多溶洞地层中路基承载力有限元极限分析
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摘要
为计算多溶洞地层中路基的极限承载力,根据上、下限定理,结合有限元方法,基于MATLAB平台编制了相关计算程序。采用修正的Hoek-Brown准则来描述岩体的非线性特点,并将其嵌入到计算程序中。在此基础上,用一个无量纲参数N_σ来衡量单个溶洞对路基承载力的影响,而用折减系数ξ来描述两个溶洞的影响。将计算结果以表格的形式展示出来,并对路基承载力的各影响因素进行探讨。结果表明:N_σ随着H/R(岩层厚度/溶洞半径)、地质力学指标GSI的增大而非线性增大,与岩石类别参数mi大致成线性关系;岩体的重度对N_σ的影响可忽略不计;当X(溶洞间水平距离)/R≤2时,ξ随H/R的增大而增大;当X/R≥3时,ξ随H/R的增大而减小;当Y(溶洞间垂直距离)/R<3时,ξ在X/R=2时取最小值;当Y/R≥3时,ξ趋近于1,此时只需要考虑单个溶洞对承载力的影响;单个溶洞的极限破坏模式与H/R有关,随着H/R的增大,破坏面由溶洞的顶部逐渐向底部发展,且影响范围逐渐扩大;两个溶洞的埋深一致,当X/R较小时,会有拱效应产生,当X/R较大时,溶洞间岩层会产生"X"形破坏面;两个溶洞的埋深不一致时,溶洞间贯穿的破坏面随着Y/R的增大而逐渐消失。将条形基础作用在岩层的承载力的结果与已有成果进行对比,误差在4%以内,验证所提方法的正确性。同时,为便于实际工程设计,提供具体的设计步骤及说明,基本能满足大部分工程需求。
In order to calculate the ultimate bearing capacity of subgrade above the stratum with multiple karst caves, based on the upper bound and the lower bound theorems of the limit analysis as well as finite element method, the computational program was coded in MATLAB. The modified Hoek-Brown criterion was adopted to describe the non-liner characteristic of the rock mass, and then was incorporated into the computational program.On this basis, a dimensionless parameter N_σ and a reduction factor ξ were defined to estimate the effects of single cave and two caves on the bearing capacity of subgrade, respectively. The numerical results were presented in the form of tables, and the impacts of different parameters on the bearing capacity of subgrade were investigated.The results reveal that N_σ may increase non-linearly with the increase of the values of H/R(thickness of rock/radius of cave) and geological strength index(GSI), and it has a roughly linear relation with rock type parameter mi. The influence of the rock mass weight on N_σ could be neglected. The value of ξ increases with the increase of H/R, when X(horizontal distance between two caves)/R≤2. The value of ξ increases with the decrease of H/R, when X/R≥3. ξ has the minimum value at X/R=2, when Y(vertical distance between two caves)/R<3. ξ approaches to 1 when X/R≥3, and in this case it is only necessary to consider the effect of one single cave on bearing capacity. The ultimate failure mechanism of single cave is related to the value of H/R.With the increase of H/R, the failure surface may extend from the top to the bottom of the cave, and the influence range may gradually extend. When the burial depths of two caves are the same, the arching effect may occur if X/R is small, X-shaped rupture planes may occur if X/R is relatively large. When the burial depths of two caves are different, the running-through rupture planes between two caves may gradually disappear with the increase of Y/R. The results for bearing capacity of strip footing located on a rock stratum are compared with those in the previous study, and the difference is within 4%, validating the correctness of the method proposed in this work.Meanwhile, for the convenience of design in engineering practice, the design steps and instructions are provided,which basically meet the requirements in most projects.
