岩溶区桥梁桩基承载力试验与合理嵌岩深度
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摘要
为研究岩溶区桥梁桩基的承载特性,依托平顶山市西斜立交桥实体工程,进行了桩基静载试验,通过在桩端和桩顶布设应变传感器和位移计,测得了桩身内力,分析了岩溶区桥梁桩顶荷载(Q)-沉降(s)规律;考虑现有桩基设计的局限性,结合静载试验结果,采用不同函数模型预测了单桩竖向极限承载力;基于岩-桩体系宽梁力学模型和溶洞顶板拉-弯破坏模式,探讨了桩基嵌岩深度的计算方法,提出了一种适于岩溶区桥梁桩基嵌岩深度的优化方法。研究结果表明:各级荷载作用下桩基Q-s曲线呈缓变型发展,当桩顶荷载较小时,曲线基本呈线性,当桩顶荷载大于6 000kN时,曲线逐渐变为非线性,虽然桩已嵌入灰岩较深,但仍表现为典型的摩擦桩承载性状,当加载到8 400kN时,桩顶沉降为3.69mm,远小于0.03D(D为桩径)或40mm的破坏标准,桩端阻力为122.9kN,仅占桩顶荷载的1.6%,桩的承载力尚有富余;在静载试验全过程中,桩的受力状态处于Kulhawy理论的第1阶段,桩侧阻力和桩端阻力同步发挥;双曲线模型拟合精度在0.99以上且预测值偏安全,建议在同类工程中优先考虑采用;在同时满足溶洞顶板安全厚度和桩基承载力与稳定性要求的前提下,采用提出的计算方法可使桩的嵌岩深度减小2.4m。
To study the bearing characteristics of bridge pile foundations in karst areas,with relevance to Xixie Overpass Project in Pingdingshan City,the pile foundatiion static load test was carried out.By placing strain sensors and displacement meters at the pile top and bottom,the internal forces of pile were measured,and pile top load(Q)-settlement(s)law of bridge in karst area was analyzed.Considering the limitations of current pile foundation design and combiningthe static load test results,the vertical ultimate bearing capacity of a single pile was predicted using different function models.Based on the wide beam mechanical model of rock-pile system and tensile-bending failure mode of karst cave roof,the calculation methods for the rock-socket depth of pile foundation were discussed,and an optimized calculation method for the rocksocketed depth of bridge pile foundations in karst areas was proposed.Research result shows that under different load levels,the Q-s curve exhibits slowly-varying developments.When the pile top load is small,the Q-s curve is basically linear.When the pile top load is greater than 6 000 kN,the Q-s curve becomes nonlinear.Although the pile is embedded deeply into limestone,it still shows typical friction pile characteristics.When loaded to 8 400 kN,the pile top settlement is3.69 mm,far less than the failure criterion of 0.03D(Dis the pile diameter)or 40 mm.The pile tip resistance is 122.9kN,accounting for only 1.6%of the pile top load.Bearing capacity of pile is still sufficient.Over the entire static load testing process,the stress state of pile is in the first stage of Kulhawy theory.The pile tip and side resistances develop synchronously.The fitting accuracy of hyperbola model is above 0.99 and the predicted value is determined to be safe.Thus,the model is recommended for use in similar projects.The proposed calculation method can reduce the rock-socketed depth of pile by 2.4m,while meeting the requirements for safe thickness of karst cave roof,as well as bearing capacity and stability of pile foundation.
