直墙拱结构的设计计算步骤及实例

一、计算步骤

1. 计算结构的几何尺寸； 2. 计算作用在结构上的荷载； 3. 计算拱顶变位；

4. 计算侧墙的弹性特征值及换算长度=h，判别侧墙所属类型5. 计算墙顶的单位变位 6. 计算墙顶力Mh，Hh

7. 计算拱顶未知力X1P，X2P，X1，X2 8. 考虑弹性抗力时，计算j 9. 计算拱圈截面内力 10.计算侧墙截面内力 11.绘制内力图，校核计算结果 12.计算拱圈、侧墙截面的内力和偏心距 13.计算拱圈和侧墙各截面的安全系数 14.计算配筋量 15.绘制结构施工图 16.计算工程量 二、计算实例 （一）设计基本资料 结构断面如图所示。

1、岩石特征

软质石灰岩，隧道埋深H=50m，岩石坚硬系数fi2，内摩擦角=650，容重=24kN/m3，侧向岩层地基系数K1=2.5×105kPa/m，基底岩层地基系数K2=3.125×105kPa/m。 2被覆材料

采用C20砼，弹性模量E=2.55×104MPa，容重=25 kN/m3，砼轴心抗压强度设计值fc10MPa，弯曲抗压强度设计值fcm11MPa，抗拉强度设计值ft1.1MPa。钢筋采用25MnSi钢：强度设计值fy340MPa，弹性模量Es200GPa。

3结构尺寸

净跨l012m，拱顶砼厚度（预先确定）d00.7m，拱脚厚度dj1.0m，内拱矢高f03.75m，侧墙厚度dg1.0m，内墙高h011.25m，底板厚度dB0.12m，墙基埋深dm0.2m （二）结构几何尺寸计算 1拱圈内圆几何尺寸

内圆净跨：l012m，内圆矢高f03.75m

2l04f02l022内圆半径计算：(R0f0)R0，从而有：R06.675m

8f2022拱圈轴线圆的几何尺寸 拱脚截面与拱顶截面厚度之差：

djd010.70.3m

轴线圆与内圆的圆心距：

2R0(R00.5)26.6752(6.6750.50.3)m00.275m

2(f00.5)2(3.750.50.3)2轴线圆半径：

RR0m0d07.30m 2sinjl0/20.88235

Rdj/20 j61.92715cosj0.47059 dhdjsinj0.88235m

dvdjcosj0.47059m

计算跨度：ll0dh120.8823512.88235m 计算矢高：ff03拱圈外圆几何尺寸

外圆跨度：l1l02dh13.7647m 外圆矢高：f1f0d0dv3.97941m

l124f12外圆半径：R17.94118m

8f1d0dv3.864705m 22外圆与轴线圆的圆心距：

m1R1R0d00.29118m 24校核公式

外圆与轴线的圆心距：mm0m10.56618m

2djR12(m1sinj)2R0(m0sinj)2mcosj1.0m

5侧墙的几何尺寸

拱脚中心到侧墙中心线的垂直距离：

eh(dgdh)/20.058825m

侧墙的计算长度（从拱脚中心算起）：

hyh0dBdmdv/211.80795m

结构总高：hkhydv/2f116.0226m （三）计算拱顶单位变位

采用分块总和法计算变位，将半拱轴线长分10等分，计算过程列于表1，故拱顶单位变位：

0j1s61.92715115R7.33.14159261.03138103E103E1801032.55104s10ni111.03138105742.85277.661634103 3Ei0Ii122122s10niyi1.03138105737.09597.602259103 3Ei0Iis10ni210ni(yicos2i)16.0392103 3Ei0Iii0Fi校核计算：

sssnn[(1y)2cos2]38.90534910-33EIF1121222(7.5827.344815.229)10738.90535210-3

判别：1121222-ss31090 说明单位变位计算结果正确。 表1 变截面圆拱拱顶单位变位计算（见附图） （四）计算拱顶载变位 1计算荷载

（1）岩石坚固系数fi2，隧道半跨度：

h1al1/213.76473.44m（考虑到有一定的间隙） fifi4隧道埋深H=50m>(2-2.5)h1，属于深埋 因此围岩竖直压力：

q地h1240003.482560Pa（采用均布荷载模式） q地f1h124(3.979413.44)12.945kpa

（2）自重计算

q自hd0250.717.5kPa

q自h(djcosjd0)25000(10.7)35.6kPa

0.47059因此：qq地q自100.06kPa

qq自35.6kPa

在实际设计中，外载还应包括超挖回填引起的拱顶荷载，一般取30cm回填高度，可忽略不计。 2计算载变位

先分别计算在均布荷载和三角形荷载作用下的载变位，然后叠加，计算过程列于表2。

表2 变截面圆拱拱顶载变位计算(见附图) 2计算载变位

先分别计算在均布荷载作用下河三角形荷载作用下的载变位，然后叠加，计算过程列于表2. 在均布荷载q作用下的载变位：

'1ps10'niMpi4.764543Ei0Ii2p'10s10'ni'ni(MpiyiNpicosi)-9.40586 3Ei0IiFi0i在三角荷载q作用下的载变位：

