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沙头坡
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沙头坡高架桥施工图设计,沙头坡,高架桥,施工图,设计
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span concrete bridges represent about 163,000 with 23% of thosesolete. The majority are short spans, averaging less than 15 m50 ftin length. The high number of decient bridges means thata considerable number of bridges are being recommended forweight limiting posting, rehabilitation, or decommissioning andreplacement. Reinforced concrete slab bridges may offer eco-nomic alternatives for short-span bridges in the United States andparticularly in developing countries where cast-in-place concreteis common practice. The main advantage of cast-in-place concreteslab bridges is the ability to eld adjustment of the roadway pro-le during construction. Typically, the design of highway bridgesin the United States must conform to the AASHTO StandardsSpecications for Highway Bridges2003or AASHTO LRFDDesign Specications2004. The analysis and design of anyhighway bridge must consider truck and lane loading. However,truck loading provisions govern for short-span structures whenconsidering AASHTO Standard Specications. AASHTO speci-es a distribution width for highway loading to reduce the two-way bending problem into a beamone-waybending problem.Alternately, an empirical expression for live-load bending mo-ment is provided. Therefore, reinforced concrete slab bridges aredesigned as a series of beam strips. The AASHTO design proce-dures were originally developed in the 1940s based on researchwork by Westergaard1926, 1930and Jensen1938, 1939.Mabsout et al.2004reported the results of parametric inves-tigation using the nite-element analysis of straightnonskewed,single-span, simply supported reinforced concrete slab bridges.The study considered various span length and slab widths, num-Inuence of Skew Angle on Reinforced ConcreteSlab BridgesC. Menassa1; M. Mabsout2; K. Tarhini3; and G. Frederick4Abstract:The effect of a skew angle on simple-span reinforced concrete bridges is presented in this paper using the nite-elementmethod. The parameters investigated in this analytical study were the span length, slab width, and skew angle. The nite-element analysisFEAresults for skewed bridges were compared to the reference straight bridges as well as the American Association for State Highwayand Transportation OfcialsAASHTOStandard Specications and LRFD procedures. A total of 96 case study bridges were analyzedand subjected to AASHTO HS-20 design trucks positioned close to one edge on each bridge to produce maximum bending in the slab. TheAASHTO Standard Specications procedure gave similar results to the FEA maximum longitudinal bending moment for a skew angle lessthan or equal to 20. As the skew angle increased, AASHTO Standard Specications overestimated the maximum moment by 20% for30, 50% for 40, and 100% for 50. The AASHTO LRFD Design Specications procedure overestimated the FEA maximum longitudinalbending moment. This overestimate increased with the increase in the skew angle, and decreased when the number of lanes increased;AASHTO LRFD overestimated the longitudinal bending moment by up to 40% for skew angles less than 30 and reaching 50% for 50.The ratio between the three-dimensional FEA longitudinal moments for skewed and straight bridges was almost one for bridges with skewangle less than 20. This ratio decreased to 0.75 for bridges with skew angles between 30 and 40, and further decreased to 0.5 as the skewangle of the bridge increased to 50. This decrease in the longitudinal moment ratio is offset by an increase of up to 75% in the maximumtransverse moment ratio as the skew angle increases from 0 to 50. The ratio between the FEA maximum live-load deection for skewedbridges and straight bridges decreases in a pattern consistent with that of the longitudinal moment. This ratio decreased from one for skewangles less than 10 to 0.6 for skew angles between 40 and 50.DOI:10.1061/ASCE1084-0702200712:2205CE Database subject headings:Bridges, skew; Concrete slabs; Finite element method; Bridges, highway; Concrete, reinforced.Introductionbridges considered to be structurally decient or functionally ob-Skewed bridges are often encountered in