To study the bearing characteristics of bridge pile foundations in karst areas,with relevance to Xixie Overpass Project in Pingdingshan City,the pile foundatiion static load test was carried out.By placing strain sensors and displacement meters at the pile top and bottom,the internal forces of pile were measured,and pile top load(Q)-settlement(s)law of bridge in karst area was analyzed.Considering the limitations of current pile foundation design and combiningthe static load test results,the vertical ultimate bearing capacity of a single pile was predicted using different function models.Based on the wide beam mechanical model of rock-pile system and tensile-bending failure mode of karst cave roof,the calculation methods for the rock-socket depth of pile foundation were discussed,and an optimized calculation method for the rocksocketed depth of bridge pile foundations in karst areas was proposed.Research result shows that under different load levels,the Q-s curve exhibits slowly-varying developments.When the pile top load is small,the Q-s curve is basically linear.When the pile top load is greater than 6 000 kN,the Q-s curve becomes nonlinear.Although the pile is embedded deeply into limestone,it still shows typical friction pile characteristics.When loaded to 8 400 kN,the pile top settlement is3.69 mm,far less than the failure criterion of 0.03D(Dis the pile diameter)or 40 mm.The pile tip resistance is 122.9kN,accounting for only 1.6%of the pile top load.Bearing capacity of pile is still sufficient.Over the entire static load testing process,the stress state of pile is in the first stage of Kulhawy theory.The pile tip and side resistances develop synchronously.The fitting accuracy of hyperbola model is above 0.99 and the predicted value is determined to be safe.Thus,the model is recommended for use in similar projects.The proposed calculation method can reduce the rock-socketed depth of pile by 2.4m,while meeting the requirements for safe thickness of karst cave roof,as well as bearing capacity and stability of pile foundation.
引文
[1]伍四明,石汉生,朱照宇.广佛放射线岩溶对桩基的影响[J].岩石力学与工程学报,2006,25(增2):3559-3564.WU Si-ming,SHI Han-sheng,ZHU Zhao-yu.Effects of karst on pile foundations of Guangdong-Foshan Lines[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(S2):3559-3564.(in Chinese)
[2]方云飞,孙宏伟,阚敦莉.唐山岩溶地区桩基工程问题分析与设计要点[J].岩土工程学报,2013,35(增2):1117-1121.FANG Yun-fei,SUN Hong-wei,KAN Dun-li.Analysis and design of pile foundation engineering in karst areas of Tangshan[J].Chinese Journal of Geotechnical Engineering,2013,35(S2):1117-1121.(in Chinese)
[3]李春生.岩溶地层灌注桩的施工工艺与质量通病[J].中南公路工程,2003,28(2):73-75.LI Chun-sheng.Construction techniques and common defects on pile foundation in karst strata[J].Central South Highway Engineering,2003,28(2):73-75.(in Chinese)
[4]张龙起,陈景,曲朝雷.唐山岩溶区桩基施工中几个问题的分析与探讨[J].岩土工程学报,2011,33(增1):318-323.ZHANG Long-qi,CHEN Jing,QU Chao-lei.Analysis and discussion of several issues of pile foundation construction in karst region of Tangshan[J].Chinese Journal of Geotechnical Engineering,2011,33(S1):318-323.(in Chinese)
[5] GONCHAROV B V,ZHILIN A N,KOVALEV V F,et al.Monitoring system for the pile bed of a slab foundation at a site prone to karst formation[J].Soil Mechanics and Foundation Engineering,2001,38(3):95-100.
[6] WONG C T,YIM K P,LEUNG M K,et al.Rock-socketed large diameter bored pile and geophysical survey in cavernous karst area:Tin Shui Wai public library cum indoor recreation centre[J].Procedia Engineering,2011,14:1744-1751.
[7]黎斌,范秋雁,秦凤荣.岩溶地区溶洞顶板稳定性分析[J].岩石力学与工程学报,2002,21(4):532-536.LI Bin,FAN Qiu-yan,QIN Feng-rong.Analysis on roof stability of karst cave in karst areas[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(4):532-536.(in Chinese)