''1p10s10''nis10''ni''''niM-0.828152p(MpiyiNpicosi)-1.86707 pi3Ei0Ii3Ei0IiFii0拱顶总载变位：

' 1p'1p'1p5.59269 2p'2p''2p11.27293校核计算：

'sps'n'n[Mp(1y)Npcos]14.1704 3EIF'1p'2p'sp0，可见计算正确。

''sps''n''n[Mp(1y)Npcos]2.69522 3EIF''''''1p2psp0 , 满足要求！

（五）在荷载作用下多余未知力的计算 1判别侧墙类型

K414EI42.5108142.5510101.03120.414

侧墙特征长度：hy0.41411.807954.89，故侧墙属于长梁 2计算墙顶单位变位

nK21.25 K14340.414310.114105 5K12.5102220.41425 u120.137105K12.510u2220.4145 0.331105K12.5103由外载引起的墙顶弯矩与水平力

1l1lMhpql(eh)lq(eh)21.1233q7.10424q

2446Hhp0

4计算多余未知力

a111117.66278103

a12122f17.6059103 a2222u22f2f2116.07013103a101pMhp1Hhp247.64089q23.2707q)103

a202pf(Mhp1Hhp2)Mhpu1Hhpu2(94.1242q52.487q)103X1a22a10a12a200.76113q0.3867q 2a12a11a22a11a20a12a105.49685q3.4491q 2a12a11a22X2（六）弹性抗力作用下多余未知力的计算 1计算j1时引起的墙顶截面内力及变位 通过积分可得到：

MjR222[cossinj2(sinjcosj)]0.801678j23(12cosj)R22[cossinj2(sinjcosj)]0.1102 j23(12cosj)NjQjR[4cosjsinj2(sinjcosj)]1.10267

3(12cos2j)因此墙顶内力（要考虑偏心距）：

MhMj(NjsinjQjcosj)eh1.41782 HhNjcosjQjsinj0.92108

VhNjsinjQjcosj0.61614

墙顶变位：

Mh1Hh20.2878210uMhu1Hhu20.499105

52计算j1时的拱顶载变位

采用分块总和法计算，将弹性抗力所分布拱轴线长对应圆心角

j450四等分

061.927154504.23180

414R34ni[cos2isin2i2(sinicosi)]30.139210E(12cos2j)3i0[d0m(1cosi)]324R44ni[cos2isin2i2(sinicosi)](1cosi)23E(12cosj)3i0[d0m(1cosi)]R24nicosi[cos2isin2i2(sinicosi)]d0m(1cosi)3E(12cos2j)3i00.421610-33计算j1时的多余未知力

a111117.66278103 a12122f17.6059103

a2222u22f2f2116.07013103

110.1421103

22fu0.4377103

X1a22a1a12a20.016 2a12a11a22a11a2a12a10.035 2a12a11a22X2

4计算弹性抗力

根据hK1u0及jhsinju0K1sinj

u0(X1X1j)u1(X2X2j)u2(X2X2j)fu1Mhpu1Hhpu2uj1.9402q1051.9416q1050.471j105从而可求得 j2.099q2.101q

