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1、FEM limit analysisTraditional method of limit analysis is suitable to engineering design.But the suitability is low that the failure surface should be obtained beforehand.传统极限方法要事先知道滑面，适用性差While FEM can not work out the safety factor in spite of its wide use.有限元数值方法无法计算安全系数FEM limit analysis can be
2、widely used in engineering design,especially in the design of geotechnical engineering(slopes,foundations and tunnels).有限元极限分析法适用性广，又能计算安全系数第1页/共40页Principles of FEM limit analysis有限元极限分析法原理（用数值计算求极限分析）Definition of safety factorAnti-slide force is related to soil strength.Vertical slice method of s
3、lope（条分法）第2页/共40页 1975,Zienkiewicz put forward FEM strength reduction.Through reducing c and of the rock and soil until failure happens,calculations show that the safety factor is the multiple of reducing the strength.通过不断折减强度，使其自动达到破坏，折减的倍数等于安全系数Zheng Yingren developed FEM limit analysis where fail
4、ure surface can be formed automatically and both safety factor and failure surface can be worked out.自动生成破坏面，并能求出破坏面形态第3页/共40页Traditional failure mechanism of tunnels（传统的隧道破坏机理）（传统的隧道破坏机理）Based on loose soil theories（松散体理论）Failure mechanism of deep-buried tunnels Theory of pressure arch of .Failure
5、mechanism of shallow-buried tunnels The soil above the tunnel collapses and stress transfer exists.Pressure arch of deep-buried tunnel（普氏压力拱）Shallow-buried theory by Terzaghi（太沙基）第4页/共40页Modern failure mechanism of tunnels（现代的隧洞破坏机理）（现代的隧洞破坏机理）Based on elastic-plastic theories Rabcewicz crack wedge
6、拉布希维兹破坏楔体理论拉布希维兹破坏楔体理论 2004,Zheng Yingren calculate the safety factor of surrounding rock of tunnels by FEM limit analysis（破裂面）（破裂面）第5页/共40页Crack surface of deep-buried tunnels（深埋隧洞的破裂面）（深埋隧洞的破裂面）DensityDensity（kgkg/mm3 3）E E（MPaMPa）c c（kPakPa）（）1.781.7870700.320.3211611621.821.8Tunnel span:8 Tunnel
7、 span:8 cmcmTunnel height:12 Tunnel height:12 cmcmTunnel depth:15Tunnel depth:15cmcm模型试验尺寸模型试验尺寸Model size:Model size:405215cm405215cm第6页/共40页 Tunnel modelTunnel model Experiment schemesExperiment schemes（模型试验（模型试验方案）方案）SchemesSchemesTunnel span/Tunnel span/cmcmHeight of Height of sidewall/sidewall/
8、cmcmHeight of Height of arch/arch/cmcm1 18 88 82 22 28 88 83 33 38 88 84 44 48 86 64 45 58 84 44 4第7页/共40页Analysis on the comparison of model tests and numerical simulation（模型试验和数值模拟的对比分析）（模型试验和数值模拟的对比分析）Scheme 1Scheme 1Scheme 2Scheme 2第8页/共40页 Results of model tests and numerical simulations（模型试验的数
9、值模拟和试验结果的对比分析）（模型试验的数值模拟和试验结果的对比分析）极限荷载极限荷载 最大距离最大距离SchemeSchemeSpinSpinHeightHeightLimit Limit loadload of of modelmodeltests/tests/kNkNLimit load Limit load ofof numerical numerical/kNkNMaximum distance of Maximum distance of crack surface and wallcrack surface and wallModel Model testtestNumerica
10、lNumerical1810625713.413.12811595514.314.13812565315.615.34810616012.912.8588686611.711.5第9页/共40页Identification of the location of crack surface of deep-buried tunnels（深埋隧洞破裂面位置的确定方法）（深埋隧洞破裂面位置的确定方法）Crack happens where the displacement or Crack happens where the displacement or plastic strain plasti
11、c strain mutatesmutates.（破裂面位置在应变突变（破裂面位置在应变突变处）处）第10页/共40页Failure state of homogeneous tunnels（均质隧洞的破坏状态）（均质隧洞的破坏状态）第11页/共40页Monitoring of failure of deep-buried tunnels（深埋隧洞破坏的现场监测）（深埋隧洞破坏的现场监测）（1）The anchor on the arch top bear small force,while that in the two sides bear large force.（2）The avera
12、ge horizontal displacement of steel frame is 19mm.The average difference of settlement between arch top and arch foot is 9.8mm.顶部锚杆受力小侧面锚杆受力大侧向位移大垂直位移小第12页/共40页Crack surface shallow-buried tunnels（浅埋隧洞的破裂面在拱顶）（浅埋隧洞的破裂面在拱顶）Crack surface of shallow-buried tunnelsCrack surface of shallow-buried tunnels
