Design Of A Skew, Reinforced Concrete Box-girder Bridge Model
Transportation Research Record 8715216.17.18.19.CELL: Vol. 1. California Department of Transportation, Sacramento, Rept. CA-DOT-DS-1129-176-3, Oct. 1976.R.E. Davis and M.R. Wallace.Skew ParameterStudies--An Implementation of the Finite Element Program CELL: Vol. 2.California Department of Transportation, Sacramento, Rept. CADOT-DS-1129-2-76-4, Oct. 1976.R.E. Davis.Structural Behavior of a Skew,Reinforced Concrete Box Girder Bridge Model:Vol. 1, Design.Division of Structures, California Department of Transportation,Sacramento, Rept. FHWA-CA-ST-4187-78-01, Jan. 1978.A.C. Scordelis, J.G. Bouwkamp, S.T. Wasti, andD. Anicic. Structural Behavior of a Skew, TwoSpan, Reinforced Concrete Box Girder BridgeModel: Vol. 1, Design, Construction, Instrumentation, and Loading.Univ. of California,Berkeley, Rept. UC SESM 80-1, June 1980.A.C. Scordelis, J.G. Bouwkamp, S.T. Wasti, andD. Anicic. Structural Behavior of a Skew, TwoSpan, Reinforced Concrete Box Girder BridgeModel: Vol. 2, Reduction, Analysis, and Inter-20.21,22.pretation of Results.Univ. of California,Berkeley, Rept. UC SESM 80-2, June 1980.A.C. Scordelis, J .G. Bouwkamp, S.T. Wasti, andD. Anicic.Structural Behavior of a Skew, TwoSpan, Reinforced Concrete Box Girder BridgeModel:Vol. 3, Response During Ultimate Loading to Failure. Univ. of California, Berkeley,Rept. UC SESM 80-3, June 1980.A.C. Scordelis, J.G. Bouwkamp, S.T. Wasti, andD. Anicic.Structural Behavior of a Skew, TwoSpan, Reinforced Concrete Box Girder BridgeModel: Vol. 4, Detailed Tables of Experimentaland Analytical Results.Univ. of California,Berkeley, Rept. UC SESM 80-4, June 1980.R.E. Davis, V. Dinh Bon, and F.M. Semans.Transverse Distribution of Loads in Box GirderBridges:Vols. 1-6. Office of Structures Design, California Department of Transportation,Sacramento, Rept. FHWA/CA/SD-80/2, 80/3, 80/4,80/5, 80/6, and 80/7, Sept. 1980.Publication of this paper sponsored by Committee on Concrete Bridges.Design of a Skew, Reinforced Concrete Box-Girder BridgeModelRAYMONDE. DAVISA 1: 2.82 scale model of a two-span, continuous, reinforced concrete box-girderbridge, which has supports skewed at 45 , was constructed and tested at theUniversity of California, Berkeley. The cross section and significant dimensionswere similar to those of two previously tested models, one straight on orthogonal supports and one curved on radial supports. The objective of the researchwas to compare behavior of the three models and to verify an analytically pre·dieted diminution of longitudinal stringer moments that result from skewingsupports. All three models were designed by the California Department ofTransportation. Because traditional design criteria make no provision forskewed supports, the skew model was designed by means of a sophisticatedfinite-element computer code called CELL. Girder moments proved to be significantly less than those in the orthogonally supported model and had a 19percent reduction in the main longitudinal reinforcing steel. Distribution ofgirder shears was changed significantly from that of the model on normal bear·ings. As a basis for implementation, this paper discusses some features of theskew model design process.For many years, the California Department of Transportation {Caltrans) has been interested in anomalies that characterize the structural behavior ofreinforced concrete box-girder bridges with skewedsupports.Initially, interest was centered on effects of skew on girder shears.Excessive crackingof webs observed at obtuse corners suggested enhancement of girder reactions that had commensurateincreases in diagonal tension.Complexities in the analysis of skew boxes restricted early efforts toward mitigation of observedexcessive web cracking to establishment of curvesfor augmentation of exterior and first interiorgirder shears at obtuse corners of such boxes.