Abstract：Based on the theory of three-dimensional continuum mechanics, an analytical model was developed to investigate the seismic response of large-diameter, double-layer concrete-lined shafts in unsaturated soil subjected to vertically incident P-waves. The three-dimensional dynamic governing equations for unsaturated soils were established, taking into account capillary pressure and a saturation-dependent dynamic shear modulus. The double-layer shaft lining was idealized as two parallel Rayleigh-Love rods to account for their transverse inertial effects.
Initially, the free-field response of the unsaturated soil was determined using one-dimensional wave propagation theory. Subsequently, the wave equations of the soil were solved through the potential function method, operator decomposition, and the separation of variables. A closed-form solution for the seismic response of the shaft was derived by applying the boundary conditions of the coupled soil-shaft system. The model proposed in this study was validated through comparison with existing results and followed by a parametric sensitivity analysis to evaluate the influence of soil saturation, shaft geometry, and material properties on the seismic response. The results indicated that the seismic response of large-diameter, double-layered shafts is significantly influenced by the excitation frequency: the low-frequency behavior is governed by the static characteristics of the shaft, while the high-frequency response is dominated by the soil-structure dynamic interaction. As soil saturation increases, both the natural frequency of the system and the kinematic amplification factor at the shaft head decrease. Additionally, the transverse inertial effect of the double-layer lining has a negligible impact within the primary frequency range of vertical seismic loading. The proposed analytical framework provides a theoretical basis for the seismic design and safety assessment of shafts in unsaturated ground.
[1] DIAS T G S，FARIAS M M，ASSIS A P. Large diameter shafts for underground infrastructure[J]. Tunnelling and Underground Space Technology，2015，45：181–189.
[2] ZHANG B，LU L，ZHONG Z，et al. Analytical solution for seismic response of vertical shaft with primary and secondary linings under SH wave excitation[J]. Computers and Geotechnics，2023，164：105813.
[3] WANG W L，WANG T T，SU J J，et al. Assessment of damage in mountain tunnels due to the Taiwan Chi-Chi Earthquake[J]. Tunnelling and Underground Space Technology，2001，16(3)：133–150.
[4] KONTOE S，ZDRAVKOVIC L，POTTS D M，et al. Case study on seismic tunnel response[J]. Canadian Geotechnical Journal，2008，45(12)：1 743–1 764.
[5] 煤炭工业部规划设计院. 唐山地震开滦煤矿井巷工程的震害[J]. 地震工程与工程振动，1982，2(1)：67–76.(Coal Mines Planning and Design Institute，Ministry of Coal Industry. Damage to structures and installations in the underground excavations of the Kailuan colliery during the Tangshan Earthquake[J]. Earthquake Engineering and Engineering Dynamics，1982，2(1)：67–76.(in Chinese))
[6] 刘仁训. 1985年墨西哥大地震述评[J]. 工程抗震，1987，9(2)：44–49.(LIU Renxun. Review of the 1985 Mexico earthquake[J]. Earthquake Resistant Engineering and Retrofitting，1987，9(2)：44–49.(in Chinese))
[7] 杜修力，李 洋，许成顺，等. 1995年日本阪神地震大开地铁车站震害原因及成灾机理分析研究进展[J]. 岩土工程学报，2018，40(2)：223–236.(DU Xiuli，LI Yang，XU Chengshun，et al. Review on damage causes and disaster mechanism of Dakai subway station during 1995 Osaka-Kobe Earthquake[J]. Chinese Journal of Geotechnical Engineering，2018，40(2)：223–236.(in Chinese))
[8] 禹海涛，张敬华，袁 勇，等. 盾构隧道–工作井节点振动台试验设计与分析[J]. 中国公路学报，2017，30(8)：183–192.(YU Haitao，ZHANG Jinghua，YUAN Yong，et al. Design and analysis of shaking table test on shield tunnel-shaft junction[J]. China Journal of Highway and Transport，2017，30(8)：183–192.(in Chinese))
[9] KEYKHOSROPOUR L，LEMNITZER A. Experimental studies of seismic soil pressures on vertical flexible，underground structures and analytical comparisons[J]. Soil Dynamics and Earthquake Engineering，2019，118：166–178.
