L=40m空腹式悬链线无铰拱石拱桥计算修改版

1、L二40m空腹式悬链线无较拱石拱桥计算（修改版）L=50m空腹式悬链线无较拱石拱桥计算1. 设计资料某等截面空腹式悬链线无较拱石拱桥上部结构为等跨50m的石砌板拱，下部结构为重力式墩和U型桥台, 均置于非岩石土上。(1)设计标准1) 设计荷载公路 II级汽车荷载，人群荷载3kN/m2o2) 跨径及桥宽净跨径Lo=5Om ,净矢高fo=lOm ,净矢跨比fo/Lo=l/5o桥面净宽为净9+2X 1.5, Bo=12mo(2)材料及其数据1) 拱上建筑 拱顶填料厚度hd=0.5m,包括桥面系的计算厚度为0.68m,换算平均重力密度幷=20kN/m3。护拱为浆砌片石，重力密度A=23kN/m3o腹孔

2、结构材料重力密度r3=24kN/m3 o 主拱拱腔填料为砂、砾石夹石灰炉渣黄土，包括两侧侧墙的平均重力密度/4 =20kN/m3 o2庄拱M7.5砂浆砌MU80块石，重力密度Z5=24kN/m 拱圈材料抗压强度设计值爲=4.37MPa。拱圈材料抗剪强度设计值/,rf=0.075MPa o弹性模量Em=7300MPa。拱圈设计温度差为+22C, -15Co(3)设计依据1) 公路桥涵通用设计规范(JTG D60-2004),简 称桥规D60；2) 公路垢工桥涵设计规范)(JTG D61-2005),简 称桥规D61；3) 公路桥涵设计手册一一拱桥上册(石绍甫)、下 册(顾安邦)，简称拱桥。2.

3、主拱圈计算(1)确定拱轴系数拱轴系数m值的确定，一般采用“五点重合法”， 先假定一个m值，定出拱轴线，拟定上部结构各种几何 尺寸，计算出半拱恒载对拱脚截面形心的弯矩W和自拱 顶至1/4跨的恒载对1/4跨截面形心的弯矩EMg。其比值 XMj/XMll4 = yyi/f o 求得 皿仃值后,可由肌m = (1/2)(/y1/4-2)2-l中 反求m值，若求出的m值与假定的舰值不符，则应以 求得的肌值作为假定值，重复上述计算，直至两者接近 为止。1) 拟定上部结构尺寸主拱eiA J几何尺寸a.截面特性截面高度：d = mk = 4.8xl.2x5000 = 98495c?主拱圈横桥向取lm单位宽度计

4、算 9横截面面积：A=0.90m2 ；惯性矩：/= = lxo=O.O6O75；截面抵抗距:Vy = =0.135m3；6截面回转半径：=0.2598 ob.计算跨径和计算矢高假定m=2.814,相应的仏仃=o.2i o查拱桥表(皿)20(8) 得sin =0.70097, cos =0.71319；计算跨径：I = + d sin（!）=50+0.90 x 0.70097=50.63087/?/x（l-0.71319）=10.1291/n计算矢臥 / =九+/2（1 -cosg）=10+WC.拱脚截面的投影水平投影:x = d sin =0.90 X 0.70097=0.63087/;/ ；

5、竖向投影：y = cos =0.90 x 0.71319=0.641 87/h od计算主拱圈坐标（图3.4-63）沿跨径24等分每等分长A/- 1 -50,6?87-2.10962(w)； 以拱顶截面形心为坐标原点，拱轴线上各截面的纵坐标片表(皿)1值Xf,相应拱背及拱 腹纵坐标：其数值见表3.4-9 o主拱圈截面坐标表表 3.4-9截面号x=d*a 1yi/ 表 -iyicoa叽養 CUD -20 (7)d/2cos(ch kg-1)lg0=2fksh/(m-l)lcoscpd/2(l-l/cosp)h=y+(l/cos01#腹拱墩22.23087 |().8781551.488127.4

