第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 (L=180 m), 20 1,, (60~, U V 300m), C,,, U +V =C (1) bridgesastherigidframebridges.2tabs,2figs,15refs. K

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1 第 33 卷第 3 期 2013 年 5 月长安大学学报 ( 自然科学版 ) JournalofChang anuniversity(naturalscienceedition) Vol.33 No.3 May.2013 : (2013) 周勇军, 张晓栋, 宋一凡, 赵煜 (1., ; 2., ) : 为了研究高墩连续刚构桥基频函数表达式, 以某一典型桥梁为例, 构造了多种不同的桥梁结构形式, 建立了 64 种有限元分析模型, 获取了桥梁结构自振特性规律, 并提出了桥梁纵向振动的基本振型函数 ; 在此基础上利用瑞利法推导了高墩连续刚构桥纵向振动基频的理论计算公式, 最后对此公式的可行性进行了算例验证, 讨论了现有抗震规范在此类桥梁结构基频计算上的适用性结果表明 : 高墩连续刚构桥第一阶振动以桥墩的纵向振动为主 ; 给出的由能量法得到的基频计算值与有限元模型计算值误差为 5%; 抗震设计细则中的自振频率计算公式不适用于高墩刚构桥此类非规则桥梁 : 桥梁工程 ; 高墩连续刚构桥 ; 基频 ; 振型函数 ; 能量法 ; 抗震设计细则 :U :A Calculationformulaoflongitudinalfundamentalvibrationfrequencyfor continuousrigidframebridgewithhighpiersbasedonenergymethod : ZHOU Yong-jun,ZHANG Xiao-dong,SONG Yi-fan,ZHAO Yu (1.EngineeringResearchCenterforLargeHighwayStructureSafetyoftheMinistryofEducation,Chang an University,Xi an710064,shaanxi,china;2.keylaboratoryofbridgedetectionreinforcement Technology MinistryofCommunications,Chang anuniversity,xi an710064,shaanxi,china) Abstract:Inordertoobtainthefunctionoffundamentalvibrationfrequencyoftherigidframe bridge,firstly,atypicalrigidframebridgewithhighpierswastakenasanexample,and64simi- larbridgeswerefabricatedandtheirfiniteelement(fe)modelsweresetuptoobtainthestruc- turalnaturalvibrationcharacteristics,then,thefundamentalmodeshapefunctionofthistypeof structurewasputforwardsoastodeducethetheoreticalfundamentalfrequencyformulabyray- leighmethod.finaly,thepresentedtheoreticalformulaoffundamentalfrequencywasvalidated bytheengineeringandthefundamentalfrequencyformulainthecurrentguidelinesforseismicde- signofhighwaybridgeswasdiscussedtherein.theresultsindicatethatthebasicoscilationof therigidframebridgewithhighpiersconsistsofthelongitudinalvibrationofpiers,andthatthe fundamentalfrequencycalculatedbytherecommendedmethodhasanerrorof5% comparedwith theferesult,andthefrequencycalculationintheguidelinescannotbeappliedtosuchirregular : ( ); (10-39K); (NCET ); (CHD2012JC030) : (1978-),,,,, zyj@chd.edu.cn

2 第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 (L=180 m), 20 1,, (60~, U V 300m), C,,, U +V =C (1) bridgesastherigidframebridges.2tabs,2figs,15refs. Keywords:bridgeengineering;continuousrigidframebridge withhighpier;fundamentalfre- quency;fundamentalmodeshapefunction;energymethod;guidelineforseismicdesignofhigh- waybridge, 0,, [1], 0,, 0+V max =C (2), La-, 0, grange 4, 0, ; [2], U max +0=C (3), (2) (3), U max, V max, ; U max =V max (4),,,, y(x,t) [3] ; y(x,t)=φ ( x)sin(ωt+θ) (5) : [4], φ (x),, ;ω, 0 m(x) y 2 dx+ 1 j 2,, v = 1 2 L M i [ y (x i,t)] 2 = 0 ; [5] 1 (ωt+θ)[ 3 ; 2 L m(x) cos2 φ2 (x)dx+ 0 j [6], 1 M, 2 iφ2 (x i )]ω 2 (6) :m(x) ;M i ;L, U = 1 2 L EI(x)( d2 y dx = 0 dx 2)2, 1 [7] (ωt+θ)[, 2 L EI(x)( d2 φ ( x) sin2 0 dx 2 )]2 dx, (7), :EI(x), (6) (7) (4),,