[10] 赵武胜，何先志，陈卫忠，等. 盾构隧道与竖井连接处管片及接头震害分析[J]. 岩石力学与工程学报，2012，31(增2)：3 847–3 854. (ZHAO Wusheng，HE Xianzhi，CHEN Weizhong，et al. Analysis of seismic damage of segments and joints at the junction of shield tunnel and shaft[J]. Chinese Journal of Rock Mechanics and Engineering，2012，31(Supp.2)：3 847–3 854.(in Chinese))
[11] 陈向红，陶连金，陈 曦. 水下隧道附属竖井的横向地震响应研究[J]. 科学技术与工程，2016，16(13)：273–278.(CHEN Xianghong，TAO Lianjin，CHEN Xi. Study on the transverse seismic response of underwater tunnel shaft[J]. Science Technology and Engineering，2016，16(13)：273–278.(in Chinese))
[12] CHEN Z Y，JIA P. Three-dimensional analysis of effects of ground loss on static and seismic response of shafts[J]. Tunnelling and Underground Space Technology，2019，92：103067.
[13] 申玉生，朱双燕，资晓鱼，等. 穿越上软下硬地层公路隧道竖井结构抗震性能分析[J]. 北京交通大学学报，2019，43(3)：98–104. (SHEN Yusheng，ZHU Shuangyan，ZI Xiaoyu，et al. Analysis on seismic performance of highway tunnel shafts crossing soft and hard stratum[J]. Journal of Beijing Jiaotong University，2019，43(3)：98–104.(in Chinese))
[14] 陈国兴，卢艺静，王彦臻，等. 海底盾构隧道-竖井连接部位三维非线性地震反应特性[J]. 岩土工程学报，2021，43(8)：1 382–1 390. (CHEN Guoxing，LU Yijing，WANG Yanzhen，et al. 3D nonlinear seismic response characteristics for the junction of undersea shield tunnel-shaft[J]. Chinese Journal of Geotechnical Engineering，2021，43(8)：1 382–1 390.(in Chinese))
[15] 张 卜，姬若愚，钟紫蓝，等. 穿越岩土交界面竖井衬砌壁厚对其水平地震响应影响研究[J]. 土木工程学报，2022，55(增1)：250–256.(ZHANG Bu，JI Ruoyu，ZHONG Zilan，et al. Study on the influence of lining thickness of shaft passing through geotechnical interface on its horizontal seismic response[J]. China Civil Engineering Journal，2022，55(Supp.1)：250–256.(in Chinese))
[16] ZHANG B，CHEN Z Y. General Winkler model for kinematic responses of shafts in linear soil[J]. International Journal of Computational Methods，2020，17(5)：1940004.
[17] 张 卜，卢立东，钟紫蓝，等. 竖井结构初衬及二次衬砌竖向地震响应解析解[J]. 西安建筑科技大学学报：自然科学版，2023，55(2)：196–204.(ZHANG Bu，LU Lidong，ZHONG Zilan，et al. Analytical solution of vertical seismic response of primary lining and secondary lining of shaft structure[J]. Journal of Xi?an University of Science and Technology：Natural Science，2023，55(2)：196–204.(in Chinese))
[18] 张 卜，卢立东，钟紫蓝，等. 竖井结构初衬及二次衬砌水平地震响应解析解[J]. 振动工程学报，2024，37(12)：2 003–2 011.(ZHANG Bu，LU Lidong，ZHONG Zilan，et al. Analytical solution for horizontal seismic response of shaft structure primary lining and secondary lining[J]. Journal of Vibration Engineering，2024，37(12)：2 003– 2 011.(in Chinese))
[19] 张 卜，卢立东，陈之毅，等. 考虑土层切向剪力作用的大直径竖井结构水平地震响应解析解[J]. 振动工程学报，2024，38(10)： 2 416–2 428.(ZHANG Bu，LU Lidong，CHEN Zhiyi，et al. Analytical expression of horizontal seismic response of large-diameter shaft structure considering soil tangential shear force[J]. Journal of Vibration Engineering，2024，38(10)：2 416–2 428.(in Chinese))
[20] 张 卜，卢立东，钟紫蓝，等. 城市地下嵌岩型和摩擦型双层衬砌竖井结构竖向地震响应对比分析[J]. 世界地震工程，2023，39(4)：32–39.(ZHANG Bu，LU Lidong，ZHONG Zilan，et al. Comparative study on vertical seismic response of rock-socketed and frictional underground shaft with double layered lining in urban city[J]. Journal of Vibration Engineering，2023，39(4)：32–39.(in Chinese))
[21] ANOYATIS G，LAORA R D，MYLONAKIS G. Axial kinematic response of end-bearing piles to P waves[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2013，37(17)： 2 877–2 896.