highway design whenthe geometry cannot accommodate straight bridges. Highwaybridges are characterized by the angle formed with the axis of thecrossed highway. The skew angle can be dened as the anglebetween the normal to the centerline of the bridge and the center-line of the abutment or pier cap, as described in Fig. 1.According to the U.S. Federal Highway Administrations 2004National Bridge Inventory data, about 25% of the nations594,470 bridges are structurally decient or functionally obsoleteas reported in Better Roads MagazineNovember 2004. Single-1Dept. of Civil and Environmental Engineering, American Univ. ofBeirut; and, Project Engineer at Dar Al-Handasah, Beirut, Lebanon.E-mail: carolmenassa2Dept. of Civil and Environmental Engineering, American Univ. ofBeirut, Lebanon. E-mail: mounir.lb3Gannett Fleming, Inc., One Penn Plaza, Suite 2222, New York, NY10019; formerly, STV, Inc., 80 Ferry Blvd., Stratford, CT 06615. E-mail:kassim, E-mail: ktarhini4Dept. of Civil and Environmental Engineering, Univ. of Nevada, LasVegas, NV 89154.Note. Discussion open until August 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be led with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on August 23, 2002; approved on July 15, 2005. Thispaper is part of theJournal of Bridge Engineering, Vol. 12, No. 2,March 1, 2007. ASCE, ISSN 1084-0702/2007/2-205214/$25.00.JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007 /205Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visit1aM= 10001.30S 202aM= 100019.5S 90Description of a skewed bridge2. AASHTOvalue by twice the distribution width,EE= 4 + 0.06S 7.0 ftwhich, in SI units, is equivalent toE= 1.2 + 0.06S 2.1 m3. Analysis and design of a unit wide strip using the appropriatewheel loads. For HS20 loading, the wheel loads are 18 KN4 Kips,72KN16 Kips, and 72 KN16 Kipswith axlespacing of 4.2 m14 ft. The appropriate wheel loads arethen divided by the distribution widthEEq.3. This ap-proach is generally used for continuous spans and currentlyadopted in the AASHTO LRFD Design Specications.This paper considered only the AASHTO empirical formula,given in Eq.1or Eq.2, when compared with the nite-element results. AASHTO Section 3.2.6suggests that slabbridges with a skew angle less than 30 be designed as a typicalslab at right angles with no modications. However, if the skewangle exceeds 30, AASHTO suggest the use of an alternate su-perstructure conguration.AASHTO requires edge beams along the free edges of thebridge slabs. The live-load bending moment in an edge beam isspecied by the expression: 0.1PSwhereP=72 KN or 16 kipsfor HS20 truck. AASHTO does not specify a width for the edgebeam. However, some departments of transportation such asOhiouse an edge beam width of 450 mm18 in. The maxi-mum FEA live-load deection was compared with the AASHTOdeection criterion ofS/800. The slab thickness was calculatedto control the live-load deection according to AASHTO Section8.9.2; the minimum thicknesshftfor a slab with main rein-forcement parallel to trafc is 1.2S+10/30, which is equivalent,in SI unitsmm,to1.2S+3000/30. Finally, AASHTO givesforS 50 ftforS 15 mAppendix Agives live-load bending moment per3a3b1b2bobtained from the nite-element analysis supported the AASHTOStandard Specications recommendations for skewed bridges.This paper presents the results of a parametric study which evalu-ated the effect of the skew angle on the load distribution in single-span, simply supported, reinforced concrete slab bridges. Bendingmoments and deections of skewed bridges were compared withstraight bridges, AASHTO Standard Specications, and LRFDprocedures.AASHTO Standard SpecicationsFor simply supported slab bridges, AASHTO Standard Specica-tions2003suggest three approaches to determine the live-loadbending moment for HS20 design truck loading.1. AASHTOSection provides empirical equationsM= 900S forS 50 ftorwhich, in SI units, are equivalent toM= 13500S forS 15 morwhereSspan lengthft in Eq.1ormin Eq.2; andMlongitudinal bending moment per unit widthlb-ft/ft inEq.1or N-m/m in Eq.2due to live load.Fig. 1.lane for span length up to 90 m300 ft. The live-load bend-ing moment per foot of width is obtained by dividing thisber of lanes, and live loading conditions for bridges with andwithout