbers 1 3 5 and 6 The study compared the distribution of live load bending moments and shear forces with those suggested by the LRFD specifi cations for a typical range of parameters encountered in concrete box girder bridges in California Methodology Bridges outside the limits of the current LRFD specifi cations were analyzed using refi ned and simple beam line models so that distribution factors for bending moment and shear force could be determined Distribution factors thus obtained are denoted as ganalysisto signify that the distribution factor is obtained from the analysis instead of the formula ganalysis5 F refi ned Fbeamline 3 where F refi nedcorresponds to the largest bending moment or shear force in the girder from the refi ned analysis while Fbeamlinecor responds to the maximum bending moment or shear force from the simple beam line model Barker and Puckett 1997 subjected to one lane of traffi c The distribution factor ganalysisis then mul tiplied by the appropriate multiple presence factor m and com pared with the distribution factor obtained using the LRFD speci fi cations formulas which is denoted as mgformula in order to allow an assessment of the level of conservatism in the formulas Refi ned Model The commonly used grillage model is used as the refi ned analysis tool in this study Selection of properties and implementation of the grillage model for box girder bridges follows the general guidance available in the literature e g Hambly 1991 Barker and Puckett 1997 O Brien and Keogh 1999 Before proceeding with the actual usage of the grillage model for distribution factors the grillage model was fi rst calibrated against a fi nite element model in order to gain confi dence in the model The basic dimensions of a continuous box girder bridge as well as one of the loading conditions used for the calibration of the grillage model are shown in Fig 1 a The box girder has two equal spans each with a length of 30 3 m and has four cells with equal spacing between girders The overall depth of the box girder is 1 263 m and the edge to edge width is 12 21 m The loading condition shown in Fig 1 a consists of two point loads applied off center to induce signifi cant twisting of the box girder in addition to bending moment and shear force Figs 1 b and c show the distribution of bending moment and shear force in one of the longitudinal girders girder 2 The con tinuous line corresponds to the grillage model whereas the dot symbol corresponds to the fi nite element model As can be seen in Fig 1 b a sharp increase in bending moment occurs under the point load resulting in the largest bending moment occurring at thatlocation Ingeneral thebendingmomentfrom Table 1 Number of Loaded Lanes and Corresponding Multiple Presence Factor Number of loaded lanesMultiple presence factor 11 20 21 00 30 85 4 or greater0 65 Fig 1 Comparison between grillage and fi nite element models a two span continuous bridge structure b bending moment distribu tion girder 2 c shear force distribution girder 2 274 JOURNAL OF BRIDGE ENGINEERING ASCE SEPTEMBER OCTOBER 2003 the grillage model compares very well with the bending moment from the fi nite element model even though the bending moment from the fi nite element model is slightly larger However the dif ference in bending moment between the two models is within 5 Similarly the shear force distribution shown in Fig 1 c also indicates very good agreement between the two models The gril lage model is able to simulate very well the discontinuity in the shear force under the point load and at the continuous support as can be seen by the good agreement between the grillage model and the fi nite element model Although not shown in Fig 1 c the comparison of vertical shear forces in the exterior girder between the grillage and fi nite element models is also very good despite the inclined web in the exterior girder Although not presented in this paper four additional load cases were used for the calibration of the grillage model These additional load cases were selected to simulate the large negative bending moment over the support the large positive bending mo ment in the loaded span and associated negative bending moment in the adjacent span and the large shear force near the support In general the simulation of bending moment and shear force by the grillage model compares very well with the fi nite element model Details of the calibration of the grillage model can be found in Song et al 2001 Traffi c Loads Unlike the calibration of the grillage model where point loads were used the distribution factors were calculated using the AASHTO HL 93 design vehicular loads AASHTO 1998 in this study The HL 93 loads consist of a single design truck combined with a design lane load referred to as HL93K or a single design tandem combined with a design lane load referred to as HL93M Extreme load effects as characterized by the largest positive and negative bending moments and shear forces are determined using the HL 93 load combinations per LRFD specifi cations AASHTO 1998 In addition dual trucks and dual tandems are used for the maximum negative bending moment AASHTO 1998 Figs 2 a and b show the elevation of the axle loads and lane load associ ated with HL93K and HL93M while Figs 2 c and d show the axle loads associated with the dual trucks and the dual tandems It should be noted that a 33 dynamic load allowance is applied to the design truck or design tandem but not to the design lane load For dual trucks and dual tandems on the other hand their mag nitudes