3 50 长安大学学报 ( 自然科学版 ) 2013 年 EI(x)[d 2 φ ( x)/dx 2 ] 2 dx ω = L 0 j 0m(x) L φ2 (x)dx+ M iφ2 (x i ) (8) (Rayleigh),,, [8] (8),, 2 ( )m, 6.5m, 2.5m, 12.0m, 6.5 m, 3.0 m, 58m, 6.5m, 3. 0m, 0.6 m, 1.0 m C50, C40 ANSYS Beam188 Fig.1 Modeshapeofrigidframebridge ( ), :f1,f2,f3,,, y, n, ( ) z( ) x,, 3, 1 [9],,, φ (z)= 3z2 i 2-2z3 h i h, 3 0 z h i (9) i :, ( ), φ i(z)= z h i h, 3 0 z h i (10) i, (9) (10) (11),,, EI i ( ω 2 0 φ i ) 2 dz = n-1 = 64, Mφ 2 i(h i )+ 1 [10-11] h i m i ( 0 φ i) 2 dz n-1, 64 (12EI i /h i) 3, n-1, M + (13m ih i /35), [9,12~14], :h i i ;EI i i 3, EI i m i 3,, /h ω 2 = 12EI EI 2 /h 3 2 (12) 13m 1h 1 /35+13m 2h 2 /35+M 1 n-1 h i (11) ;m i i ;M,

4 第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 51 h /m 1 Tab.1 ParametersofFE modelsofrigidframebridgewithhighpiers l1/m l2/m hb/m hc/m I/m 4 A/m 2 /Hz

52 长安大学学报 ( 自然科学版 ) 2013 年 h /m l1/m l2/m hb/m hc/m I/m 4 A/m 2 /Hz 49 120 91.0 160.0 8.5 4.0 11.358 9.7 0.115 50 120 101.0 180.0 9.5 4.0 11.358 9.7 0.108 51 120 106.0 190.0 9.5 4.5 11.358 9.7 0.105 52 120 111.

5 52 长安大学学报 ( 自然科学版 ) 2013 年 h /m l1/m l2/m hb/m hc/m I/m 4 A/m 2 /Hz ( )m, 9.5 m, 3.5 m, 1/ 12.9m, 7m Hz, 90m, 4.0m, 7.0m, 3.0 m, 0.7 m, 1.1m C50, C40 5%, M= kg, m i=67905kg/m,, EI i= N m 2 (14), Hz, ANSYS, (14) 1 64, 2 ( 1 ), (JTG/TB ), [15] 2 Tab.2 Longitudinalfrequenciesofbridgewithdifferentmethods / / 2/Hz (2/1) 3/Hz (3/1) , Fig.2 FrequenciescomparisonbetweenpresentedmethodandFE methodfor64bridgemodels

6 第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 53 2, 64, (1): WU Hui-qin,WANG Zan-zhi,TU Hui,etal.Finite- 9%, diferencemethodsforcalculatingdynamicproperties, ofvariationalcontinuousbeams[j].journalofguan- gxiuniversity:naturalscienceedition,2010,35(1): (inChinese) [5]. 5 [J].,2000(2): (1), CHENG Xiang-yun.Commentsonthevibrationalfre- ( 50 m) quencyequationofthree-spansimplesupportedsus-, pensionbridge[j].journalofchongqingjiaotongin- (2), stitute,2000(2):22-24.(inchinese), [6],,., [J].,2006,39(1): ZHOUJian-min,WU Ding-jun,LIqi,etal.Variation (3) offreevibrationcharacteristicanalysisforpier-beam systembyenergymethod[j].chinacivilengineering Journal,2006,39(1):60-64.(inChinese) : [7] JTG D , [S]. JTG D ,Generalspecificationsfordesignof References: highwaybridgesandculverts[s].(inchinese) [1],,. Lagrange [8]. [M]. : [J]. :,2000.,2019(1): SONG Yi-fan.Kineticsofhighwaybridges[M].Bei- HUANG Yong-yu,ZHAO Yong-gang,ZHAO Wei- jing:chinacommunicationspress,2000.(inchinese) dong.analysisoncantileverbeam snaturalfrequency [9],,. basedon Lagrange motionequation[j].journalof [J].,2007,27 QinghaiUniversity:NatureScience,2019(1):12- (3): (inChinese) ZHOU Yong-jun,HE Shuan-hai,SONG Yi-fan.Dy- [2],. namictestofcurvedcontinuousrigidframebbridge [J].,2008,30(6):590- modelusinghammerexcitationmethod[j].journalof 593,619. Vibration,Measurement & Diagnosis,2007,27(3): ZHANG Huai-jing,PAN Dan-guang.Semi-analytic (inChinese) solution to dynamic characteristics of non-uniform [10],,. [M]. : continuousbeams[j].journalofuniversityofscience,2011. andtechnologybeijing,2008,30(6): ,619. YANG Yun,ZHOU Lie-mao,ZHOU Yong-jun. (inchinese) Curvedbridge with high pier[m ].Beijing:China [3],,. CommunicationsPress,2011.(inChinese) [J]. :, [11]. 2008,28(2): [D]. :,2011. ZHOU Yong-jun,HEShuan-hai,SONG Yi-fan,etal. ZHANG Xiao-dong.Practicalmethodoffundamental Longitudinalseismic vibration ofcontinuous beam frequencyforlarhgespanprestressedconcretegirder bridgeonelasticfoundation[j].journalofchang an bridgewithhighpiers[d].xi an:chang an Universi- University:NaturalScienceEdition,2008,28(2):49- ty,2011.(inchinese) 52.(inChinese) [12],,. [4],,. [J]. : [J]. :,2010,35,2010,42(5):