[8]赵明华,程晔,曹文贵.桥梁桩基桩端溶洞顶板稳定性的模糊分析研究[J].岩石力学与工程学报,2005,24(8):1376-1383.ZHAO Ming-hua, CHENG Ye, CAO Wen-gui. Fuzzy method for the stability analysis of cave roof under pile tip in karst region[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(8):1376-1383.(in Chinese)
[9]赵明华,蒋冲,曹文贵.岩溶区嵌岩桩承载力及其下伏溶洞顶板安全厚度的研究[J].岩土工程学报,2007,29(11):1618-1622.ZHAO Ming-hua,JIANG Chong,CAO Wen-gui.Study on bearing capacity of rock-socked piles and safe thickness of cave roofs in karst region[J].Chinese Journal of Geotechnical Engineering,2007,29(11):1618-1622.(in Chinese)
[10]曹文贵,颜艳芬,张永杰.桩基桩端岩溶顶板稳定性模糊能度可靠性分析方法[J].岩石力学与工程学报,2009,28(1):88-94.CAO Wen-gui,YAN Yan-fen,ZHANG Yong-jie.Fuzzy possibilistic reliability analysis method for stability of karst roof under pile tip[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(1):88-94.(in Chinese)
[11]赵明华,曹文贵,何鹏祥,等.岩溶及采空区桥梁桩桩基端岩层安全厚度研究[J].岩土力学,2004,25(1):64-68.ZHAO Ming-hua,CAO Wen-gui,HE Peng-xiang,et al.Study on safe thickness of rock mass at end of bridge foundation's pile in karst and worked-out mine area[J].Rock and Soil Mechanic,2004,25(1):64-68.(in Chinese)
[12]赵明华,雷勇,张锐.岩溶区桩基冲切破坏模式及安全厚度研究[J].岩土力学,2012,33(2):524-530.ZHAO Ming-hua, LEI Yong, ZHANG Rui. Study of punching failure mode and safe thickness of pile foundation in karst region[J].Rock and Soil Mechanic,2012,33(2):524-530.(in Chinese)
[13]汪华斌,刘志峰,赵文锋,等.桥梁桩基荷载下溶洞顶板稳定性研究[J].岩石力学与工程学报,2013,32(增2):3650-3657.WANG Hua-bin,LIU Zhi-feng,ZHAO Wen-feng,et al.Research on stability of cave roof under pile loading in bridge construction engineering[J].Chinese Journal of Rock Mechanics and Engineering,2013,32(S2):3650-3657.(in Chinese)
[14]汪稔,孟庆山,罗强,等.桥基岩溶洞穴顶板稳定性综合评价[J].公路交通科技,2005,22(6):76-80.WANG Ren,MENG Qing-shan,LUO Qiang,et al.The comprehensive evaluation on roof stability of karst cave under bridge foundation[J].Journal of Highway and Transportation Research and Development,2005,22(6):76-80.(in Chinese)
[15]黄生根,梅世龙,龚维明.南盘江特大桥岩溶桩基承载特性的试验研究[J].岩石力学与工程学报,2004,23(5):809-813.HUANG Sheng-gen, MEI Shi-long, GONG Wei-ming.Testing study on bearing behavior of piles for Nanpan River Great Bridge in karst areas[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(5):809-813.(in Chinese)
[16]张慧乐,马凛,张智浩,等.岩溶区嵌岩桩承载特性影响因素试验研究[J].岩土力学,2013,34(1):92-100.ZHANG Hui-le,MA Lin,ZHANG Zhi-hao,et al.Test research on factors influencing bearing capacity of rocksocketed piles in karst area[J].Rock and Soil Mechanics,2013,34(1):92-100.(in Chinese)
[17]张智浩,张慧乐,马凛,等.岩溶区嵌岩桩的破坏模式与工程设计探讨[J].岩石力学与工程学报,2013,32(增2):4130-4138.ZHANG Zhi-hao,ZHANG Hui-le,MA Lin,et al.Discussion on failure mode and engineering design of rock-socketed pile in karst area[J].Chinese Journal of Rock Mechanics and Engineering,2013,32(S2):4130-4138.(in Chinese)
[18]尹凯丽,郝世龙.武汉地区岩溶对桩基承载力影响数值模拟研究[J].水文地质工程地质,2015,42(6):96-102.YIN Kai-li,HAO Shi-long.A study of the effect of karst on the bearing capacity of pile foundation in Wuhan district based on numerical simulation[J].Hydrogeology and Engineering Geology,2015,42(6):96-102.(in Chinese)
[19] PANNO S V,KELLY W R,ANGEL J C,et al.Hydrogeologic and topographic controls on evolution of karst features in Illinois'sinkhole plain[J].Carbonates Evaporites,2013,28(1/2):13-21.
[20]龚平,廖辉煌,周栋梁,等.岩溶地质桩基自平衡静载试验研究[J].中外公路,2012,32(3):220-224.GONG Ping,LIAO Hui-huang,ZHOU Dong-liang,et al.Self-balanced static load test of karst geological pile foundation[J].Journal of China and Foreign Highway,2012,32(3):220-224.(in Chinese)
[21] PELLS P J N,TURNER R M.Elastic solutions for the design and analysis of rock-socketed piles[J]. Canadian Geotechnical Journal,1979,16(3):481-487.