(Traditionally, skewed boxes in California have beendesigned as structures of the same spans on orthogonal supports and detailed with skewed supports.)Curves for shear augmentation were established withlittle scientific basis and furnished, at best, onlyestimates.A request in 1959 by design management for a moredefinitive study of this problem initiated a protracted study of reinforced concrete cellular structures performedjointly by Caltrans'StructuralResearch Unit and the University of California,Berkeley.The research effort included tests offull-scale prototypes and small and large-scalemodels.Structures of increasing complexity werestudied on a progressive basis, as follows:{a)simple span boxes without diaphragms on normal supports; {b) simple span boxes with rigid intermediatediaphragms, or continuous boxes without intermediatediaphragms on normal supports; (c) continuous boxeswith intermediate diaphragms, which consider effectsof bent and diaphragm flexibility; (d) curved boxeswith radial supports; (e) nonprismatic boxes; (f)skewed boxes; (g) prestressed boxes; and (h) composite concrete and steel boxes.Analytic methodsemployed in the development of computer codes by theUniversity of California relied heavily on , and finite-element methods.A valuable computer code developed as part of theresearch effort employs a finite-element analysis toassess behavior of cellular structures of arbitraryplan geometry. This program, called CELL, was firstused within Caltrans to analyze a heavily skewed,and curved, box-girder bridge to carry rail trafficand to assess the influence of intermediate diaphragms on that behavior. The program has been usedin studies of boxes of varying skews and aspectratios to establish functional relations betweenskew angle and shear augmentation factors.Estimated curves of such factors previously used byCaltrans were proved to be unconservative.A serendipitous result of these studies was the
51Transportation Research Record 871Figure 2. Cross section ofstructure for sampleanalysis.sored by Caltrans and the Federal Highway Administration (FHWA) .Opinions, findings, and conclusions expre,;sed inthis paper are ours and do not necessarily reflecto ff i cial views or policies of Caltrans or FHWA.This report does no t constitute a standard, spe cification, or regulation.Figure 3. Sample nomographic analysis.REFERENCES1."· 9 r· .---.,-1-, .--- , s;. ;; 1·1-':. :.:.:.:i---1-.::::i.:::,,.i.::.,,3.Figure 4 . Sampleinfluence-line analysis.4.405.20106.7.25 ft, and slope factor (K) (2.81/4.69) 0.60.By using the (typical) nomograph shown in Figure3, and noting carefully the correction factors indicated by aster is ks, a designer can obtain D-factors, where values of S/D (or, for exterior girders,We/D) represent distribution factors.The typical influence-line analysis is illustrated (in part) in Figure 4.Interpolations arerequired for span and slope factors and eight separate diagrams must be considered (for three girderseach).Resulting moment percentages are obtainedfor P-series and H-series trucks separately.Influence-line analysis (which, incidentally, isa misnomer) may be easily applied without a computer, and it is compatible with arbitrary loadingconditions.It is probably the only simple approachto distribution of such loads.The method becomescumbersome in a production environment, so the computer code has been written in FORTRAN IV language.Caltrans' program (LANELL) is immediately accessiblein the time-share option system via a cathode-raytube (CRT) remote terminal.The designer entersseven parameters from the keyboard and obtains asreadout suitably interpolated values for the numberof wheel lines of P-series and H-series trucks to bedistributed to each girder and to the whole structure. Factors have also been established for curvature correction.For the rare design that cannot be treated by thefirst three methods, the Berkeley programs may beused, e.g., for spine beams, MUPDI3 and SIMPLA2; forheavily skewed structures or boxes with arbitraryplan geometry, CELL; for short radius structures,CURD!; and for composite steel boxes, FINPLA.7.8.9.10.11.12.13.14.ACKNOWLEDQ.IENTWork described herein was performed as part