shoulders. Longitudinal bending moments and deectionsin the slab were evaluated and compared with procedures recom-mended by AASHTO. The FEA results for bridges subjected towheel loading placed on one edge of the slab showed that theAASHTO Standard Specications procedure overestimated thebending moment by 30% for one-lane and span lengths less than7.5 m25 ftand agreed with the FEA bending moments forlonger spans. The AASHTO expression for live-load bending mo-ment gave similar results to the FEA when considering two ormore lanes and span lengths less than 10.5 m35 ft. However, asthe span length was increased, AASHTO underestimates themaximum FEA longitudinal bending moment by 1530%. It wasalso shown that the presence of shoulders on both sides of thebridge increased the load carrying capacity of the bridge. TheAASHTO LRFD procedure gives higher bending moments thanAASHTO Standard Specications as well as the FEA results.AASHTO LRFD procedure gave closer design bending momentsto the FEA results when considering slabs with shoulders on bothsides and subject to live load in each lane in addition to a disabledtruck on one shoulder. Schickel et al.1999showed that theaverage strain values calculated using AASHTO procedures to be30% larger than the strain values obtained in the laboratory re-gardless of the skew angleup to 30when considering one truckper lane. However, placing an additional disabled truck on theshoulder, the average experimental strain was similar to the strainvalues obtained using AASHTO procedures. Therefore, the actualstrength of concrete slab bridges with a skew angle less than 30was shown to be predicted using AASHTO procedures. A prelimi-nary study by Mabsout et al.2002investigated the effect ofincreasing the skew angle on the live-load distribution for onesingle-span reinforced concrete slab bridge. The limited results206/ JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visit54 ftwith corresponding solid slab thicknesses of 450, 525, 600,and 675 mm18, 21, 24, and 27 in., respectively. One, two,three, and four lane bridges without shoulders at both free edgeswere investigated. The overall slab widths for the bridges wereselected to be the worst cases that may be encountered in practice:noted that the results could be applied to voided slabs to reducecombining the spans and lanes considered with skew angles vary-ing between 0 and 50 by increments of 10. Straight bridges withzero skew angles served as a reference for comparison withskewed bridges. In total, 96 bridge cases were analyzed and as-sessed using nite-element analysis.In a previous study by Mabsout et al.2004related to straightreinforced concrete slab bridges, two possible transverse loadingpositions of the design trucks were considered:acentered con-dition where each truck is centered in its own lane as prescribed4a4bfor the edge condition was intentionally selected for this study toproduce the worst live loading condition on the bridge. Placingthe HS20 truck wheel loads this close in the transverse directionis not in accordance with the AASHTO load-ing requirements of one truck per lane. However, it meetsAASHTO LRFD Section .1 provision of a minimum 1.2 mdistance between the wheels of adjacent design trucks.Therefore, the FEA models and HS20 wheel load arrangementsare expected to generate higher bending moments due to thisimposed live loading condition. Placing the wheel load at 0.3 mfrom the free edge could be considered critical or overesti-mating the longitudinal bending moment. However, AASHTOspecies a minimum distance of 0.6 m2ftrailing to be more realistic and practical. Therefore, bridges withEdge Load were reanalyzed further by placing the wheel load ofthe left truck 0.6 m2ftfrom the parapet0.3 m or 1 ftwhichtotals 0.9 m3ftinstead of 0.3 m1ftfrom the edge. The FEA3 ft from the free edgeshown that the “Edge Load” resulted in higher maximum mo-Since the objective of this investigation is to evaluate theload at 0.3 m or 1 ft from the free edgeand edge beam longitudinal moments. Design trucks traveling inthe same direction were considered to be severe for the live load.AASHTO Standard Specications Section 3.6Trafc Lanesas-sumes that lane loading or standard design truck occupies a widthof 3 m10 ft. Therefore, according to AASHTO