6、11330.8276900.80296-0.110436.4009002#腹拱墩18.43087 |().7280491.233754.816850.5899880.8901-0.055563.8612913#拱座中心线14.92742 |().5896570.999233.027450.4388060.94499-0.02622.101251空实腹界限14.82397 f).5855700.992312.982280.4348210.946270.025552.056727表 3.4-10注：g = 2xlLk =ln 伽+-1) = 1.6946横壤高度也可根据表349的有关数值内插计算得

7、到。2) 恒载计算恒载分主拱圈、拱上空腹段、拱上实腹段三部分进行计算，不考虑腹拱推力和弯矩对主拱的影响，其计算图式见图3.4-64 o主拱也2=表刃-19值A灯=0.55288 x 0.9 x 50.63087 x24 = 604.6444kN=却 一19值人去厂/4=0.28762 x 0.9 x 50.630872 x 24 / 4 = 3981.471 kN.mM,=表/ 一19 值A/5/2/4=0.52303 x0.9x 50.630872 x24/4 = 7240.2096kN.m图 3.4-64拱上空腹段的恒载 乩腹孔上部（图3.4-65）图 3.4-65腹拱圈外弧跨径:你二/；

8、 + 2/sin%= 3 + 2x0.30x0.689655 = 3.41379加腹拱内弧半径:Ro = 0.72500” = 2.1750m腹拱圈重:匕=冷兀（凡+ -町卜29.7664W腹拱侧墙护拱重:Pb=（R0 + dr）2/2 *2 = 16.9004kN（以上三个系数依次分别査拱桥上册表3、表1-10、表 19）填料及路面重:Pc =1； =3.41379 x 0.68 x 20 = 46.4275kN（hd = 0.68m）两腹拱之间起拱线以上部分的重力（图3.4-66）：IrXI4I814.弋丄之厂色=土竺竺=0.5856 y % 2%/50.63087k = n(m + Jm

9、2-V) = ln(2.8144-V2.8142-1) = 1.694533= 1.694533 x 0.4836 = 0.8195则悬链线曲边三角形的重量为:l.f2(m 一 l)k(M08195-0.8195) x!9 = 147.7jWV50.63087x9.94812(2.814-1)x1.694533重心位心 gmWS 75北=0.44,0.各块恒载对拱脚及拱跨1/4截面的力矩见表3.4-11o半拱恒载对拱脚和1/4拱跨截面的弯矩表3.4-11块号自 M/KN1/4截面拱脚截面力脅/m力矩/(KN-m)力和m力矩/(KNm)P(M2604.64443981. 741231427445

10、.7121P13187.40541/2-22.230873.0846578.0707块号自 M/KN1/4截面拱脚截面P14143.12141/24843087=7.00671002.808748.336614.927421-1/4=2.2697109.70961/2-14.92742=10.3880502.1206P16201.60601/4-1/2*14.82397=5.24571057.56461/2-1/2*14.82397=17.90353609.4530?17147.70001/4-10.5=2.1177312.78431/2-10.5=14.81542188.2346合计1332

11、.81381480.058515326.3997由表得3）验算拱轴系数JNMg = 15326.3997 / 1480.0585 = f / yl/4(丄一2尸 一1 =丄(10 3553-2)2-1=2.843 儿4212.814-2.8431=0.027 -.5 14 _ q | （满足半带宽要求）2故可确定2.814为设计拱轴系数。（2）弹性压缩内力I）弹性中心儿=af= 0.333431x10.1291= 3.38m2）弹性压缩系数系数:(7)2 = (S2 = 000066查表计算得:“ =11 1272( )2 = 0.0734“ =9.18594(=)2 = 0.006061 +