of a research project entitled, Development of a Box GirderLoad Distribution Formula, which was jointly spon-15.R.E. Davis, J.J . Kozak,and C.F. Scheffey.Structural Behavior of a Box Girder Bridge.HRB, Highway Research Record 76, 1965, pp. 3282.R.E. Davis, J.J. Kozak, and C.F. Scheffey.Structural Behavior of a Box Girder Bridge, andAppendices.BridgeDepartment,CaliforniaDivision of Highways, Sacramento, Rept. SSR2-65, 1965.R.E. Davis, C.F. Scheffey, G.A. Castleton, andE.E. Evans.Box Girder Model Studies.BridgeDepartment, California Division of Highways,Sacramento, Rept. R&D 4-70, 1970.R.E. Davis, C.F. Scheffey, G.A. Castleton, andE.E. Evans.Model and Prototype Studies of aBox Girder Bridge.Journal of the StructuralDivision, ASCE, Vol. 98, No. STl, Jan. 1972.A.C. Scordelis.Analysis of Simply SupportedBox GirderBridges.Univ.of California,Berkeley, Structural Engineering and StructuralMechanics Rept. SESM 66-17, Oct. 1966.NTIS:PB 175646.A.C. Scordelis.Analysis of Continuous BoxGirder Bridges.Univ. of California, Berkeley,Structural Engineering and Structural MechanicsRept. SESM 67-25, Nov. 1967. NTIS: PB 178355.A.C. Scordelis, R.E. Davis, and K.S. Lo.LoadDistribution in Concrete Box Girder Bridges.Proc., American Concrete Institute First International Symposium on Concrete Bridge Design,Toronto, Ontario, Canada, April 1967, AC! Publ.SP-23, 1969.W.W. Sanders, Jr., and H.A. Elleby. Distribution of Wheel Loads on Highway Bridges. NCHRP,Rept. 83, 1970.A.C. Scordelis and C. Meyer.Wheel Load Distribution in Concrete Box Girder Bridges. Univ.of California, Berkeley, Structural Engineeringand Structural Mechanics Rept. SESM 69-1, Jan.1969. NTIS: PB 183923.R.C.Cassano and R.J.LeBeau.CorrelatingBridge Design Practice with Overload PermitPolicy.TRB, Transportation Research Record664, 1978, pp. 230-238.K.J. Willam and A.C. Scordelis.Computer Program for Cellular Structures of Arbitrary PlanGeometry. Univ. of California, Berkeley, StructuralEngineeringandStructuralMechanicsRept. UC SESM 70-10, Sept. 1970.NTIS: ures of Arbitrary Plan Geometry. Journalof the Structural Division, ASCE, Vol. 98, No.ST7, July 1972.M. Aslam and W.G. Godden.Model Studies ofMulticell Curved Box Girder Bridges.Journalof the Engineering Mechanics Division, ASCE,Vol. 101, No. EM3, June 1975.H.D. Nix.Analysis of a Skewed Concrete BoxGirder Bridge (Floral Park Underpass) --An Implementation of the Finite Element ProgramCELL.Bridge Department, California Divisionof Highways, Sacramento, Rept. BD 73-8, June1973.M.R. Wallace.Skew Parameter Studies--An Implementation ofthe Finite Element Program
Transportation Research Record 871demonstration that significant diminutions of longitudinal bending moments might be realized in skewedstructures.A potential for significant economieswas manifested by curves drawn for simple spanstructures, which suggested the possibility of reducing dead load resisting moments by nearly onehalf in structures skewed 45 and by nearly 70 percent for skews of 60 .The reduction of resisting moments is usuallyexplained as the result of a tendency to span thenormal distance between supports.This explanationis overly simplistic.Diminution of longitudinalresisting moments in the girders is realized at theexpense of increasing torsional moments that act onthe structure.As orthogonal supports are changedto skewed, formerly symmetrical reactions becomeasymmetrical as those at obtuse corners are increased and those at acute corners are decreased inmagnitude.Resultants of support reactions moveaway from the centroidal axis of the structure, andtorsional forces are introduced.Closed cellularsections possess high torsional rigidity, and increases in torsional moments are less significantthan diminutions of longitudinal moments.It wasdesirable to evaluate qualitatively the influence oftorsional forces in skewed structures.Tests conducted by the University of Californiaof small-scale aluminum models that had varyingskews and aspect ratios verified the accuracy ofCELL.However, Cal trans' Structures Design management