LRFD Section.1 the minimum tolerable distance between wheel loads ofadjacent trucks was set to be 1.2 m4ftworst wheel loading conditions in the slab. A typical layout of atwo-lane skewed bridge with edge loading is shown in Fig. 2.In this paper, selected two-lane bridges with 10.8 m36 ftspan lengths were rst analyzed for centeredand edge wheel loadings, considering six skew angles0, 10, 20,30, 40, and 50. The FEA results of skewed bridges reconrmedby AASHTO, andbedge condition where the design trucks areplaced close to one edgeleftof the slab, such that the center ofthe left wheel of the left-most truck is positioned at one foot fromthe free edge of the slab. The distance between the adjacent trucks5a1.2 m or 4 ft apart4ft1ftfrom the curb orshowed only a 5% difference. It was also6edge loading condition isin order to produce theand 16.2 m54 ftbe 1.2 m4ftcenter-to-center spacing, in order toor 3 m10 ftments than the “Centered Load.”special provisions for transverse reinforcement placed perpen-dicular to the main steel reinforcement in bridge slabs. Theamount of distribution reinforcement is given as a percentage ofthe main reinforcement equal to 54.8/S, where S is in m100/S,Sin ft, and shall not exceed 50%.4.2 m14 ftfor one lane, 7.2 m24 ftfor two lanes, 10.8 m36 ftfor three lanes, and 14.4 m48 ftfor four lanes. It isAASHTO LRFD Bridge Specicationsthe dead load of the bridge. The effect of skewness is studied byAASHTO LRFD Section provides an equivalent stripwidth to design slab bridges similar to the AASHTO StandardSpecications. This simplistic approach to divide the total staticalmoment by the bridge width to achieve a moment per unit width.The moments are determined by establishing the structural widthper design lane. The equivalent widthEof longitudinal strips perlane for both shear and moment is determined using the followingformulas:Width for one lane loaded isE= 250 + 0.42L1 W11/2or E=10+5L1W11/2Width for multilanes loaded isE= 2100 + 0.12L1W11/2or E= 84 + 1.44L1W11/2 5bwhereEis in mm in Eqs.4aand5ainches in Eqs.4band5b;L1span length in mmft, the lesser of the actual span or18,000 mm60 ft;W1edge-to-edge width in mmftof bridgetaken to be the lesser of the actual width or 18,000 mm60 ftformultilane loading, or 9000 mm30 ftfor single-lane loading.AASHTO LRFDSection live load HL93 requires theconsideration of lane loading plus design truckHS20or laneloading plus tandem. The bending moment is determined for thedesign lane and is then divided by the widthEto determine thedesign moment per unit width. AASHTO LRFD TableA.3-1 provides the minimum slab thicknesshto be 1.2S+3000/30, wherehandSare in mm, which is similar to theAASHTO Standard Specications equation 1.2S+10/30ft.For skewed bridges, AASHTO LRFD -3 reduces theresults of the two edge loading conditions E1wheel load atlongitudinal force effects by a factorrwhich is a function of the0.3 m or 1 ft from the free edgeand E3wheel load at 0.9 m orskew angler= 1.050.25 tan1whereis the skew angle in degrees.AASHTO procedure for calculating live-load bending momentsin reinforced concrete slab bridges. Therefore, only the E1wheelDescription of Bridge Casesconsidered further in this study in order to produce higher slabA typical skew bridge crossing another highway is shown in Fig.1. The skew angle denotes the angle between the axes ofsupportabutment or pierrelative to a line normal to the lon-gitudinal axis of the bridge. Based on this notation, a straightbridge is, therefore, dened as having a 0 skew angle.Typical simply supported, one-span, multilane bridge studycases were considered in this investigation. The simple supportcondition is used even though concrete slab bridges are tied di-rectly to abutments, thus creating partial xity which may impactthe results. The assumed condition, however, will not affect therelative comparison between straight and skewed cases, which isthe focus of this study. Four span lengths were considered in thisparametric study as 7.2, 10.8, 13.8, and 16.2 m24, 36, 46, andJOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007 /207Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visit0.3 m11ftmaximum longitudinal moment was dened as the rst peakvalue occurring after the maximum value near the left-most edge.beam