12、 0,07341+0.0()606=0.0729主拱圈截面内力计算大跨径拱桥应验算拱顶、3/8拱跨、1/4拱跨和拱脚四个截面，必要时应验算1/8拱跨截面。为节省篇幅,本例只验算拱顶、1/4拱跨和拱脚三个截面的内力。其 余截面，除不计弹性压缩的内力必须在影响线上直接布载求得以外，其步骤和1/4拱跨相同。1）恒载内力计算计算拱圈内力时，为了利用现有的表格，一般采用所确定的拱轴线进行计算。但是在确定拱轴系数时，计算得的恒载压力线与确定的拱轴线很难在5五点”完全重合，本例中二者相差0.21001-0.21=0.00001 o当这个偏差较大时，要用“假载法”计入其影响。本例不计偏离的影响。不计弹性压缩的

13、恒载推力=15326 3997 -1513.1058./10.1291在不考虑偏离影响时，拱为纯压构件，弯矩、剪力为零，轴力为皿=旦cos ri / m0.(M)000.60662.8888y = ” 一 几-1.2011-0.54051.6877coscp1.00000.94210.7132“ /(1 +0.07290.07290.0729H k / kN1513.15081513.15081513.1508S = H：/kN I + “11.029911.029911.0299H 严 H；-SlkN1502.12991502.12991502.1299N.=-Scos/kN cos1502

14、.12091595.75542113.7695A/t =S(y y)/ kN m29.896511.0674-59.7666sin(p0.(H)000.33530.7010Q =Ssin/AAf0.000()3.69837.73202）公路 II级活载内力计算汽车荷i设计荷载：公路II级、人群荷载3kN/m公路【I级一车道荷载：弘=10.5x0.75 = 7875（kN/m）单位拱宽承担均布荷载：= 1.96875 / /n12P =360V单位拱宽承担集中荷载：曲仝皿NE12单位拱宽承担人群均布荷载:3.0x2xl.512=075伙/加)截面项目汽车荷戟荷裁影响线面积（S-眾）汽内力粧缩）

15、飙影响钱径标人余广生内力 效应（计弹性圧缩P沪生内力 姬（不# 弹性陇）/GNn） otKXP严生内力MW性瞒）表(Hl) -H CS9)乗数(10)-13；敲 m)-：2 cb g (ID) -22乗数协注|内力飆（時ABV 劎/(MP*/(kX/a)拱顶Mmax1.96875900.750.007252563.485033.922412.92280.0540550.63087246.2939228.3390262.2614相应H10.06913253.081231.933312.16510.233024.9963104.781497.1428129.0761相应N0.3456750.630

16、8731.944412.16931.165121104.860897.2164129.1608Mmin0.004562563.485021.3360-8.1280-0.0114650.63087-52.220748.4138-69.7498相应H10.05903253.081227.267810.38770.108974.996349.000245.428172.6959相应N0.2951650.6308727.276610.39110.54487149.038345.463472.74001/4截面Mmax0.008822563.485041.268315.72130.059150.6308

17、72693056249.6732290.9415相应Hl0.04035253.081218.63897.10050.138084.996362.090057.563776.2026相应N0.2215550.6308720.47417.79960.93089183.780177.672598.1466Mmin-0.010472563.4850 48.98861&6623-0.0297350.63087135.4730-125.5970-174.5856相应Hl0.08781253.081240.562215.45220.207414.996393.265486.4664127.0285相应N0.

18、4660050.6308743.064416.40551.08493197.643790.5255133.5899拱脚Mmax0.019942563.485093.298235.54220.0540150.63087246.1116228.1701321.4682相应Hl0.09242253.081242.691716.26350.198464.996389.240959.314196.2651相应VI0.1706750.6308715.77216.00840.29351126.415917.749531.4017相应N0.4491950.6308741.510915.81370.913441

19、82.209654.566890.4984Mmin-0.014092563.4850砧.926325.1148-0.0591350.63087269.4423-144.6952-190.4897相应H10.03575253.081216.51406.29110.064494.996328.999019.274233.5677相应VI0.3293350.6308730.434311.59400.93757184.381356.697883.0416相应、0.3583250.6308733.113412.61460.88717179.845353.431682.09443）温度变化内力计算 1月平