was understandably reluctant to adopt the indicated large reductions in reinforcement quantitiesbecause they lacked verification with a larger-scalereinforced concrete model, the behavior of whichmight also be directly compared with that of the(orthogonal) straight and curved boxes tested previously.SCOPE OF PAPERResponsibility for construction, testing, and analysis of behavior of the skew model was assumed by theUniversity of California, Berkeley.Responsibilityfor design of the model and implementation of results was assumed by Cal trans' Structural ResearchUnit.This paper describes the techniques used inthe model design.[This paper is a condensation ofthe work by Davis OJ, in which the design of themodel has been described in detail.]MODEL DESCRIPTIONThe model comprised reinforced concrete elementsthat had a linear scale reduction of 1:2.82.A No.4 reinforcing bar in the model, which was built tothis scale exactly, simulates main No. 11 reinforcing bars in the prototype. A 6-mm (0.236-in) bar inthe model (smallest available with deformations)approximately simulates a No. 5 or a No. 6 bar inthe prototype with minor variations in spacing.Constructed on the test floor of Raymond E. DavisHall at the University of California, the model was3.66 m (12 ft) wide (from edge to edge of deck),25.6 m (84 ft) long (measured between acute corners), and approximately simulated a 10.4-m (34-ft)wide continuous bridge with two 31-m (101.5-ft)spans. The structure was identical in cross sectionto the curved and straight models tested previously. Transverse reinforcement in the top and bottom slabs was the same as in previous models inorder to maintain similitude.The cross section and transverse reinforcementpatterns were originally established in the designof the Harrison Street Undercrossing, a full-sizedprototype tested by the California Division of Highways in 1960 (1 ,1 ) and a 1: 3. 78 scale model tested53contemporaneously by the University of California( .? )in the initial phases of the box-girder research program.MODEL DESIGNThe methods used to design the skew model appeartedious but were characterized by much more thoroughness than would be expected for a full-sizedstructure.Cal trans' Structural Research Unit employs its own modified version of CELL, which permits storage of the decomposed stiffness matrix forfuture use in analyses of various loading conditions.An optional mesh-plotting routine is included, since errors in the geometry of large meshesare easily made.The mesh employed is depicted in plan to a smallscale in Figure 1 and, in part, to a larger scale inFigure 2.The same geometrical mesh is used for thetop and bottom slabs (this mesh-generating scheme inthe current version of CELL mandates vertical webs);however, material properties of elements in the twoslabs may differ.A current research project willremove some of the deficiencies in CELL, therebypermitting sloping webs and adding a prestressingfacility and automated girder moment integration.This last feature will eliminate much of the effortexpended in the design of this model.The mesh was made rectangular to satisfy a requirement of the CELL postprocessor (CELLPOP) ( )that all cross sections have the same number ofgirders (e.g., if girder moments within longitudinallimits of end supports are desired) .All elementsbeyond supports are made null elements (i.e., withzero thickness) in the materials properties sectionof the input data.A second study made with askewed mesh without null end elements yielded similar results.Careful choice of numerical designations of nodesand elements allows maximum use of program meshgeneration features. Although punched-card input isconceivable,the repetitive nature of the datagreatly decreases key data entry if this work isdone on a cathode-ray tube (CRT)terminal withstandard utility routines that allow rapid proliferation of data blocks (e.g., in Caltrans' IBM System,the INCLUDE routine).In all, 688 slab elementswere described on 50 card images, and 362 verticalelements, including all transverse diaphragm elements followed by longitudinal web elements, weredescribed on 22 card images.Materials propertiesfor upper and lower slabs were specified separately.Figure 1. Finite- ilementmesh for design of modelwith CELL program.CELL FINITE ELEMENT MESHFigure 2. Partial finiteelement mesh showingskew reference lines forvehicle location, orthogonal reference linesfor moment , and shearcalculations and wheelpaths.GSG4G3 - -- -i l l"-7'C,.1--.,.C ., ;,.:;. - "ll--- -T--G2 r-l?'',.f.-'--l-,.L.--l-'- .,.'- --,#- -o-/- GI34SKE W SCCTIO/IS