13、The span is 8m and buried depth is 3The span is 8m and buried depth is 3mm.Failure of the numerical Failure of the numerical simulationsimulation（Pressure is 26Pressure is 26kNkN）Failure of the experiment Failure of the experiment model model（Pressure is 28Pressure is 28kNkN）第13页/共40页Failure process
14、 and divide between shallow-and deep-bury of different buried depth（不同埋深隧洞的破坏过程与深浅埋分界线）不同埋深隧洞的破坏过程与深浅埋分界线）（1）rectangle cavern Buried depth:1Buried depth:1mm safety factor:0.3 safety factor:0.3 The tunnel span is 12The tunnel span is 12mm and the height is 5 and the height is 5mm.The buried depth is
15、1The buried depth is 1mm and failure happens and failure happens in the middle of tunnel top.in the middle of tunnel top.When the buried depth is 3When the buried depth is 3mm,failure happens,failure happens on the shoulder of tunnel and safety factor on the shoulder of tunnel and safety factor incr
16、eases with the increase of buried depth.increases with the increase of buried depth.Buried depth:3Buried depth:3mm safety factor:0.52safety factor:0.52跨度跨度12m，高度，高度5m埋深埋深1m（浅埋状态）（浅埋状态）埋深埋深3m（浅埋状态）（浅埋状态）第14页/共40页With the increase of buried depth,Pressure arch of shallow-bury forms gradually.（形成浅埋压力拱）
17、Pressure arch of shallow-bury forms when the buried depth is 9m.Buried depth:7m safety factor:0.65 Buried depth:9m safety factor:0.66 Buried depth:10m safety factor:0.69 Pressure arch of shallow-bury disappears gradually and arch of deep-bury forms when the buried depth is 10m.So,the buried depth of
18、 9-10m is the dividing line between deep-and shallow-bury.（形成深埋压力拱）埋深埋深7m（浅埋状态）埋深埋深9m（浅埋状态）埋深埋深10m第15页/共40页Failure on the two sides Failure on the two sides appear obviously and appear obviously and safety factor reduces safety factor reduces afterafter the the depth is 18depth is 18mm.埋深18m后，两侧先破坏W
19、hen the buried depth is When the buried depth is between 10between 10mm and 18 and 18mm,the,the safety factor keeps constant safety factor keeps constant basically and two crack basically and two crack surfaces appear respectively surfaces appear respectively on the arch top and side wall.on the arc
20、h top and side wall.1018m，两侧逐渐形成破裂面 Buried depth:Buried depth:1212mm safety factor:0.safety factor:0.7 7 Buried depth:Buried depth:1818mm safety factor:0.safety factor:0.7 7 Buried depth:Buried depth:3030mm safety factor:0.safety factor:0.6767 第16页/共40页Failure mechanism of rectangle tunnel（矩形隧洞的破坏机理
21、矩形隧洞的破坏机理）Shallow-buried pressure archShallow-buried pressure arch forms gradually forms gradually and the maximum height of arch is 9and the maximum height of arch is 9mm-10-10mm.（逐渐（逐渐（逐渐（逐渐形成浅埋压力拱，最大拱高形成浅埋压力拱，最大拱高形成浅埋压力拱，最大拱高形成浅埋压力拱，最大拱高9 9mm1010mm）When the buried depth is between 10When the buri
22、ed depth is between 10mm and 18 and 18mm,pressure arch of shallow-burypressure arch of shallow-bury disappears and safety disappears and safety factor keeps constant.So 10factor keeps constant.So 10mm can be regarded as the can be regarded as the dividing linedividing line between deep-and shallow-b
23、ury.between deep-and shallow-bury.（10181018mm，浅埋压力拱消失，形成深埋压力拱，安全，浅埋压力拱消失，形成深埋压力拱，安全，浅埋压力拱消失，形成深埋压力拱，安全，浅埋压力拱消失，形成深埋压力拱，安全系数不变）系数不变）系数不变）系数不变）If the buried depth is over 18If the buried depth is over 18mm,failure moves from,failure moves from the arch to the two sides and safety factor reduces the ar
24、ch to the two sides and safety factor reduces with the increase of buried depth.with the increase of buried depth.（1818mm后，破坏从拱后，破坏从拱后，破坏从拱后，破坏从拱顶转至两侧）顶转至两侧）顶转至两侧）顶转至两侧）第17页/共40页 Collapsed balance arch forms after the lateral failure.破坏后形成平衡拱破坏后形成平衡拱 When the buried depth is 9m,pressure arch of shal
25、low-bury forms and failure happens on the arch top.浅埋压力拱形成（埋深浅埋压力拱形成（埋深9m）After the buried-depth is over 9m,pressure arch of does not exist when lateral failure happens and safety factor reduces with the increase of depth.不存在普氏压力拱（2）Carven of arch shape 拱形隧洞 Buried depth:9m safety factor:0.82 Buried