[22]刘晓明,何青相,赵明华.岩溶地基上桥梁桩基设计优化方法研究与实例[J].中南大学学报:自然科学版,2014,45(5):1653-1658.LIU Xiao-ming, HEQing-xiang, ZHAOMing-hua.Optimization design of bridge pile foundation in karst region:methodology and examples[J].Journal of Central South University:Science and Technology,2014,45(5):1653-1658.(in Chinese)
[23]张建华.岩溶区桥梁桩桩基长确定方法研究[J].公路工程,2009,34(4):1-4,15.ZHANG Jian-hua.Determination method of pile length of bridge pile in karst area[J].Highway Engineering,2009,34(4):1-4,15.(in Chinese)
[24] DALERCI G,BOVOLENTA R.A new method for the evaluation of the ultimate load of piles by tests not carried to failure[J].Geotechnical and Geological Engineering,2014,32(6):1415-1426.
[25]修朝英,李大展.单桩垂直静载试验P-S曲线的数学描述和极限荷载的预测[J].岩土工程学报,1988,10(6):64-73.XIU Chao-ying,LI Da-zhan.Mathematical description of P-S curve and prediction of ultimate load for vertical static load test of single pile[J].Chinese Journal of Geotechnical Engineering,1988,10(6):64-73.(in Chinese)
[26] KULHAWY F H.Stress deformation properties of rock and rock discontinuities[J].Engineering Geology,1975,9(4):327-350.
[27]谢富贵.岩溶地区桥梁桩基承载特性研究[D].西安:长安大学,2007.XIE Fu-gui.A study on properties of bridge piles in karst areas[D].Xi'an:Chang'an University,2007.(in Chinese)
[28]龚成中.岩溶地区嵌岩桩基承载特性研究[D].南京:东南大学,2005.GONG Cheng-zhong.Study of the characteristic of bearing capability of pile foundation in karsts area[D].Nanjing:Southeast University,2005.(in Chinese)
[29]唐咸力.岩溶区基桩下伏溶洞顶板竖向承载机理及其试验研究[D].长沙:湖南大学,2016.TANG Xian-li. The bearing capacity and experimental research of karst cave roof under pile tip foundation in karst area[D].Changsha:Hunan University,2016.(in Chinese)
[30] JENSEN V P,HOLL D L.An application of derivatives of non-analytic functions in plane stress problems[J].Bulletin of the American Mathematical Society,1937,43(4):256-260.
[31]柏华军.考虑溶洞顶板自重时桩端持力岩层安全厚度计算方法[J].岩土力学,2016,37(10):2945-2952.BAI Hua-jun.A method for calculating the safety rock thickness of pile bearing strata with considering deadweight of karst cave roof[J].Rock and Soil Mechanics,2016,37(10):2945-2952.(in Chinese)
[2]方云飞,孙宏伟,阚敦莉.唐山岩溶地区桩基工程问题分析与设计要点[J].岩土工程学报,2013,35(增2):1117-1121.FANG Yun-fei,SUN Hong-wei,KAN Dun-li.Analysis and design of pile foundation engineering in karst areas of Tangshan[J].Chinese Journal of Geotechnical Engineering,2013,35(S2):1117-1121.(in Chinese)
[3]李春生.岩溶地层灌注桩的施工工艺与质量通病[J].中南公路工程,2003,28(2):73-75.LI Chun-sheng.Construction techniques and common defects on pile foundation in karst strata[J].Central South Highway Engineering,2003,28(2):73-75.(in Chinese)
[4]张龙起,陈景,曲朝雷.唐山岩溶区桩基施工中几个问题的分析与探讨[J].岩土工程学报,2011,33(增1):318-323.ZHANG Long-qi,CHEN Jing,QU Chao-lei.Analysis and discussion of several issues of pile foundation construction in karst region of Tangshan[J].Chinese Journal of Geotechnical Engineering,2011,33(S1):318-323.(in Chinese)
[5] GONCHAROV B V,ZHILIN A N,KOVALEV V F,et al.Monitoring system for the pile bed of a slab foundation at a site prone to karst formation[J].Soil Mechanics and Foundation Engineering,2001,38(3):95-100.
[6] WONG C T,YIM K P,LEUNG M K,et al.Rock-socketed large diameter bored pile and geophysical survey in cavernous karst area:Tin Shui Wai public library cum indoor recreation centre[J].Procedia Engineering,2011,14:1744-1751.