Transportation Research Record 87154Specifications of nodal coordinates required separate card images for each of 737 nodes: however,because of the rectangular nature of the mesh, ordinate values are repetitive in blocks of 11, and theproliferating routine was used to advantage.Asingle block with 11 different ordinates was established. The block successively proliferated to 10,100, and 800, followed by deletion of 63 card images.Numerical nodal designations (different foreach card image) and the abscissae (all the same foreach card image in an 11-card group) may be readilyentered.Materials properties were described on eight cardimages and included separate elastic moduli in thex- and y-directions, shear moduli, mean values ofPoisson's ratio in two directions, and elementthicknesses.One null element with zero thicknesswas included to represent nonexistent elements inthe bottom slab that correspond geometrically tocantilever upper slab elements and nonexistent elements outside abutment bearings.Eleven boundary-condition cards specified zerodisplacements in the z-direction at supports, in thex- and y-directions (also) at the center pier toprohibit rigidbodytranslation,andinthey-direction (also) at the two central abutment supports (to prevent rotation of the whole structureabout the z-axis).An additional entry on all ofthese cards located reactions at bottom slab nodes.A designer who has developed skill in using a CRTand utility routines can establish data-input fileswithout key data personnel.A partial listing ofinput for geometric and physical properties is shownin Figure 3.CELL INPUT FOR MODEL DESIGN:LOADSGeneralFigure 3. Partial list of input to CELL program./ 114Determination of Longitudinal Girder MomentsFinite-element output from CELL was translated intoUB5& s l2C.120ZlH'i7H7l8395E711913132'1: 33 :.''J '1 'iHZ JH0.H ll2 41 344j55 .3S6Z!i7 3!!.'IJl109ll25H49617385' 7109121310:3214!8.321l2l3B.) .',53576SJ J7465657Il65. 10966129l 2412 361474 17 160201722"C425596282 108J!J'l 1203J6 JIJ9337 321)4 5 4 ! 7J',"J 3:!135' lZU I 3324,8 3;4'd5 356lJ2512124.!64860;28496l'J8120132321322t6o : )' l))los1.-2 6C.!l i Sl6 2 Hl6(.4 675f.76 6e1tr.26646766! '.\M3 pr; i}"!11 .Il l '.!897 10 f9 l ia i 1110 l :') ISO11 ;:I) 20112 2 01 2oc;.ll 2:.'6 2Z1l'o :21 2:3Deck El !mcnt"?I26;:J'52930'I32H 41.1;"!: II'1lIIn""" ::4041 ll431:454'IiCt :6l " 012223'411''"l: n ; 61!6 11 H J. n : u 4 63i"6'.l!15.i6!):, ,n"ti11111111)'1lbl1 t,O !.:.'')1314151617Ia19111111it.4.l3l6:?9 u.1 (:,,:!) 0;:9725 7 37 72() 7;'' 617 6;:r. 6::1 c i11211) :t ; i ;67-56D6 )t.64' 65l6616686::4II(i4 516 H2 r,.Jl t,C4 J9lS'il '10' '115 416 4L'5Si Cl 4 l 6 4 27 4 3 4 ! 7III:' 7 910Ll.11t.t.6II))ln" . 11.slJ i i i,U1J113738""44IIIIIIIIIIIIIi'-'4'.495051H'II1 )1!1;7;"".,h,., 1 tJ 5 ; 0 lI 2:, 1 25217 32 ! 3 3324 C.05 4172:; 6 2 1 6 l32 iD i'.:'1JJ2 13j',J3 2' &9 61 62990 629 64098 ?17 7 3'i941510 I263iio2Current American Association of State Highway andTransportation Officials (AASHTO) specifications forbridge design call for 3.66-m (12-ft) traffic lanesand no fractional lanes. Tread of the AASHTO designvehicle is 1.83 m (6 ft), and minimum distance froma wheel line to edge of a lane is 0.6 m (2 ft). Sixvehicle paths were chosen as the most probable critical paths for the five girders.Four paths wouldsuffice but for asymmetry produced by having intermediate diaphragm elements in only one span.Sixvehicle paths produce 12 separate wheel paths: because those in proximity to one another are separated by only 