5.在弹性抗力作用下多余未知力计算 X1X1J0.0336q0.0336q X2X2J0.0735q0.0735q

（七）计算弹性抗力及外载共同作用下的多余未知力 X1X1X1J0.72753q0.4203q

X2X2J5.57035 X2q3.5226q

将q=100.06kpa及q=35.6kpa代入得

57.83397103N•m X168.2774104N X2（八）计算拱圈内力 1.拱圈任一截面的内力

X2yMpM MX1cosNpN NX2各截面的内力计算见表3 （九）计算侧墙内力

侧墙为长梁，其任一截面的弯矩与轴力为

Mu0K1K11MH2 304h1h23224NVhdxh

现将侧墙分为6等分：

hyhy61.96799

侧墙各截面的内力计算见表4

附表

表1 变截面圆拱拱顶单位变位计算 截面号 0 1 2 3 4 5 6 7 8 9 10 Fd1 0 6.192715 12.38543 18.57815 24.77086 30.96358 37.15629 43.34901 49.54172 55.73444 61.92715 sin 化为弧度 0 0 0.108083 0.107873 0.216167 0.214487 0.32425 0.318598 0.432333 0.41899 0.540416 0.514493 0.6485 0.603991 0.756583 0.686441 0.864666 0.760879 0.972749 0.826437 1.080833 0.88235 cos 1 0.994164688 0.976726854 0.947890007 0.907990693 0.857494561 0.796990932 0.727185921 0.648894197 0.563029472 0.470593843 yR(1cos) d0m(1cos)0 0.7 0.04259778 0.703303837 0.16989397 0.71317679 0.38040295 0.729503636 0.67166794 0.752093829 1.04028971 0.78068373 1.4819662 0.814939674 1.99154278 0.854461875 2.56307236 0.898789084 3.18988485 0.947403973 3.86466495 1 d 0.343 0.347879599 0.362736788 0.388224002 0.425418211 0.475801038 0.541223175 0.62384697 0.726061431 0.850365447 1 3112 Id334.98542274 34.49469313 33.08183898 30.90998996 28.20753719 25.22062595 22.17199958 19.23548656 16.52752714 14.11157995 12 n 1 4 2 4 2 4 2 4 2 4 1 n F1.428571429 5.687442311 2.804353742 5.483180349 2.659242666 5.123713801 2.454169386 4.681308922 ncos I34.98542274 137.1736233 64.62384099 117.1970824 51.22436248 86.50619827 35.3417652 55.95110002 n I34.9854227 137.978773 66.163678 123.63996 56.4150744 100.882504 44.3439992 76.9419462 ny I0 5.87758911 11.24080986 47.03300514 37.8921969 104.9470304 65.71630793 153.2331773 n2y I0 0.250372236 1.909745802 17.89149378 25.4509739 109.1753156 97.38934714 305.1704274 ncos2 F1.428571429 5.621259972 2.67534042 4.926612682 2.1924049 3.767450984 1.558875006 2.475473178 n(1y) I34.98542274 143.8563616 77.40448782 170.672965 94.30727127 205.8295342 110.0603071 230.1751235 n(1y)2 I34.98542274 149.984323 90.55504348 235.5974639 157.6504421 419.9518803 273.1659622 688.5787282 2.225216167 4.222063779 1 21.4492329 31.78094166 5.647126113 Σ 33.0550543 84.7224961 56.4463198 180.0572605 12 46.37597938 742.852731 737.0958526 217.1498883 574.3619276 179.2276219 1527.977114 0.936957705 1.338403451 0.221458565 27.14280829 117.7775504 236.5035803 58.37597938 419.6499348 990.9227683 283.9795807 3745.02155 表2

变截面圆拱拱顶载变位计算 垂直均布载q 截面号 0 1 2 3 4 5 6 7 8 9 10 1MPqx2 qxsin NPxRsin 2x² x³ 0.000000 0.000000 0.000000 0.000000 0.000000 0.7874725 0.620113 0.4883219 -31.02425274 8.49979527 1.5657548 2.451588 3.8385856 -122.652947 33.6035471 2.3257637 5.4091766 12.580466 -270.6211068 74.142769 3.0586295 9.3552141 28.614133 -468.0413627 128.23051 3.7557991 14.106027 52.979404 -705.7245338 193.349187 4.4091363 19.440483 85.715738 -972.6073496 266.467767 5.0110161 25.110282 125.82803 -1256.267404 344.18285 5.5544142 30.851517 171.3621 -1543.501374 422.877089 6.0329888 36.396953 219.58241 -1820.939582 498.887557 6.4411546 41.488473 267.23367 -2075.668308 568.676249 Σ n I0.000000 -4280.688312 -8115.170088 -33459.58278 -26404.58829 -71195.25797 -43129.29949 -96659.65906 -51020.5217 -102785.338 -24908.0197 -461958.1254 MPny I0.000000 -182.3478101 -1378.718456 -12728.12391 -17735.11547 -74063.69413 -63916.16408 -192501.8459 -130769.2891 -327873.3926 -96261.15066 -917409.8421 MPncos F0.000000 48.06000406 92.04305949 385.3534726 309.6212344 849.4906202 521.1978273 1171.661043 610.6048545 1185.92873 267.6155412 5441.576388 NP n(1y) MPI0.000000 -4463.036122 -9493.888544 -46187.70668 -44139.70376 -145258.9521 -107045.4636 -289161.5049 -181789.8108 -430658.7306 -121169.1704 -1379367.967 2MPqx3 30.000000 -0.899642337 -7.071879821 -23.17716909 -52.71621773 -97.6046961 -157.9153004 -231.8147295 -315.7027906 -404.5397435 -492.3283243 垂直均布载q x2qsin NPl0.000000 0.184858014 1.453125991 4.762432004 10.83209953 20.05575947 32.4483494 47.63316359 64.87043643 83.12460482 101.1633543 n I0.000000 -124.13155 -467.90158 -2865.6243 -2973.9893 -9846.6061 -7002.5959 -17836.276 -10435.573 -22834.78 -5907.9399 -80295.418 MPny I0.000000 -5.287728002 -79.4936564 -1090.091913 -1997.533302 -10243.32301 -10377.61051 -35521.70754 -26747.12845 -72840.31796 -22832.20821 -181734.7023 MPncos F0.000000 1.045234225 3.980239392 24.75251108 26.15483647 88.11611668 63.46737317 162.1519552 93.66836004 197.5993503 47.60685164 708.5428282 NPn(1y) MPI0.000000 -129.4192733 -547.3952354 -3955.716168 -4971.522646 -20089.92914 -17380.20646 -53357.98398 -37182.70133 -95675.0977 -28740.1481 -262030.12