20、均最高温度：20C月平均最低气温：0c拱圈合拢温度：15C拱圈砌体线膨胀系数=0.8xl0-5温度升高值：八=37。0-15。（7 = 22。（7温度降低值：Ar, =OC-15C = -15C2.内力计算：1.）单位温变引起弹性中心的赘余力:采用M7.5砌MU80号的块石，弹性模量:E=7300Mpao查附表有：1讐碧rraStlO.8xlO5x 1x50.63087“口一-(】+“) EI=,一032433(1 + O.O729)xO.O99373x5O-63O8：X，12917.3 xl()6x 0.060752）内力计算表拱圈温度变化内力见表3.4-16项目拱脚L/4截 面拱顶单位温变

21、（the）coscp1.00000.94210.7132y =儿一儿-1.2011-0.5405-1.6877拱圈温度内力表 3.4-16=1 (kN)0.32430.32430.3243N(kN)0.32430.30550.2313M(kN-m)0.38590.1753-0.5473N(kN)0.12630.0536-0.1266温升A/ = 5CM(kN*m)1.94750.8765-2.7365温降N(kN)-11.9991-11.3035-8.55810 = -37CM(kN*m)-14.4115-6.486120.2501注：1 温度下降时的内力与温度上升的内力符号相反；2当砂浆为小

22、石子混凝土时，应酌情计入徐变的影响。(4)主拱圈正截面强度验算1)正截面受压强度验算荷载组合根据桥规021和桥规022的规定，本例可得 如下两种荷载组合：S| =乙E 乙 = 1.2S 恒 +1.4S 汽 +1.12S 人3Hs = Y鼎Z乙门=12S恒+14S汽+ O98(s型+ S人)荷载效应不利组合的设计值见表3.4-17主拱圈荷载效应不利组合的设计值表3.4-17荷載效应系数拱顶1/4截面拱脚M/(kNm)N/kNM/(kN-m)N/kNM/(kNm)N/kN1恒載1.2(1.0)29.89651502.120911.06741595.7554-59.76662113.76952汽车M

23、max1.4262.2614129.0761290.941576.2026321.468296.2651Mmin-69.749872.6959-174.5856127.0285-190.489733.56773上升0.981.94750.12630.87650.0536-2.7365-0.1266下降-14.4115-11.9991-6.4861-11.303520.2501-8.55814人群Mmax12.922812.165115.72137.100535.542216.2635Mmin8.12810.3877-18.662315.4522-25.11486.2911荷載俎合MjNjMjN

24、jMjNj组合1S也+S心.+S 人.ntx417.51531996.8765438.20682029.5427418.14282689.5097Sfu +S汽年.聞+S人.min-50.94802201.8386-214.91012403.7982340.389530()6.6046组合2Sfl+S畑唤+(S“ 升+S人.mQ417.61471995.2972436.86482028.6011410.48512687.1087max +i, R +S人，皿367.58624154.6783227.59494385.264984.56125635.9821s也+s“，+(s“升+s人詔-47.

25、201019163649214.87312099.7574306.25592599.3323S也+S汽輸俪+(Sq.障+S人，mir?7 力.84926066.8820659.90786326.1608493.355182103712注：荷 载安 全系 数的取值，对于结构自重，当其弯矩与车辆荷载的弯矩同号取1.2,异号时则取0.9,轴向力的荷载安全系数取值与其相应的弯 矩相同，对于其它荷載都取1.4拱圈截面抗力主拱圈承载能力根据公式： 人2 Nj3902. 04733516. 54714136. 8991组合3满足强度耍求2379. 82992324. 80031930. 4436组合4352

26、4. 26383782. 06683789. 2289组合53897.96473404. 73264116.1869组合63162. 59193386. 93893234. 7748经验算：符合强度要求。6、拱圈整体“强度稳定力验算1 恒载作用效果釆用无支架施工或在拱上建筑完成前就落架的拱式拱上建筑拱桥，应进行“强度稳定”验算，其验算恒载为主拱自重和腹拱的自重。劭2+(出+ 1#腹拱墩)+(： +2#腹拱墩)(|p； + 3#拱座)=485. 2223+(29. 7664+84. 34) +(29. 7664+40. 056) + (1x29. 7664 + 4.0084) 2=688.042