0.3 m (1 ft), six compromised wheelpaths shown in Figures 2 and 4 were established bymoving truck paths a maximum distance of 0.15 m (0.5ft) transversely.Thirteen lines paralleling theskew were chosen to establish live load positions atintersections with parallel wheel paths, and 19orthogonal cross sections were established for moment calculations (see Figure 2).Exterior load paths are coincident with exteriorgirder webs, and intersections with skew lines fallon nodal points.Intersections of skew lines andinner wheel paths lie within elements, and concentrated loads may be distributed to element nodes bythe tributary-area method.Typical input for a single live load condition isshown in Figure 5. The requirement of a list of 219nodes at which stresses are to be calculated for 78live, and l dead, load cases studied again suggeststedium in the input i however,, the repetitive natureof input again permits rapid proliferation of asingle file, which is followed by a separate entryof nodal points and magnitudes of applied loads. l21l1'l 3I I I,6&2n4s157 6 ·' 2 653156 653 66!1l"" 719 no165l!i'l1 065']6I1761722 '.:Iu12I12t 6' II,.· 11IIII'iJ 109II'i.,,97"" ;7100lfl 9I0111213IOl:\nJ.:i I ConrJ i 11 .1 t 0O.' ·.: 5IOS.\rr iv10107165(l'arti;Jl),,.:: . l' ;' . )15:,.16: )l7:: )sII I II1 1 : ,.72423!"'n1"'2H . S7"73l7'47lS }!, .726:!31 . S234 ,234 .2l4 .23'1 .234 .1287JO7Jl:!l4 ,234 ';:34 , . ' ,"'"'i , u.I)).,"IIIIIIl82182282'824825I .,,9 . !1' Hl1720 : '27823f,29.,,24 27H\.1''-l'ZIOOIIIII I'"02'-l?CllfiJC.J:GOOr. J;:?OOO.r61" 4l:"C"OCI432CCQ4J2CQQ 4lJOCIO4)::'0CIO"' t'ppo81927 . SISjj 5'ti O'! a .t,.J2000 .l ll'"12511 a101.""91 a.'s109"II711,4'"'2'.,,89o.Ij87IIIIi0485II:1! ;:1 7 :;t.l?7I021'808\tlZH30! 1:19 I:'.l361 71 r, 7 25,.11".,"" aIIII305"".,II,.,.ull6(,6 6 "i 7226 6 7 63.1230 7 l 7: 22ll6I?74is2!l .SS'162'3 6 !l HI2 91, ic1 112t I 'li'.?C00 1'-32,CIO ,.C.32.)0G .16lhi.unsa .lt3050llDOSOIS3050 .!l3050183050Sl:AU-UJO)OCLU1818\6ISIa!.\I). . ,. ' "'"'G\7ill:. 0 SJ!0 !i587200R .lll.!hspl.lc:ew.cnt th:iunJ.:aryConJ it ion Arr;Jy"j)·1., 1. ri l-49850851
Transportation Research Record 87155load. This vector will list values for all loads inthe node listing at the bot toms and tops of webs.Output values at structure supports comprise combinations of upward reactions, and downward reactionsdue to the dead load of elements that frame intosupport nodes.Output values from the generalizedforce vector will not, by themselves, provide correct statics checks.A minimum number of statics checks were made forthe model design, but agreement was excellent. Sumsof these corrected reactions agreed within 0.03 percent with the calculated dead load of the superstructure, while the total internal resisting momentat cross section 6 (see Figure 2) agreed with thecalculated dead load moments within 0.23 percent.Figure 4. Critical and compromisedwheel paths.GIG2G G3G5Live Load MomentsFigure 5. Typical input for one live load case for CE LL program.Positive Moments.!"fl'[2JConcentratedNoda I LoadsS!!124:uJ #14l!!Du cpu I Gnqp l0.26272829JO3132Jl45465455;a21 J 214118 ll'.l 120 . 310 Jll21s 216 211 352 Hit312 3lJ H4 3'14 l95365 3! 6 36779soa1azSJ84as868788 111 112 llJ 114 115121 ISS 561571 8 l747Cu cl25 K I 1641J495051 l' i6921B 219 220 26 5 266 :'67 268 269 2:-0 211 212 2n 2:4 21s J091112lJl 15315 JI, Jl7 318 319 J42 J4J 344 HS 34b 347 348 349 JSO l51161715? 160 161 162 163 164 165 211J 211 212368 H 9 3!0 l7l J72 JiJ JH 336 H:' 385 369 390 l'JlH2 l'JJl,70396 419 420577 57 84'/l 472 OJH f 52 653579 580 Slll X 705 706654 655 656 10s 421 422. 