are reduced to 90 including that of the design lane load For the design truck and design tandem the transverse spacing of the wheels is 1 8 m Further details of the vehicular loads are given in Song et al 2001 Results The accuracy or conservatism of the LRFD specifi cations distri bution factor formulas may be assessed by comparing the distri bution factors from the refi ned analyses i e Eq 3 with the distribution factors calculated from the LRFD formulas To this end an acceptance ratio RA is defi ned as RA m3ganalysis mgformula 5 m3F refi ned Fbeamline3mgformula 4 where m5multiple presence factor as given in Table 1 An ac ceptance ratio of RA 1 indicates that the distribution factor rec ommended by the LRFD specifi cations formula is conservative Discussions of the results are primarily based on this parameter and are separated into the following subsections 1 nonprismatic cross section 2 small plan aspect ratio and 3 curved bridges Nonprismatic Cross Section One of the rather restrictive constraints imposed by the LRFD specifi cations for the use of its distribution factor formulas in conjunction with the simple beam line analysis is the requirement for the girders of the box girder to be parallel and for the deck width to be constant Bridges with variable width and nonparallel girders which violate constraints 1 and 3 discussed previously are frequently used in bridge structures to accommodate the sepa ration or merging of traffi c lanes For the study of possible application of the LRFD distribution factor formulas to bridges with a nonprismatic cross section two continuous box girder bridges with dimensions shown in Fig 3 were analyzed These bridges have two equal spans each with a span length of 30 3 m The nonprismatic cross section bridge shown on the left hand side of Fig 3 is typical of an on off ramp structure where widening of the bridge is often required In this case the widening of the bridge is facilitated by a fl are of 6 25 starting from 0 2L in one span and continuing full length in the second span It is worth noting that the rate of widening is close to the practical geometric limit of 6 7 for widening of bridges in California California Department of Transportation 2001 An additional girder is added to the widened portion of the bridge resulting in nonparallel girders For comparison of results a pris matic bridge with parallel girders shown on the right hand side of Fig 3 was also analyzed Except for the deck width the pris matic bridge is similar to that of the bridge with the nonprismatic cross section Note that a nominal deck width of 11 36 m as adopted for these structures can only accommodate two traffi c lanes at most after allowing for the width of the barriers In order to create the most severe load condition while maintaining a rea Fig 2 AASHTO design vehicular live loads a HL 93K design truck with lane load b HL 93M design tandem with lane load c dual trucks d dual tandems JOURNAL OF BRIDGE ENGINEERING ASCE SEPTEMBER OCTOBER 2003 275 sonable number of analyses traffi c lanes are positioned as close to the bridge girder as possible and follow the closest fl are girder when the bridge widens Acceptance ratios for bridges with prismatic and nonprismatic cross sections are presented in Table 2 In all cases acceptance ratios are less than unity irrespective of whether one or two traf fi c lanes are loaded For interior girders of the nonprismatic bridge acceptance ratios are 0 72 and 0 77 for positive bending moment 0 90 and 0 88 for negative bending moment and 0 95 and 0 96 for shear force for one and two lanes loaded respec tively For exterior girders of the nonprismatic bridge acceptance ratios are 0 52 and 0 74 for positive bending moment 0 66 and 0 86 for negative bending moment and 0 61 and 0 78 for shear force for one and two lanes loaded respectively The LRFD dis tribution factor formula is thus more conservative for the exterior girder than for the interior girder The acceptance ratios presented in Table 2 indicate that the distribution factor formulas in the LRFD specifi cations are applicable to the two span bridge in this study which has nonparallel girders or a nonprismatic cross sec tion Asimilar level of conservatism in the LRFD distribution factor formulas is also observed for the prismatic bridge even though the acceptance ratio varies somewhat between the exterior and interior girders For the interior girder the acceptance ratio RA varies between 0 8 and 1 0 as can be seen in Table 2 for both positive and negative bending moments and for shear force An acceptance ratio of 1 0 is calculated for the shear force of the interior girder when the structure is subjected to a one lane load ing For the exterior girder the acceptance ratio RAvaries be tween 0 54 and 0 83 for both bending moment and shear force and it is therefore smaller than that of the interior girder Although not presented in this paper actual values of the distribution factor are available in Song et al 2001 Fig 3 Sections and plan views of two span continuous box girder bridge with prismatic and nonprismatic cross section Table 2 Comparison of Acceptance Ratios between Bridges with Prismatic and Nonprismatic Cross Section Bridge One lane loaded m51 2Two lanes loaded m51 Positive momentNegative momentShear forcePositive momentNegative momentShear force Interior girders Nonprismatic cross section0 720 900 950 770 880 96 Prismatic cross section0 850 931 000 880 890 83 Exterior girders Nonprismatic cross section0 520 660 610 740 860 78 Prismatic cross secti