In order to calculate the ultimate bearing capacity of subgrade above the stratum with multiple karst caves, based on the upper bound and the lower bound theorems of the limit analysis as well as finite element method, the computational program was coded in MATLAB. The modified Hoek-Brown criterion was adopted to describe the non-liner characteristic of the rock mass, and then was incorporated into the computational program.On this basis, a dimensionless parameter N_σ and a reduction factor ξ were defined to estimate the effects of single cave and two caves on the bearing capacity of subgrade, respectively. The numerical results were presented in the form of tables, and the impacts of different parameters on the bearing capacity of subgrade were investigated.The results reveal that N_σ may increase non-linearly with the increase of the values of H/R(thickness of rock/radius of cave) and geological strength index(GSI), and it has a roughly linear relation with rock type parameter mi. The influence of the rock mass weight on N_σ could be neglected. The value of ξ increases with the increase of H/R, when X(horizontal distance between two caves)/R≤2. The value of ξ increases with the decrease of H/R, when X/R≥3. ξ has the minimum value at X/R=2, when Y(vertical distance between two caves)/R<3. ξ approaches to 1 when X/R≥3, and in this case it is only necessary to consider the effect of one single cave on bearing capacity. The ultimate failure mechanism of single cave is related to the value of H/R.With the increase of H/R, the failure surface may extend from the top to the bottom of the cave, and the influence range may gradually extend. When the burial depths of two caves are the same, the arching effect may occur if X/R is small, X-shaped rupture planes may occur if X/R is relatively large. When the burial depths of two caves are different, the running-through rupture planes between two caves may gradually disappear with the increase of Y/R. The results for bearing capacity of strip footing located on a rock stratum are compared with those in the previous study, and the difference is within 4%, validating the correctness of the method proposed in this work.Meanwhile, for the convenience of design in engineering practice, the design steps and instructions are provided,which basically meet the requirements in most projects.
引文
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[5]程晔,曹文贵,赵明华.高速公路下伏岩溶顶板稳定性二级模糊综合评判[J].中国公路学报,2003,16(4):21-24(Chen Ye,Cao Wengui,Zhao Minghua.Synthetie judgement on the two stage fuzzy of the stability of karst top slab beneath expressway[J].China Journal of Highway and Transport,2003,16(4):21-24(in Chinese))
[6]曹文贵,程晔,赵明华.公路路基岩溶顶板安全厚度确定的数值流形方法研究[J].岩土工程学报,2005,27(6):621-625(Cao Wengui,Chen Ye,Zhao Minghua.Study of numerical manifold method for determination of safe thickness of karst roof in roadbed[J].Chinese Journal of Geotechnical Engineering,2005,27(6):621-625(in Chinese))
[7]Kiyosumi M,Kusakabe O,Ohuchi M,et al.Yielding pressure of spread footing above multiple voids[J].Journal of Geotechnical and Geoenvironmental Engineering,2007,133(12):1522-1531
[8]Kiyosumi M,Kusakabe O,Ohuchi M.Model tests and analyses of bearing capacity of strip footing on stiff ground with voids[J].Geotechnical and Geoenvironmental Engineering,2011,137(4):363-375
[9]Lee J K,Jeong S,Ko J.Undrained stability of surface strip footings above voids[J].Computers and Geotechnics,2014,62:128-135
[10]Lee J K,Jeong S,Ko J.Effect of load inclination on the undrained bearing capacity of surface spread footings above voids[J].Computers and Geotechnics,2015,66:245-252
[11]Sloan S W.Geotechnical stability analysis[J].Géotechnique,2013,63(7):531
[12]Lyamin A V,Sloan S W.Upper bound limit analysis using linear finite elements and non-linear programming[J].International Journal for Numerical and Analytical Methods in Geomechanics,2002,26:573-611
[13]Lyamin A V,Sloan S W,Krabbenhoft K,et al.Lower bound limit analysis with adaptive remeshing[J].International Journal for Numerical Methods in Engineering,2005,63(14):1961-1974