423 424,25 4 26 427 428 429 46] 464 465 466 467 'i68G 518 519 520 521 522 523 524 525 526 527 528 571 574 575 576"'"20212223,.688 689 690 691 692 69J"61 109 321 417 6Z9 677 7012629582 583 617 618 619 620 6:'1 622 623 624 625 626 6Z7 650 65126657 658 659 6f· 683 6M 685 6866 7 i 25707 i03 709 710 711 712 llJ 714 715lll727Li.st of Nodes foeInternal Forc:esNot :Dnta in Columns ) to lD ofcard i.rn.:ig es with !! in Columns 1 JnJ 2 belong. inCo lums 7J to 80 of following cardlongitudinal bending moments by the CELLPOP program( 6) .The program choos e s significant values fromthe matrices of deck a nd web stresses in two CELLdata-output files, takes mean values of longitudinalmembrane stresses (Nxxl at webs and midbays, converts them into element forces, and integrates incremental moments about computed neutral axes ofwebs to determine total longitudinal moments. Output from CELLPOP was plotted as influence lines fortwo adjacent load paths per girder.Sta tics ChecksAny finite-element analysis requires statics checksto prove that equilibrium is achieved.A poorlychosen mesh or elements of poor configuration mayproduceincorrectresults.Membranestresses(Nxxl are given in units of FL- 1 Total forcein an element is the product of the mean value ofNxx and the element length.Zero stress pointsare found by propor tion from membrane stresses atthe tops and bottoms of webs.Unit stresses atjoints in adjoining elements may be compared by dividing Nxx by the element thickness.Membranestresses at the lower, outer edges of the cross section are double those listed in the CELL output,since the program averages values at this joint between real and zero thickness slabs; doubling wasperformed in the postprocessor.Forces and moments in slabs and webs are found inthe usual manner and are sununed.Longitudinalforces should sum to zero.Moment sums at crosssections were used in statics checks.Reactions at supports may be taken from the generalized force vector in the CELL output for deadOutput influence coefficients for live load wereplotted with a programmable desk calculator and attached plotter for two load paths (on each of twographs) for a total of 19 cross sections for fivegirders. Compar iso ns of mirror images of influencelines plot ted for l oad paths 1 and 2 and for paths 5and 6 demonstrated that influence of the intermediate diaphragm in span 1 was small, that mirroredinfluence lines differed inappreciably from oneanother, and that a portion of the plotting could beeliminated by considerations of symmetry.Two influence lines were plotted on each sheet,since any vehicle axle would be expected to be orthogonal to the structure centerline and the wheelsof this axle at the same abscissa.Influence lineswere digitized to obtain total moments; the twoadjoining load paths are read simultaneously.Influence coefficients for pairs of load paths in adjacent lanes were plotted on separate cross sections.The vehicles on these two load paths neednot be at the same abscissae.Calculations of live load moments were performedwith more precision at many more cross sections thanwould be required in designs of full-sized structures.Conversion of influence coefficient plotsinto l ong itud i .na l resisting mome nt s was pe rformed bydigitiz at ion on a transparent ove rlay drawn t o thesame hori zontal scale as t hat use d in the i n f luenceline plots.This overlay c o mprised a single horizontal line for reference and t hr ee vertical linesspaced at 4.3 m (14 ft) on this scale to facilitatereading influence ordinates at the axle abscissae onadjacent load paths. Values of these ordinates wereinput to a desk calculator program to compute totallive load moments for a standard AASHTO design vehicle moving in either direction; the output providesthe maximum of these two values.The plastic overlay was placed in successive ho r izo nt a l positionsuntil a maximum moment had been calc ulated.Maximawere tabulated for load paths 1 and 2, sununed withthose of load paths 3 and 4, and augmented by theimpact factor.Negative MomentsMaximum negative moments over the center bent resultfrom the alternative lane loading and imposition oftwo concentrated loads.The lane loading requiresmeasurements of areas under influence lines and maximum negative ordinates. A program that integratesareas on a digitizer by circumnavigating theirboundaries with the digitizer's crosshair probe wasused.Certain positions of the live load produced smallnegative moments in girders over abutment supportsnear obtuse corners of the structure due to the fact