[7]黎斌,范秋雁,秦凤荣.岩溶地区溶洞顶板稳定性分析[J].岩石力学与工程学报,2002,21(4):532-536.LI Bin,FAN Qiu-yan,QIN Feng-rong.Analysis on roof stability of karst cave in karst areas[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(4):532-536.(in Chinese)
[8]赵明华,程晔,曹文贵.桥梁桩基桩端溶洞顶板稳定性的模糊分析研究[J].岩石力学与工程学报,2005,24(8):1376-1383.ZHAO Ming-hua, CHENG Ye, CAO Wen-gui. Fuzzy method for the stability analysis of cave roof under pile tip in karst region[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(8):1376-1383.(in Chinese)
[9]赵明华,蒋冲,曹文贵.岩溶区嵌岩桩承载力及其下伏溶洞顶板安全厚度的研究[J].岩土工程学报,2007,29(11):1618-1622.ZHAO Ming-hua,JIANG Chong,CAO Wen-gui.Study on bearing capacity of rock-socked piles and safe thickness of cave roofs in karst region[J].Chinese Journal of Geotechnical Engineering,2007,29(11):1618-1622.(in Chinese)
[10]曹文贵,颜艳芬,张永杰.桩基桩端岩溶顶板稳定性模糊能度可靠性分析方法[J].岩石力学与工程学报,2009,28(1):88-94.CAO Wen-gui,YAN Yan-fen,ZHANG Yong-jie.Fuzzy possibilistic reliability analysis method for stability of karst roof under pile tip[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(1):88-94.(in Chinese)
[11]赵明华,曹文贵,何鹏祥,等.岩溶及采空区桥梁桩桩基端岩层安全厚度研究[J].岩土力学,2004,25(1):64-68.ZHAO Ming-hua,CAO Wen-gui,HE Peng-xiang,et al.Study on safe thickness of rock mass at end of bridge foundation's pile in karst and worked-out mine area[J].Rock and Soil Mechanic,2004,25(1):64-68.(in Chinese)
[12]赵明华,雷勇,张锐.岩溶区桩基冲切破坏模式及安全厚度研究[J].岩土力学,2012,33(2):524-530.ZHAO Ming-hua, LEI Yong, ZHANG Rui. Study of punching failure mode and safe thickness of pile foundation in karst region[J].Rock and Soil Mechanic,2012,33(2):524-530.(in Chinese)
[13]汪华斌,刘志峰,赵文锋,等.桥梁桩基荷载下溶洞顶板稳定性研究[J].岩石力学与工程学报,2013,32(增2):3650-3657.WANG Hua-bin,LIU Zhi-feng,ZHAO Wen-feng,et al.Research on stability of cave roof under pile loading in bridge construction engineering[J].Chinese Journal of Rock Mechanics and Engineering,2013,32(S2):3650-3657.(in Chinese)
[14]汪稔,孟庆山,罗强,等.桥基岩溶洞穴顶板稳定性综合评价[J].公路交通科技,2005,22(6):76-80.WANG Ren,MENG Qing-shan,LUO Qiang,et al.The comprehensive evaluation on roof stability of karst cave under bridge foundation[J].Journal of Highway and Transportation Research and Development,2005,22(6):76-80.(in Chinese)
[15]黄生根,梅世龙,龚维明.南盘江特大桥岩溶桩基承载特性的试验研究[J].岩石力学与工程学报,2004,23(5):809-813.HUANG Sheng-gen, MEI Shi-long, GONG Wei-ming.Testing study on bearing behavior of piles for Nanpan River Great Bridge in karst areas[J].Chinese Journal of Rock Mechanics and Engineering,2004,23(5):809-813.(in Chinese)
[16]张慧乐,马凛,张智浩,等.岩溶区嵌岩桩承载特性影响因素试验研究[J].岩土力学,2013,34(1):92-100.ZHANG Hui-le,MA Lin,ZHANG Zhi-hao,et al.Test research on factors influencing bearing capacity of rocksocketed piles in karst area[J].Rock and Soil Mechanics,2013,34(1):92-100.(in Chinese)
[17]张智浩,张慧乐,马凛,等.岩溶区嵌岩桩的破坏模式与工程设计探讨[J].岩石力学与工程学报,2013,32(增2):4130-4138.ZHANG Zhi-hao,ZHANG Hui-le,MA Lin,et al.Discussion on failure mode and engineering design of rock-socketed pile in karst area[J].Chinese Journal of Rock Mechanics and Engineering,2013,32(S2):4130-4138.(in Chinese)
[18]尹凯丽,郝世龙.武汉地区岩溶对桩基承载力影响数值模拟研究[J].水文地质工程地质,2015,42(6):96-102.YIN Kai-li,HAO Shi-long.A study of the effect of karst on the bearing capacity of pile foundation in Wuhan district based on numerical simulation[J].Hydrogeology and Engineering Geology,2015,42(6):96-102.(in Chinese)
[19] PANNO S V,KELLY W R,ANGEL J C,et al.Hydrogeologic and topographic controls on evolution of karst features in Illinois'sinkhole plain[J].Carbonates Evaporites,2013,28(1/2):13-21.