moment. Figs. 4 and 5 show the typical longitudinal bend-ing moment across the critical width of two- and four-lane bridgesspan length and various skew angles. TheAASHTO Standard Specications empirical bending moment andLRFD procedure are also plotted in Figs. 4 and 5 for comparison.For the loading condition considered in the study, the maximumvalue of the transverse moment occurs underneath the concen-trated wheel loads in the slab.The FEA results for three- and four-lane short-span bridges7.2 m24 ftfor three lanes and 30 and higher for four lanesand reported in the tables, but were excluded from the discussionof results which follows. Those few cases, very short, wide, andhighly skewed, will rarely be considered in practice.AASHTO Standard Specications and AASHTO LRFDprocedures.Maximum Longitudinal Bending Moment and EdgeBeam MomentThe maximum slab and edge beam longitudinal bending momentsare summarized in Tables 1 and 2, respectively, for all bridgesanalyzed along with the corresponding AASHTO bending mo-ments. The AASHTO moments are computed using Eq.1orfor 10.8 m36 fthaving relatively high skew angles40 and higherwere analyzedFig. 3. Typical nite-element model for a 10.8 m 36 ft span,two-lane bridge, with 30 skewnessgitudinal bending moments. A typical square element size of 0.3was tested and adopted for the slab discretiza-tion. Quadrilateral and triangular elements were additionally usedat the supports to accommodate for skewness. Fig. 3 illustratesthe FEA discretization of a two-lane slab bridge subject to edgeloading and 30 skewness.The FEA results were obtained and reported in terms of themaximum longitudinal bending moment, edge beam moment,transverse moment, and live-load deection in the bridge. Themaximum longitudinal bending moments and edge beam mo-ments were obtained at the critical cross section of each slab. TheFig. 2.Typical two-lane skewed bridge with edge loadingThe maximum moment near the edge was assumed to be resistedby an edge beam, and will be referred to thereafter as the edgethe ndings from straight bridges that “Edge Load” gave higherbending moments than the “Centered Load.” Therefore, the cur-rent study was limited to placing truck loads closer to one edge. Itis also worth noting that only the left-most truck was centeredlongitudinally, while adjacent trucks were aligned with the edgetruck as shown in Fig. 2. This condition resulted in slightly highermoments than for the case where each adjacent truck was cen-tered longitudinally in its own lane.The material properties used in modeling the highway bridgeswere normal-strength reinforced concrete. The compressivestrength of the concrete was 27,500 KPa4,000 psi, the modulusof elasticity of 25106KPa3.6106psi, and the Poissonsratio 0.2. Grade 60 reinforcing steel could be assumed in thedesign of slab reinforcement but the FEA models did not includesuch property in the analysis.FEA Results versus AASHTOFinite-Element AnalysisThe FEA results of skewed bridges were primarily compared withThe general FEA program, SAP20001998, was used to generatethe three-dimensional3Dnite-element models. This study con-sidered all elements to be linearly elastic and the analysis as-sumed small deformations and deections. SAP2000 was used togenerate nodes, elements, and 3D meshes for the slab bridgesinvestigated. The concrete slabs were modeled using quadrilateralshell elementsSHELLwith 6 of freedom at each node. Hingeswere assigned at one bearing location and rollers at the other tosimulate simple support conditions. AASHTO HS20 wheel loadswere applied at isolated nodes in order to produce maximum lon-208/ JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visitmoment for all two-lane bridges. For three- and fourlanebridges, AASHTO underestimates the maximum moment byabout 20% for bridges with spans less than 7.5 m25 ft, and25% for bridges with span lengths greater than 10.5 m35 ft.The FEA edge beam moment results are similar to the AASHTOprocedure recommended moment for all three- and four-lanebridges. Considering the skew angle between 30 and 40, and forspan lengths between 7.5 and 16.5 m 25 and 55 ft, theAASHTO procedure overestimates the maximum longitudinalmoment and edge beam moment by about 40% for one-lanebridges, 30% for two-lane bridges, and 25% for three- and four-lane bridges. At relatively high skew angles, reaching 50,AASHTO overestimates the maximum longitudinal moment andedge beam moment by about 50% for all span lengths and slabFig. 