Transportation Research Record 87156total shear out in the span.At the pier, deviations of CELL shears are evident and result from thefact that orthogonal sections intersect the bent capwhere there is no convenient way to determineshears.Lesser deviations are evident at the abutments, but shear predictions break down at supportednodes.Statics checks were greatly improved when crosssections were taken parallel to supports.The totalload of the superstructure in span 1 was calculatedwith precision, and the total reaction at abutment 1was deducted to provide a total bent shear of 200 t(440 kips): the sum of CELL reactions at the bentwas 180 t (397 kips).Figure 6. Dead load shears from CE LL program.DEAD LOADGIRDERSHEARS (KIPS)12060-GIRDER--·-GIRDER-- ·GIRDER--GIRDER---GIRDER23q5-50Live Load Shears·120Calculation of live load shears required an influence-line approach, since the CELL program wasrun for 78 separate locations of the unit load onthe deck. The plotting of influence lines was automated, again with curves for two adjacent wheelpaths on each plot, and t he maximum shear valueswere obtained by trial-and-error digitization withthe transparent overlay.Envelopes of live loadshears for each girder were added to dead loadshears, and stirrup spacings were de
Publication of this paper sponsored by Committee on Concrete Bridges. Design of a Skew, Reinforced Concrete Box-Girder Bridge Model RAYMONDE. DAVIS A 1: 2.82 scale model of a two-span, continuous, reinforced concrete box-girder bridge, which has supports skewed at 45 , was constructed and tested at the University of California, Berkeley.
Sep 15, 2020 · Two nonparallel lines in space that do not intersect are called skew lines. Skew lines are non-coplanar lines. Therefore, they are neither parallel nor intersecting Examples of Skew Lines are skew lines in the figure shown. Solved Example on Skew Lines Which of the following are skew
c3-12 jointless bridge abutment inline ww zero skew . c3-13 jointless bridge abutment inline ww 45 skew-right . c3-14 jointless bridge abutment inline ww 45 skew-left . c3-15 jointless bridge abutment flared ww zero skew . c3-16 jointless bridge abutment flared ww 45 skew-right . c3-17 jointless bridge abutment flared ww 45 skew-left
in the case of an asymmetrically laminated skew plate. The maximum is obtained near to the plate centre. 2 Geometrically Non-Linear Vibrations of Fully Clamped Laminated Composite Skew Plates 2.1 Constitutive Equation of a Laminated Skew Plate at Large Deflections Consider the skew plate with a skew angle shown in Fig.1.
Any two non-parallel and non-intersecting lines are called skew lines. Skew lines are non coplanar. Shortest distance between two skew lines is difficult to visualize. Shortest distance between two skew lines equals the distance between their intersections with their common normal line. For every pair of
Skew Lines Skew lines are lines that are and do not . In this diagram, planes R and W are parallel. DEand FGare lines. Perpendicular lines are not skew lines, because they're in the same . Parallel lines are skew lines,
Common perpendicular XY between skew lines AB and CD in view #1 AB is in true length in view #3 Common perpendicular XY between . shortest horizontal distance between skew lines parallel Shortest horizontal distance between the two skew lines Horizontal projection plane X Y Y X. shortest horizontal distance between skew lines 1 2 B A D C A B D C.
they are used in field experiments and other campaigns. Figure 1 is an example skew-t-log P diagram. Figure 1: Skew-T - Log P Thermodynamic Diagram Let’s take a closer look at Figure 1 and identify the lines on the skew-t diagram. Figure 2 is a close up of the lower right corner of the diagram in Figure
Skew is the deviation of propagation delay from required reference timing. Skew is important in traditional parallel synchronous busses between the data lines of the parallel bus and a separate timing signal. In such cases skew was usually between the transition of the timing