26、 depth:30m safety factor:0.77 第18页/共40页Failure process of joint tunnelGeometry modelGeometry modelMaterialsMaterialsUnit Unit weight weight of soilof soilElastic Elastic modulusmodulusPoissons Poissons ratioratioInner Inner cohesioncohesionc cfriction friction angleanglekNkN/mm3 3PaPaMPaMPa RockRock
27、岩石岩石岩石岩石25251E+101E+100.20.21.01.03838JointJoint节理节理节理节理17171E+91E+90.30.30.120.12242417171E+91E+90.30.30.240.24272717171E+91E+90.30.30.360.363030Mechanic parameters of rock and jointMechanic parameters of rock and joint（节理隧道的破坏机制）（节理隧道的破坏机制）岩石与节理力学参数岩石与节理力学参数第19页/共40页Joint of 45 Model test Numerica
28、l calculation Failure state of joint tunnel（节理隧洞的破坏机制）（节理隧洞的破坏机制）不同节理隧洞的破坏形态第20页/共40页Safety factors of tunnel under different joint angles（不同节理倾角时隧洞的安全系数）（不同节理倾角时隧洞的安全系数）Angles()030456090Safety factors3.38 3.323.373.333.31The safety factor of homogeneous tunnel is 4.6 without joints.第21页/共40页Problem
29、s existing in present tunnel design（现行隧洞设计存在的问题）（现行隧洞设计存在的问题）Design methods are not in accordance with practice,so loose pressure and load-structure methods are used.松散压力与荷载-结构法不符合实际 Deformation pressure is adopted and there is no Objective failure standard.变形压力计算无破坏标准，采用经验标准：以洞周位移为标准；以塑性区大小为标准。Plas
30、tic zone after excavation第22页/共40页Problem Problems of taking the displacement of tunnel surrounding or convergence displacement as criteria.以洞周位移或收敛位移为破坏判据存在的问题 In mechanic analysis,there is no failure criteria which is expressed by displacement.So the limit displacement is identified according to e
31、xperience.力学分析中没有位移与破坏的严格力学关系第23页/共40页Elastic modulus has great influence on the displacementElastic modulus Elastic modulus E E（MPaMPa）20203030404050506060Maximum vertical Maximum vertical displacement on the arch displacement on the arch top top（cmcm）9.49.47.37.34.74.74.44.43.63.6Maximum horizonta
32、l Maximum horizontal displacement of side wall displacement of side wall（cmcm）7.67.65.15.13.83.83.03.02.52.5Safety factor Safety factor 安全系数安全系数安全系数安全系数1.621.621.621.621.621.621.621.621.621.62 Calculation results of different elastic modulus Calculation results of different elastic modulusThe elasti
33、c modulus of rock and soil is hard to The elastic modulus of rock and soil is hard to test,so the calculation results would be test,so the calculation results would be influenced seriously.influenced seriously.弹性模型对位移影响很大弹性模型对位移影响很大第24页/共40页Problem Plastic zone of surrounding rock after excavation（=
34、0.20）Plastic zone of surrounding rock after excavation （=0.25）Plastic zone of surrounding rock after excavation （=0.30）Problems of taking the size of plastic zone as failure criteria.以塑性区大小为破坏判据的问题 In mechanic analysis,there is no failure standard of taking the size of plastic zone.没有位移与破坏的严格力学关系，泊松
35、比影响很大 Different poissons ratio has great influence on the size of plastic zone.第25页/共40页以安全系数为围岩稳定标准的优点Adopting the safety factor of surrounding rock as standards:It has strict mechanic basis.有严格的力学基础 Stability has unique standards.稳定有统一标准 It would not be affected by deformation parameters of rock a
36、nd soil.不受岩土参数影响第26页/共40页Calculation of shear safety factor of loess tunnels（黄土隧洞剪切安全系数的计算）黄土隧洞剪切安全系数的计算）The material failures are divided into shear failure and tensile failure.（剪破坏与拉破坏）The safety factors of loess tunnel can be divided into shear safety factor and crack safety factor.（分为剪切安全系数与拉裂安全
37、系数）Failure of loess tunnels in construction and operation are mainly caused by the reduce of soil strength.So strength conserve safety factor is：第27页/共40页 Calculation examples dwelling carven（居民洞室）Engineering conditions Loess tunnel without lining:the span is 3m;the side wall is 1.5m high and buried