4.Longitudinal bending moment distribution at critical section for 10.8 m36 ftspan, two-lane bridgesEq.2for the standard specications, and Eqs.4and5forLRFD after applying the necessary reduction for all skew anglesEq.6.The FEA maximum longitudinal bending moment and edgebeam moments for skew angles less than 30 were rst comparedto the AASHTO standard specications equations. For one-lanebridges with span lengths less than 10.5 m 35 ft, AASHTOoverestimates the maximum longitudinal bending moment byabout 25%, and the edge beam moment by about 20%. AASHTOgives similar results to the FEA for both moment values forbridges with span lengths greater than 13.5 m45 ft. For two-lane bridges, AASHTO gives longitudinal bending moment simi-lar to the FEA results for span lengths less than 13.5 m45 ft,and underestimates the maximum moment by about 20% whenthe span length is greater than 13.5 m45 ft. The FEA edgebeam moment results are similar to the AASHTO recommendedFig. 5.Typical longitudinal bending moment at critical section for 10.8 m36 ftspan, four-lane bridgesJOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007 /209Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. VisitAASHTOstandardNumberoflanes FEA LRFD FEA LRFD FEA LRFD FEA LRFD FEA LRFD FEA FEAbridges, 20% for three-lane bridges, and gives similar results forfour-lane bridges. The LRFD overestimate of the maximum lon-gitudinal moment increases almost linearly with the skew angleas it varies from 30 to 50. The LRFD overestimatesthe bending moment by about 55% for one-lane bridges,50% for two-lane bridges, and 45% for three- and four-lanebridges.AASHTONumber oflanes 0 10 20 30 40 50specicationsmomentKN-m/medge momentKN-m/mTable 1.Comparison of FEA Maximum Longitudinal Bending Moment with AASHTOFEA and LRFD maximum momentKN-m/mSkew angle0 10 20 30 40 50Spanlengthm7.2 1 74.5 126.5 72.6 126.5 67.7 121.4 60.5 113.9 53.0 106.3 43.3 94.9 97.22 92.3 108.5 89.6 108.5 86.0 104.2 74.3 97.7 62.1 91.1 58.5 81.43 97.7 101.7 95.4 101.7 88.7 97.6 76.1 91.5 77.4 85.4 83.3 76.34 100.4 96.8 99.0 96.8 90.5 92.9 90.5 87.1 106.7 81.3 94.5 72.610.8 1 131.0 212.4 130.5 212.4 120.6 203.9 104.4 191.2 87.8 178.4 69.3 159.3 145.82 158.9 205.2 155.3 205.2 144.0 197.0 124.2 184.7 102.6 172.4 81.9 153.93 168.8 190.4 167.4 190.4 154.4 182.8 131.9 171.4 106.2 159.9 81.9 142.84 176.0 180.0 174.2 180.0 159.3 172.8 134.1 162.0 105.8 151.2 81.9 135.013.8 1 189.0 283.1 184.1 283.1 168.8 271.8 146.7 254.8 121.1 237.8 96.3 212.3 186.32 226.8 293.9 220.5 293.9 201.6 282.1 172.8 264.5 139.1 246.9 108.9 220.43 238.1 271.8 233.6 271.8 212.0 260.9 181.8 244.6 145.4 228.3 110.7 203.94 248.4 269.1 243.0 269.1 219.6 258.3 189.9 242.2 146.3 226.0 111.2 201.816.2 1 234.9 338.9 229.1 338.9 210.2 325.3 183.6 305.0 152.6 284.7 121.5 254.2 225.92 281.7 367.7 275.4 367.7 252.5 353.0 221.4 330.9 179.6 308.9 139.1 275.83 292.5 347.0 285.8 347.0 261.5 333.1 222.8 312.3 180.5 291.5 140.0 260.34 304.7 347.0 300.2 347.0 271.4 333.1 231.3 312.3 183.2 291.5 141.8 260.3widths considered in the analysis. It is worth noting here that thisoverestimation is counter balanced by a signicant increase in themaximum transverse moment as detailed later.The maximum FEA longitudinal bending moments were alsocompared to the AASHTO LRFD moments. For skew angles lessthan 30, LRFD overestimates the maximum longitudinal mo-ment by about 40% for one-lane bridges, 25% for two-laneTable 2.Comparison of FEA Edge Beam Moment with AASHTOFEA edge beam momentKNm/mSkew angleSpanlengthm7.2 1 90.5 88.7 83.7 76.1 70.2 58.5 115.22 104.0 101.7 98.1 88.7 77.4 74.33 108.5 106.2 100.8 90.5 95.9 99.04 110.3 109.4 101.3 106.2 124.7 109.810.8 1 147.2 143.6 133.7 117.5 101.3 83.3 172.82 173.3 168.8 158.0 138.6 117.0 97.23 182.7 180.9 168.3 146.3 121.1 94.14 189.5 187.7 173.3 149.0 118.4 94.113.8 1 205.2 200.7 185.4 163.4 138.6 111.6 220.82 241.7 235.8 217.8 190.4 157.5 125.13 254.3 250.2 229.1 198.9 164.3 126.94 264.6 259.2 237.2 206.1 164.7 127.416.2 1 251.6 245.3 226.4 200.3 169.2 138.2 259.22 293.0 287.6 264.6 234.0 193.1 152.63 309.6 302.4 