[22] DAI D H，EI NAGGAR M H，ZHANG N，et al. Kinematic response of an end-bearing pile subjected to vertical P-wave considering the three-dimensional wave scattering[J]. Computers and Geotechnics，2020，120：103368.
[23] ZHANG S P，XU Z，DENG C. Kinematic responses of a pipe pile embedded in a poroelastic soil to seismic P waves[J]. Acta Geotechnica，2022，17(12)：5 533–5 556.
[24] ZHENG C J，KOURETZIS G，DING X M. Seismic response of end-bearing piles in saturated soil to P-waves[J]. Acta Geotechnica，2023，18(10)：5 519–5 533.
[25] ZHENG C J，LUO T，DING X M. Axial kinematic response of single floating pipe piles to vertically propagating P-waves[J]. Ocean Engineering，2023，273：114040.
[26] DAI D H，XUE K，GAO Y F，et al. Axial kinematic response of an end-bearing pile subjected to seismic P-wave excitation in a double-layered soil[J]. Soil Dynamics and Earthquake Engineering，2024，187：109012.
[27] 吕述晖，王奎华，吴文兵. 考虑横向惯性效应时黏弹性支承桩纵向振动特性研究[J]. 振动工程学报，2016，29(4)：679–686.(LV Shuhui，WANG Kuihua，WU Wenbing. Longitudinal vibration of viscoelastic bearing piles considering transverse inertia effect[J]. Journal of Vibration Engineering，2016，29(4)：679–686.(in Chinese))
[28] 李振亚，王奎华. 考虑横向惯性效应时非均质土中大直径桩纵向振动特性及其应用[J]. 岩石力学与工程学报，2017，36(1)：243–253. (LI Zhenya，WANG Kuihua. Vertical vibration of a large diameter pile embedded in inhomogeneous soil considering the transverse inertia effect and its applications[J]. Chinese Journal of Rock Mechanics and Engineering，2017，36(1)：243–253.(in Chinese))
[29] MA W J，SHAN Y，WANG B L，et al. Torsional dynamic response of a pipe pile in homogeneous unsaturated soil[J]. Computers and Geotechnics，2022，143，104607.
[30] MA W J，SHAN Y，WANG B L，et al. Vertical dynamic impedance of end-bearing pile groups embedded in homogeneous unsaturated soil[J]. International Journal for Numerical and Analytical Methods in Geomechanics，2022，46(6)：1 154–1 176.
[31] 马文杰，王长丹，王炳龙，等. 层状非饱和土中端承桩竖向振动响应的半解析算法[J]. 岩石力学与工程学报，2023，42(增1)：3 767–3 777.(MA Wenjie，WANG Changdan，WANG Binglong，et al. Semi-analytical method for vertical vibration response of end-bearing pile in layered unsaturated soils[J]. Chinese Journal of Rock Mechanics and Engineering，2023，42(Supp.1)：3 767–3 777.(in Chinese))
[32] 徐明江. 非饱和土地基与基础的动力响应研究[博士学位论文][D]. 广州：华南理工大学，2010.(XU Mingjiang. Investigation on dynamic response of unsaturated soils and foundation[Ph. D. Thesis][D]. Guangzhou：South China University of Technology，2010.(in Chinese))
[33] 邹新军，杨紫健，吴文兵. 非饱和土地基中端承桩对SH波的水平地震响应[J]. 岩土工程学报，2024，46(1)：72–80.(ZOU Xinjun，YANG Zijian，WU Wenbing. Horizontal seismic response of end-bearing pile in unsaturated soil foundation under SH wave[J]. Chinese Journal of Geotechnical Engineering，2024，46(1)：72–80.(in Chinese))
[34] LIU H B，DAI G L，ZHOU F X，et al. Vertical kinematic response of an end-bearing pipe pile in fractional viscoelastic unsaturated soil under vertically-incident P-waves[J]. Applied Mathematical Modelling，2023，120：686–710.
[35] LIU H B，DAI G L，ZHOU F X，et al. Kinematic response of pipe pile embedded in fractional-order viscoelastic unsaturated soil subjected to vertically propagating seismic SH-waves[J]. Acta Geotechnica，2023，18(12)：6 803–6 830.
[36] ZHANG M，WANG X H，YANG G C，et al. Solution of dynamic Green's function for unsaturated soil under internal excitation[J]. Soil Dynamics and Earthquake Engineering，2014，64：63–84.
[37] MYLONAKIS G，GAZETAS G. Kinematic pile response to vertical P-wave seismic excitation[J]. Journal of Geotechnical and Geoenvironmental Engineering，2002，128(10)：860–867.
2026, Vol. 45 