表3 拱圈各截面的M N值 截面号 0 1 2 3 4 5 6 7 8 9 10 X1 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397 57.83397  y•X20 29.084655 115.99918 259.72924 465.5974 750.2828 1050.848 1389.774 1754.9912 2121.652 2464.693 M 0 0 0 0 0 0 0 0 0 -9.00303 -16.80556 NP NPNP0 8.684653287 35.05667312 78.90520099 139.0626099 213.4049468 298.9161164 391.8160141 487.7475251 582.0121615 669.8396031 NX2cosNPN 682.774 687.4744539 701.9403738 726.0998528 759.0150472 798.8799379 843.0808027 888.319654 930.7956115 967.2031565 1001.230994  X2yMPM N MP MX1MPMP0 57.83397 0 -31.92389508 54.9947301 0 -129.7248268 44.108328 0 -293.7982759 23.7649361 0 -520.7575805 2.67378952 0 -803.3292299 4.78754006 0 -1130.52265 -21.84068 0 -1488.082134 -40.474164 0 -1859.204164 -46.378994 0 -2225.479325 -54.996385 0.76911 -2567.996633 -62.275223 10.08215 nnyM M II2023.345889 0 7588.10536 323.236427 2918.369209 495.813328 2938.29575 1117.73636 150.8420348 101.315759 482.9790286 502.438113 -968.5030939 -1435.2888 -3114.160925 -6201.9847 -1533.060178 -3929.3442 -3104.343553 -9902.4985 -747.3026701 -2888.0744 6634.566851 -21816.651  cosX21682.774 678.7898007 666.8837007 647.1946518 619.9524374 585.4749911 544.1646863 496.50364 443.0480863 384.421885 321.3092404 ncosN F975.3914286 3887.155395 1922.676179 3773.869031 1832.693134 3509.924315 1649.024524 3024.001723 1344.003575 2299.183446 471.1731407 24689.09589 表4 侧墙各截面M 、N值 2 1 截面 X X 11 12 13 14 15 16 17 0 1.96799 3.93598 5.90397 7.87196 9.83995 11.80794 12 20.0000 815.2599 1269.5308 -332.7357 -7326.7328 -20877.4925 -27280.2026 Hh3 0.0000 0.6606 2.4484 3.6717 -1.5218 -23.5890 -65.3461 4 0.0000 0.3598 2.7880 8.1102 11.3767 -6.0891 -76.9812 Mh1 -73.7794 -68.3666 11.4643 328.2651 954.7908 1292.8884 -857.9870 0.00000 0.81475 1.62950 2.44424 3.25899 4.07374 4.88849 K13 220.0000 -621.7281 -2304.4615 -3455.8482 1432.3220 22202.0916 61504.1317 uo1.0000 0.9266 -0.1554 -4.4493 -12.9412 -17.5237 11.6291 K14 430.0000 155.8861 1207.9031 3513.7676 4928.9835 -2638.1044 -33352.3151 0.0000 1.6056 2.5002 -0.6553 -14.4293 -41.1162 -53.7257 截面 11 12 13 14 15 16 17 0Mx(KN•m) -73.7794 281.0512 184.4366 53.4489 -10.6365 -20.6169 13.6270 dhx 0 49.19975 98.3995 147.59925 196.799 245.99875 295.1985 Nx(KN) 934.6459 983.84565 1082.2452 1229.8444 1426.6434 1672.6422 1967.8407