278.6 240.3 199.4 156.24 321.8 312.3 288.9 248.9 202.5 159.8210/ JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. VisitNumber oflanes 0 10 20 30 40 50deection results range from 1/5 to 1/2 the limiting valueS/800given by AASHTO, and the percent difference with the AASHTOlimiting criteria increases with the skew angle. The percent dif-ference is higher for short spans, and decreases as the span lengthincreases to 16.5 m55 ftfor a given skew angle. Moreover, thebasic assumption of the FEA model is the elastic section behavior,an actual cracked section analysis would yield higher deectionsin the slabs. The results will increase to approximately 2/5 to 1 ofAASHTO limiting deection value.FEA Results of Skewed versus Straight BridgesThe effects of the increase in bridge slab skewness on the maxi-mum longitudinal moment, edge beam moment, transverse mo-ment, and live-load deection for a given bridge span and numberof lanes were also evaluated. Thus, the nite-element results forskew angles ranging between 10 and 50 are compared to theircorresponding FEA values for straight bridges. The FEA bendingmoments are presented in the form of the ratioM/M0, whereMis the maximum FEA moment in the bridge for a given skewangle between 10 and 50, andM0 is the FEA moment fornonskewed bridges0 skewness. Similarly, the ratio/0iscalculated from the FEA deection results.Beam MomentThe ratiosM/M0 for the maximum longitudinal moment areshown in Fig. 6, for each of the four span lengths considered7.2,10.8, 13.8, and 16.2 mversus the skew angle. Similar gurescould be shown for edge beam moments. Such gures indicate auniform pattern of decrease in the maximum longitudinal andedge moment values with the increase in the skew angle, com-pared to that of straight bridges regardless of the number of lanesand span length. This decrease appears to be signicant when theskew angle exceeds 20. Also, for skew angles greater than 20,Table 3.Comparison of FEA Maximum Transverse Moment with Maximum Longitudinal MomentFEA maximum transverse momentKN-m/mand ratio of maximum transverse/longitudinal momentSkew angleSpan lengthm7.2 1 22.2 0.30 22.4 0.31 23.0 0.34 23.4 0.39 35.2 0.66 23.8 0.552 28.4 0.31 29.0 0.32 31.8 0.37 31.3 0.42 33.6 0.54 41.1 0.703 31.3 0.32 31.6 0.33 32.4 0.37 38.4 0.51 51.3 0.66 51.3 0.624 32.3 0.32 33.3 0.34 36.3 0.40 45.9 0.51 56.3 0.53 88.4 0.9410.8 1 25.1 0.19 26.4 0.20 31.0 0.26 36.5 0.35 46.8 0.53 39.4 0.572 36.0 0.23 35.1 0.23 40.8 0.28 47.8 0.38 56.4 0.55 56.5 0.693 44.6 0.26 43.5 0.26 52.6 0.34 58.8 0.45 62.4 0.59 58.3 0.714 49.1 0.28 48.8 0.28 57.2 0.36 62.4 0.47 63.0 0.60 58.4 0.7113.8 1 23.5 0.12 27.0 0.15 33.1 0.20 59.9 0.41 75.2 0.62 69.2 0.722 36.0 0.16 37.2 0.17 44.6 0.22 67.1 0.39 82.2 0.59 84.3 0.773 48.1 0.20 50.2 0.21 59.9 0.28 78.5 0.43 91.6 0.63 88.5 0.804 57.3 0.23 60.3 0.25 69.4 0.32 78.1 0.41 92.2 0.63 85.9 0.7716.2 1 23.0 0.10 23.9 0.10 37.2 0.18 70.4 0.38 76.4 0.50 85.8 0.712 38.9 0.14 37.8 0.14 51.2 0.20 92.6 0.42 101.6 0.57 105.7 0.763 51.1 0.17 53.3 0.19 62.4 0.24 79.0 0.35 115.2 0.64 117.4 0.844 63.2 0.21 68.7 0.23 76.6 0.28 86.0 0.37 94.6 0.52 158.9 1.12Maximum Transverse MomentTable 3 summarizes the FEA maximum transverse moments forall case study bridges investigated. The maximum transverse mo-ment was compared with the corresponding maximum momentvalue in the longitudinal direction. The ratio of transverse to lon-gitudinal moments is also reported in Table 3.The maximum FEA transverse moments increased with theincrease in the skew angle even though the maximum longitudinalmoment is decreasing. The ratio of maximum FEA transverse tolongitudinal moment increases signicantly when the skew angleis increased, from about 20% for straight bridges up to 75% forbridges with 50 skewness. AASHTO accommodates the trans-verse bending moment by specifying a percentage of the mainreinforcement equal to 54.8/S, whereSis in m100/S,Sin ft.For the span lengths considered in this study, the percent decreaseof the main reinforcement from 20% for short-span bridges of7.5 m25 ftto 15% for the long-span bridge of 16.5 m55 ft.The results for the 0, 10, and 20 skew angle indicated a conform-ance with AASHTO requirements whereby the percent differencewith the maximum longitudinal moment is decreasing with theincrease