38、-depth is 30m.Arch height are 0.0,0.5,1.0,1.5 and 2.0m.Mechanical and physical parameters of soil massElastic modulus（MPa）Poissons ratioUnit weight of soil（kN/m3）Cohesion（MPa）Inner friction angle（）Tensile strength（MPa）400.35170.05250.02第28页/共40页Calculation results of reducing and （FLAC）Shear safety
39、factors under different span ratios Shear safety factors under different span ratios Span ratioArch height（m）Only reducing 00.01.721/60.51.711/31.01.671/21.51.642/32.01.59第29页/共40页Shear safety factors under different span ratiosShear safety factors under different span ratios（ANSYS ANSYS）CohesionCoh
40、esion（MPaMPa）Inner friction Inner friction angleangle（）Span Span（mm）Arch Arch heightheight（mm）Span Span ratioratioShear safety Shear safety factorfactor0.052530.51/61.69（1.71）11/31.65（1.67）1.51/21.62（1.64）0.021830.51/60.9311/30.901.51/20.88Calculation error of the two programs is less than 2%.第30页/共
41、40页Plastic zone after excavationPlastic zone after excavation Plastic zone when failure happensPlastic zone when failure happensThe arch is 0.5The arch is 0.5mm high.high.Equivalent plastic strain and potential failure surfaceEquivalent plastic strain and potential failure surface第31页/共40页Exploratio
42、n on the design method of homogenous tunnel（均质隧洞设计方法探索）（均质隧洞设计方法探索）Principles and requirements of tunnel design It should ensure the safety in construction and operation.考虑施工与运行的安全 As reinforcement of surrounding rock,preliminary lining mainly bear load.As safety converse,condary lining bear small p
43、art of load.初衬加固围岩，二衬作安全系数储备或承受少量荷载。第32页/共40页Design and calculation of tunnel with lining（有衬砌隧洞的设计计算）（有衬砌隧洞的设计计算）Preliminary lining should be under plastic state and as elasto-plastic body.（初衬应进入塑性状态，初衬应进入塑性状态，变按变按弹塑性体计算）弹塑性体计算）50%of load on the surrounding rock of tunnel is released and preliminary
44、 lining is imposed.Then thecalculation is carried out to obtain the safety factor oftunnel after preliminary lining.（隧洞围岩释放（隧洞围岩释放50%荷载后加初衬再算安全系数，要求大于荷载后加初衬再算安全系数，要求大于1.2）第33页/共40页Calculation of treating secondary lining as Calculation of treating secondary lining as elastic memberselastic members（二
45、衬作弹性杆件计算，承受（二衬作弹性杆件计算，承受10%10%荷载）荷载）Secondary lining bears 10%of inner force to Secondary lining bears 10%of inner force to calculate the safety factors of surrounding rock calculate the safety factors of surrounding rock and secondary lining.and secondary lining.Design requirementsDesign requiremen
46、ts：（设计要求）Safety factor of surrounding rock after Safety factor of surrounding rock after secondary lining is over 1.35.secondary lining is over 1.35.（二衬后围岩安全系数大于1.35）Safety factor of secondary lining is over 1.4.Safety factor of secondary lining is over 1.4.（二衬安全系数大于1.4）第34页/共40页Tunnel model is esta
47、blished accordingTunnel model is established according to two-row road.to two-row road.Total height of tunnel is 9.8Total height of tunnel is 9.8mm.高度高度高度高度9.8m9.8mSpan is 11.6Span is 11.6mm.跨度跨度跨度跨度11.6m11.6mBoundary is identified as 8 times of span.Boundary is identified as 8 times of span.Coverag
48、e on the tunnel top is 30Coverage on the tunnel top is 30mm thick.thick.The bottom of boundary is refrainedThe bottom of boundary is refrainedby fixed reaming and the upper part is by fixed reaming and the upper part is free boundary.The left and right sidesfree boundary.The left and right sidesare
49、horizontal refrain.are horizontal refrain.四个结点受力最大四个结点受力最大四个结点受力最大四个结点受力最大Engineering calculation example（算例算例）第35页/共40页Calculation schemeCalculation scheme（计算方案）（计算方案）（计算方案）（计算方案）Calculation resultsCalculation results（初衬安全系数）（初衬安全系数）（初衬安全系数）（初衬安全系数）:F F=1.24=1.24（scheme 1scheme 1），），），），F F=1.38=1.
50、38（scheme 2scheme 2）Working Working conditionsconditionsThickness of Thickness of preliminary preliminary lining/lining/cmcmThickness of Thickness of secondary secondary lining/lining/cmcmPercent of Percent of releasing releasing load on the load on the surround surround rockrockPercent of Percent o