27、7换算均布荷载:688.042750.63087/2= 27.1788)WV/m= (0.04035 + 0.0878 吟 x270.04035 + 0.0 河牆寻27皿= 881.54143Md = (0.0082-0.01047)x/2x27.1 788 = (0.0082 一 0.01047) x 50.630872 x 27788 = -15 &1565(QV“cos(pni = j(= 0.9284J + 4(/ / 厅Jl + 4(10.1291 / 50.63087)22活?:作用效应cos(pnt=881 54 -944.5261AW0.9284拱式拱上建筑的拱桥不必验算腹拱合

28、拢后的 “强度-稳定”，在腹拱合拢前没有活载作用的。温度上升:3 温度作用效果“ 0.7xAT,f 0.7x0.1263 八 “ “ N占 t = = 0.2445RN九COS%0.9284Mdt = 07N、,牙一儿)= 07x 0.1263 x (0.5405) = 0.0477伙 N?)温度下降:= O7x = 0.7x(-11.9991)=_9 0471 cos%0.9284M/,=07 竹(厂儿)=().7 x (-11.991) x (0.5405) = 45399伙2加)4 荷载组合 温度上升：N = 12Nd +1 ANdl =1.2x 944.526 l + 1.4x 0.2

29、445 = 1133.7736RNM = 1+1 4M 山=1.2 x (-158 1565) + 1.4x (0.0477) = 一 189.7210N 讪M189.7210八e =y N一一-0.1673加1133.7736温度下降:N = 1 帆 +1 AN姑=1.0x 944.5261 +1.4x (-9.047 ) = 957.19193M = 1 0M+1 AM dJ =1.0x(-158565) +1.4 x 4.5399 = -151 8006QVmMeK =:5仙6y N 957.19195“强度稳定”验算根据公式：(pAfcd进行验算4 = 0.90 加=437xlO*N/

30、r,% = i.oI1一( S1 + (于尸 1 + cQ(0y-3) 1 + 1.33(仝)2 vra = 0.002,匕=0,由于主拱圈为板拱其宽跨比大于1/20,可取仪=3。因此似=1。只 0、=叭 =:1 + (評2 1 + 00(0-3)1 + 133(子)2S =丄 / = 1 10575 x 50.63087 = 55.9850加i=i = 0.259&m = 8,y = 0.45,a = 0.002厶=0.36S =0.36x 55.9850 = 20.1546加1.3x20.15463.5x0.25980.45 ().1673 . 1 + (Y0.2598“002 x 28.

31、814x(28.8 3)*33(溜尸= 0.1337= 28.814温度上升:1133.7736RN = yJJ (pAfed =0.1337x0.90x4370 = 2779184RN 成立。.0.1586 M1 ()* I ,0.1586 ,1+(Y0.2598温度下降: = ().14001 + 0.002 x 28.814x(28.814-3) 1 + 1.33(；：；爲加926.1862 = yNd (pAfcd = 0.1400 x 0.90 x 4370 = 2864.5812kN成立。因此拱圈整体“强度稳定”验算符合要求。7 拱脚直接抗剪验算1 恒载作用效应Q =0, sin转

32、一匕 cos? =1513.1058x0.7010-1332.8138x0.7132 = 110A244kNN, =21137695册:作用效应1. )乞：单位拱宽承担均布荷载：1.97册/加H 屮=31.9333 + 27.2678 = 59.2773&NVyI = (0.17067+0.32933) x 50.63087 x 1.97=49.872. ) Pk :单位拱宽承担集中荷载：90.00kN(1)不考虑弹性压缩的水平推力(2)由于弹性压缩引起的水平推力：弹性压缩系数上-=0.07291+心=0.0729 xH；2 = 0.0729x148.547 = 10.83RN(3) 考虑弹性

33、压缩的水平推力：Hq=H；2_H；2 = 148.457 -10.83 = 137.627与水平推力对应的竖向反力：匕 2 =05xR =0.5x90 = 45.00W(4) Qq=HqSh0j-VqCOS0j=(59.2773 + 1.2x97.1428) x 0.7010 一 (49.87 + 1.2x45)x0.7132=123.27 - 7595 = 4732bVNq=HqCOS(Pj+VqSgj=(59.2773+97.1428) x 0.7132+(49.87 + 45) x 0.7010= 111.559 + 66.504 = 178.063W3) 单位拱宽承担人群均布荷载：0.