in span length. Generally, this transverse reinforcementis low so that shrinkage and temperature reinforcement governs.However, for skew angles equal to 40 and 50, the percent differ-ence increases with the span length and is in the range of 7080%for two-, three-, and four-lane bridges.Maximum Longitudinal Bending Moment and EdgeMaximum Live-Load DeectionTable 4 summarizes the maximum FEA live-load deection ascompared to the AASHTO criterion ofS/800. The FEA resultsare directly related to the assumed slab thickness, which was areasonable assumption for deection control. But one can alwaysassume a different thickness and obtain different deectionresults.For any given span length and its corresponding slab thick-ness, the maximum live-load deection results decrease as theskew angle increases from 0 to 50. On the other hand, the FEAJOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007 /211Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. VisitAASHTONumber oflanes 0 10 20 30 40 50Maximum Transverse MomentThe ratios M/M0 for the maximum transverse moment areshown in Fig. 7 for each of the four span lengths considered7.2,10.8, 13.8, and 16.2 mversus the skew angle. In contrast withthe longitudinal moment results, the maximum transverse mo-deectionmmTable 4.Comparison of FEA Maximum Live-Load Deection with AASHTOFEA maximum slab deectionmmSkew angleSpanlengthm7.2 1 1.7 1.6 1.4 1.2 1.0 0.7 9.02 2.0 1.9 1.8 1.5 1.2 1.53 2.1 2.0 1.9 1.6 2.7 6.64 2.2 2.1 1.9 2.6 7.7 4.210.8 1 4.5 4.4 4.0 3.2 2.5 1.7 13.52 5.5 5.3 4.8 3.9 3.0 2.13 5.8 5.8 5.3 4.1 3.2 2.14 6.1 6.1 5.4 4.2 3.0 2.113.8 1 7.3 7.0 6.2 5.0 3.9 2.7 17.32 8.8 8.4 7.5 6.2 4.7 3.33 9.3 9.2 8.1 6.5 4.9 3.34 11.7 11.2 10.0 8.1 6.0 4.116.2 1 8.9 8.6 7.6 6.2 4.8 3.4 20.32 10.6 10.4 9.3 7.0 5.9 4.13 11.0 10.8 9.8 7.8 6.0 4.34 11.7 11.2 10.0 8.1 6.0 4.1the ratioM/M0 decreases with the increase in the number oflanes from one to four. For both the maximum longitudinal mo-ment and edge beam moment, the ratioM/M0 is almost one forbridges with skew angles less than 20, decreases to about 0.75for bridges with skew angles between 30 and 40, and furtherdecreases to about 0.5 as the bridge skew angle increases to 50.Fig. 6.FEA maximum longitudinal bending momentratioM/M0212/ JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visitthan 7.5 m25 ftis constant for all bridge widths; however, thisrate decreases as the number of lanes increases from one to fourfor bridges having span lengths between 10.5 and 16.5 m35 and55 ft.For span bridges less than 10.5 m35 ftin combination withmultilane bridges, the ratioM/M0 increases to 1.5 as the skewangle increases to 40. This ratio reaches a maximum of 1.9 forFig. 7.FEA maximum transverse momentratioM/M0ment increases almost linearly as the skew angle increases from10 to 40, where it reaches a peak value. TheM/M0 ratio de-creases again for skew angles between 40 and 50. The gureindicates that the ratioM/M0 varies with the variation in thespan length and width, thus no general pattern can be deduced asfor the results discussed in previous sections. Also the rate ofincrease in theM/M0 ratio for bridges with span length lessFig. 8.FEA maximum live-load deectionratio/0JOURNAL OF BRIDGE ENGINEERING ASCE / MARCH/APRIL 2007 /213Downloaded 30 Dec 2010 to 4. Redistribution subject to ASCE license or copyright. Visit50% for 50. The ratio between the FEA longitudinal momentsfor skewed and straight bridges was almost one for bridges withskew angle less than 20. This ratio decreased to 0.75 for bridgeswith skew angles between 30 and 40, and further decreased to0.5 as the skew angle of the bridge increased to 50. This de-crease in the longitudinal moment ratio is offset by an increase byup to 75% in the maximum transverse moment ratio as the skewmum live-load deection for skewed bridges and straight bridgesdecreases in a pattern consistent with that of the longitudinal mo-ment. This ratio decreases from one for skew angles less than 10to 0.6 for skew angles between 40 and 50.In general, this research supports the AASHTO StandardSpecications as well as the LRFD procedure in recommendingthat bridges with skew angles less than or equal to 20 be de-signed as straightnonskewedbridges.
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