7 54 长安大学学报 ( 自然科学版 ) 2013 年 YIN Hai-jun,LIZi-qing,GUOqi,etal.Studyonthe dynamiccharacteristicsoflongspancontinuousrigid framebridgewithhighwalpiers[j].journalofxi an (inChinese) [14],. [J].,2012,29(8): University of Architecture & Technology:Natural ScienceEdition,2010,42(5): (inChinese) YU Jin-liang,YANG Dong-shen.Study on healthy [13],,,. monitoringofhigh-pierandlong-spancontinuousrig- [J]. :,2010,27 idframebridge[j].road Machinery & Construction (2): Mechanization,2012,29(8):28-30.(inChinese) CAO Guo-jun,HE Bo,YUAn Feng,etal.Dynamic [15] JTG/T B , characteristicsanalysisofacontinuousrigidframe [S]. bridge[j].journalofhuazhonguniversityofscience andtechnology:urbanscienceedition,2010,27(2): JTG/TB ,Guidelinesforseismicdesign ofhighwaybridges[s].(inchinese ) 欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁欁 ( 上接第 41 页 ) ChinaCommunicationsPress,2005.(inChinese) [4] Sn bjǒrnssonjt,barkercj,sigbjǒrssonar.proba- [11],,. bilisticassessmentofroadvehiclesafetyinwindyen- [J]. :,2006, vironment[j].journalofwindengineeringandindus- trialaerodynamics,2007,95(11): [5] Baker C J.Theproblemsofroad vehiclesincross winds[j].highways &Transport,1991,38(5):6-9. [6] BakerC J.Thequantificationofaccidentalriskfor roadvehiclesincrosswinds[j].journalofwindengi- neeringandindustrial Aerodynamics,1994,52(5): 33(2): HAIGui-chun,GU Zheng-qi,WANG He-yi.Research ontheefectofcrosswindsonthestabilityofhigh speedvehicles[j].journalofhunanuniversity:natu- ralsciences,2006,33(2):40-43.(inchinese) [12],. [J].,2003,16(4): namicasinica,2008,26(4): (inchinese) (6): [8],,. ZHANG Chang-sheng,MA Rong-guo.Identifying [J].,2009,31 methodoftraficaccidentblackspotson mountain (7): PANGJia-bin,LIU Xiao-hui,CHEN Li.Areviewon Reynoldsnumber,blockageandboundarylayerefects inautomotivewindtunneltests[j].automotiveengi- neering,2009,31(7): (inchinese) [9] RhondaK Y,JoelL.Estimatingtherelationshipbe- tweenmeasuredwindspeedandoverturningtruckcra- shesusingabinarylogicmodel[j].accidentanalysis searchand Development,2010,27(2): (in andprevention,2007,39(3): Chinese) [10]. [M]. : [15] JTGB , [S].,2005. GU Zheng-qi.Automobileaerodynamic[M].Beijing: [7],. PEIYu-long,MAJi.Researchoncountermeasuresfor [J]., roadconditioncausesoftraficaccidents[j].china 2008,26(4): JournalofHighwayand Transport,2003,16(4):77- HAN Wan-shui,CHEN Ai-rong.Safetyassessmentof 82.(inChinese) movingvehiclesonalong-span bridgeundercross- [13],. windandinfluencefactorsanalysis[j].actaaerody- [J]. :,2010,30 freeway[j].journalofchang an University:Natural ScienceEdition,2010,30(6):76-80.(inChinese) [14],,. [J].,2010,27(2): SONG Nian-xiu,SU Jian,WANG Dong-jie.Analysis onlateralstabilityofsemitrailertrainrunningalonga curve[j].journalofhighwayandtransportationre- JTGB ,Technicalstandardofhighwayengi- neering[s].(inchinese)

第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 49 bridgesastherigidframebridges.2tabs,2figs,15refs. 犓犲狔狑狅狉犱狊 :bridgeengineering;continuousrigidframebridgewithhi

第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 49 bridgesastherigidframebridges.2tabs,2figs,15refs. 犓犲狔狑狅狉犱狊 :bridgeengineering;continuousrigidframebridgewithhi 第 33 卷第 3 期 2013 年 5 月长安大学学报 ( 自然科学版 ) JournalofChang anuniversity(naturalscienceedition) Vol.33 No.3 May.2013 文章编号 :1671 8879(2013)03 0048 07 高墩连续刚构桥纵向振动基频的能量法计算公式周勇军, 张晓栋, 宋一凡, 赵煜 (1. 长安大学公路大型结构安全教育部工程研究中心,

第 3 期 周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 (L=180 m), 20 1,, (60~, U V 300m), C,,, U +V =C (1) bridgesastherigidframebridges.2tabs,2figs,15refs. K

第 3 期周勇军, 等 : 高墩连续刚构桥纵向振动基频的能量法计算公式 (L=180 m), 20 1,, (60~, U V 300m), C,,, U +V =C (1) bridgesastherigidframebridges.2tabs,2figs,15refs. K