34、75kN/m=12.1657 + 10.3871 = 22.5528&NVr = (07067 + 0.32933) x 50.63087 x 0.75 = 1 & 98663Qr = Hf sin p _ 匕 cos pi= 22.5528x0.7010-18.9866x0.7132= 15.81 13.541 = 2.2693Nt = Hr cos , + V sin(p=22.5528 xO.7132 + 18.9866 x 0.7010=16.085 + 13.310 = 29.3950V3 温度作用效应温升：Qt = Hf sin cp = 8.3833 x 0.6616 = 5.5

35、47Ar/V 温降：Qt =Ht sin p = -5.7159 x 0.6616kN = -3.7816kNNt = H cos 0 = 8.3833x0 .7498 = 6.286AJV Nt = Ht cos 196.91492810.418467.50001405.20921472.7092满足&主拱圈裸拱强度和稳定性验算本桥釆用早托架施工，应验算裸拱在自重内力作用下的强度和稳定性。1 弹性中心的弯矩和推力也=表= o. 8636x迴空严陛ZI = 2579.7他如Ht =表(111)-16-八空一 =0.53288X. 24 5().63(用7 = 723.86%L4(1 + /)/

36、4(1+ 0.00606)xlO.12912 截面内力1）拱顶截面内力:M = M 厂 Hsys = 2579.784 - 723.868 x 1.2011 x 2.814 = 133.1859kN.m N = Hs =723.868W2) 1/4截面内力：M = My - H,(yx 一”)一表(III)一 19(8)=2579.784 - 723.868(3.38 - 1.71) -().28762 x )” “ 沢 5()(、3(用7 = 一 92.828ORN加 4N = a cos % + 表(III) 一 19(8) Ay5l sin 久=723.868 x 0.94212 + 0

37、.39351 x 0.90 x24x 50.63087 x 0.3353 = 826.2679AJV3) 拱脚截面内力：M = M$ - (儿一儿)一表(III)一 19(8)弓-=2579.784 一 723.868 x (-4.7492) 一0.52303x“ * 二().心()8 厂=_】222.6757A;V./h4N = H、cosq +表(111)-19(8)J Ay5lsincp-= 723.868x0.7132 4- 0.55288 x 0.90 x24x 50.63087 x 0.70097 = 954.37883 强度验算承载力根据公式:心肌进行验算。0= 广亿=+ I只

38、9、i-(rA = 0.9/n2fcd = 4.37 x 10叹N / m2/0 = 1.01一(/045)1 + (“/02598尸庄=0、=3,5=0代=1,其中横桥向为X方向（每米宽度）,竖桥向为Y方向。匕=M / N、m = & y = 0.45,/ = i = 0.25981）拱顶截面M =133.1859 好/讪N = 72386%Nev = M / N = 133.1859 / 723.868 = 0.1828所以：723868 =人血=卩4為=06684 x 0.90 x 4370 = 2628.82成立。2) 1/4截面cos(pnt = - .= = f= 0.7453Jl

39、 + 4( / /)2Jl + 4(10291 / 50.63087)2z 叽 786.119、小,= = 846.7460/JVcos pm 0.92842）活载；作用效应M =-1928280N加N =8262679ZW = M / N =-192.8280/826.2679 = -0.2334y1-(0.0679/0.45/055051 +(0.0679/0.2598)2所以：826.2679=y、N& =(pAfcd = 0.5505 X 0.90 X 4370 = 2165.12成立。3）拱脚截面M =-1222.6757皿nN = 954.3788e、=M/N= -1222.6757/954.3788 = -0.1281所以：954.3788 = /0;Vz