[14]赵明华,张锐,刘猛.下限分析有限单元法的非线性规划求解[J].岩土力学,2015,36(12):3589-3597(Zhao Minghua,Zhang Rui,Liu Meng.Nonlinear programming of lower bound finite element method[J].Rock and Soil Mechanics,2015,36(12):3589-3597(in Chinese))
[15]赵明华,张锐.有限元上限分析网格自适应方法及其工程应用[J].岩土工程学报,2016,38(3):537-545(Zhao Minghua,Zhang Rui.Adaptive mesh refinement of upper bound finite element method and its applications in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):537-545(in Chinese))
[16]张锐.基于非线性规划的有限元极限分析方法及其工程应用[D].长沙:湖南大学,2015(Zhang Rui.Finite element limit analysis based on non-liner programming and its applications in engineering[D].Changsha:Hunan Unversity,2015(in Chinese))
[17]Hoek D E,Marinos D P,Hoek E,et al.A brief history of the development of the Hoek-Brown failure criterion[J].Soils and Rocks,2007,2(1):1-13
[18]Serrano,Olalla C,Gonzalez J.Ultimate bearing capacity of rock masses based on the modified Hoek-Brown criterion[J].International Journal of Rock Mechanics and Mining Sciences,2000,37(6):1013-1018
[2]Wang M C,Badie A.Effect of underground void on foundation stability[J].Journal of Geotechnical Engineering,1985,111(8):1008-1019
[3]Wang M C,Hsieh C W.Collapse load of strip footing above circular void[J].Journal of Geotechnical Engineering,1987,113(5):511-515
[4]Goodings D J,Abdulla W A.Stability charts for predicting sinkholes in weakly cemented sand over karst limestone[J].Engineering Geology,2002,65(2):179-184
[5]程晔,曹文贵,赵明华.高速公路下伏岩溶顶板稳定性二级模糊综合评判[J].中国公路学报,2003,16(4):21-24(Chen Ye,Cao Wengui,Zhao Minghua.Synthetie judgement on the two stage fuzzy of the stability of karst top slab beneath expressway[J].China Journal of Highway and Transport,2003,16(4):21-24(in Chinese))
[6]曹文贵,程晔,赵明华.公路路基岩溶顶板安全厚度确定的数值流形方法研究[J].岩土工程学报,2005,27(6):621-625(Cao Wengui,Chen Ye,Zhao Minghua.Study of numerical manifold method for determination of safe thickness of karst roof in roadbed[J].Chinese Journal of Geotechnical Engineering,2005,27(6):621-625(in Chinese))
[7]Kiyosumi M,Kusakabe O,Ohuchi M,et al.Yielding pressure of spread footing above multiple voids[J].Journal of Geotechnical and Geoenvironmental Engineering,2007,133(12):1522-1531
[8]Kiyosumi M,Kusakabe O,Ohuchi M.Model tests and analyses of bearing capacity of strip footing on stiff ground with voids[J].Geotechnical and Geoenvironmental Engineering,2011,137(4):363-375
[9]Lee J K,Jeong S,Ko J.Undrained stability of surface strip footings above voids[J].Computers and Geotechnics,2014,62:128-135
[10]Lee J K,Jeong S,Ko J.Effect of load inclination on the undrained bearing capacity of surface spread footings above voids[J].Computers and Geotechnics,2015,66:245-252
[11]Sloan S W.Geotechnical stability analysis[J].Géotechnique,2013,63(7):531
[12]Lyamin A V,Sloan S W.Upper bound limit analysis using linear finite elements and non-linear programming[J].International Journal for Numerical and Analytical Methods in Geomechanics,2002,26:573-611
[13]Lyamin A V,Sloan S W,Krabbenhoft K,et al.Lower bound limit analysis with adaptive remeshing[J].International Journal for Numerical Methods in Engineering,2005,63(14):1961-1974
[14]赵明华,张锐,刘猛.下限分析有限单元法的非线性规划求解[J].岩土力学,2015,36(12):3589-3597(Zhao Minghua,Zhang Rui,Liu Meng.Nonlinear programming of lower bound finite element method[J].Rock and Soil Mechanics,2015,36(12):3589-3597(in Chinese))
[15]赵明华,张锐.有限元上限分析网格自适应方法及其工程应用[J].岩土工程学报,2016,38(3):537-545(Zhao Minghua,Zhang Rui.Adaptive mesh refinement of upper bound finite element method and its applications in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):537-545(in Chinese))
[16]张锐.基于非线性规划的有限元极限分析方法及其工程应用[D].长沙:湖南大学,2015(Zhang Rui.Finite element limit analysis based on non-liner programming and its applications in engineering[D].Changsha:Hunan Unversity,2015(in Chinese))
[17]Hoek D E,Marinos D P,Hoek E,et al.A brief history of the development of the Hoek-Brown failure criterion[J].Soils and Rocks,2007,2(1):1-13
[18]Serrano,Olalla C,Gonzalez J.Ultimate bearing capacity of rock masses based on the modified Hoek-Brown criterion[J].International Journal of Rock Mechanics and Mining Sciences,2000,37(6):1013-1018