[20]龚平,廖辉煌,周栋梁,等.岩溶地质桩基自平衡静载试验研究[J].中外公路,2012,32(3):220-224.GONG Ping,LIAO Hui-huang,ZHOU Dong-liang,et al.Self-balanced static load test of karst geological pile foundation[J].Journal of China and Foreign Highway,2012,32(3):220-224.(in Chinese)
[21] PELLS P J N,TURNER R M.Elastic solutions for the design and analysis of rock-socketed piles[J]. Canadian Geotechnical Journal,1979,16(3):481-487.
[22]刘晓明,何青相,赵明华.岩溶地基上桥梁桩基设计优化方法研究与实例[J].中南大学学报:自然科学版,2014,45(5):1653-1658.LIU Xiao-ming, HEQing-xiang, ZHAOMing-hua.Optimization design of bridge pile foundation in karst region:methodology and examples[J].Journal of Central South University:Science and Technology,2014,45(5):1653-1658.(in Chinese)
[23]张建华.岩溶区桥梁桩桩基长确定方法研究[J].公路工程,2009,34(4):1-4,15.ZHANG Jian-hua.Determination method of pile length of bridge pile in karst area[J].Highway Engineering,2009,34(4):1-4,15.(in Chinese)
[24] DALERCI G,BOVOLENTA R.A new method for the evaluation of the ultimate load of piles by tests not carried to failure[J].Geotechnical and Geological Engineering,2014,32(6):1415-1426.
[25]修朝英,李大展.单桩垂直静载试验P-S曲线的数学描述和极限荷载的预测[J].岩土工程学报,1988,10(6):64-73.XIU Chao-ying,LI Da-zhan.Mathematical description of P-S curve and prediction of ultimate load for vertical static load test of single pile[J].Chinese Journal of Geotechnical Engineering,1988,10(6):64-73.(in Chinese)
[26] KULHAWY F H.Stress deformation properties of rock and rock discontinuities[J].Engineering Geology,1975,9(4):327-350.
[27]谢富贵.岩溶地区桥梁桩基承载特性研究[D].西安:长安大学,2007.XIE Fu-gui.A study on properties of bridge piles in karst areas[D].Xi'an:Chang'an University,2007.(in Chinese)
[28]龚成中.岩溶地区嵌岩桩基承载特性研究[D].南京:东南大学,2005.GONG Cheng-zhong.Study of the characteristic of bearing capability of pile foundation in karsts area[D].Nanjing:Southeast University,2005.(in Chinese)
[29]唐咸力.岩溶区基桩下伏溶洞顶板竖向承载机理及其试验研究[D].长沙:湖南大学,2016.TANG Xian-li. The bearing capacity and experimental research of karst cave roof under pile tip foundation in karst area[D].Changsha:Hunan University,2016.(in Chinese)
[30] JENSEN V P,HOLL D L.An application of derivatives of non-analytic functions in plane stress problems[J].Bulletin of the American Mathematical Society,1937,43(4):256-260.
[31]柏华军.考虑溶洞顶板自重时桩端持力岩层安全厚度计算方法[J].岩土力学,2016,37(10):2945-2952.BAI Hua-jun.A method for calculating the safety rock thickness of pile bearing strata with considering deadweight of karst cave roof[J].Rock and Soil Mechanics,2016,37(10):2945-2952.(in Chinese)