Analytical solution for dynamic response of curved tunnels under travelling wave effect
-
摘要: 隧道曲率变化段是制约隧道结构抗震安全性的关键控制区段,但目前隧道抗震设计仅以横断面剪切变形为主,未考虑隧道沿纵向的曲率半径变化,缺乏针对曲线隧道的纵向抗震简化分析方法。将曲线隧道沿纵向简化为作用在黏弹性地基上的变曲率有限长均质Euler-Bernoulli梁,基于Hamilton原理及黏弹性地基梁理论建立了结构的微分动力控制方程及边界条件,并通过模态叠加法进行求解,推导出任意动载作用下曲线隧道的位移、速度、加速度、弯矩、剪力等动力响应的解析表达式,以行波荷载为例,给出行波效应下曲线隧道动力响应的退化解答。通过与有限元基准模型在相同条件下的对比分析,验证了所推导解析解的正确性。最后应用该解析公式进行参数敏感性分析,揭示了隧道曲率半径、行波波速、行波频率及地层-结构相对刚度比等关键因素对曲线隧道结构动力响应的影响规律。Abstract: The tunnel section with curvature variation that restricts the seismic safety of a tunnel is one of the critical sections for structural design. However, the current seismic design of tunnel structures only focuses on the shear deformation of the tunnel cross-section and the longitudinal curvature is not considered. The most important issue is that there are no available simplified methods in the current literatures for longitudinal seismic analysis of curved tunnels. It is therefore necessary to solve the forward problem with the purpose of obtaining an analytical solution for the dynamic response of curved tunnels under travelling loads. Firstly, a curved tunnel is assumed as a finite homogeneous beam with variable curvature resting on a viscoelastic foundation, and the governing differential equation and boundary conditions of the dynamic problem are established based on the Hamilton principle and the viscoelastic foundational beam theory. Then, the modal superposition method is employed to solve the dynamic problem, and thus the analytical solutions of dynamic responses for curved tunnels subjected to arbitrary dynamic loads are derived. Finally, the degraded solution for travelling loads is obtained with the proposed solution. The solutions of tunnel responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The validation of the analytical solution is verified by providing comparisons between its results and those from the finite element method. The parametric analyses are performed to investigate the influences of the radius of curvature, the velocity and frequency of travelling loads and soil-structure relative stiffness ratio on the dynamic responses of the curved tunnel.
-
-
![]()
图 1 任意动载作用下黏弹性地基上曲线隧道简化模型
Figure 1. Curved tunnel on viscoelastic foundation subjected to arbitrary dynamic loads
![]()
图 2 x=0处隧道动力响应对比
Figure 2. Dynamic responses at position of x =0 of tunnel under travelling loads
![]()
图 4 不同曲率下隧道最大加速度响应
Figure 4. Peak accelerations of tunnel under travelling loads with different curvatures
![]()
图 5 不同行波波速下隧道加速度峰值
Figure 5. Peak accelerations of tunnel under travelling loads with different velocities
![]()
图 6 不同行波频率下隧道加速度响应幅值-频率曲线
Figure 6. Frequency spectra at position of x =0 of tunnel under travelling loads with different frequencies
![]()
图 7 不同行波频率下隧道最大弯矩响应
Figure 7. Peak bending moments of tunnel under travelling loads with different frequencies
-
[1] 张玉娥, 白宝鸿, 张耀辉, 等. 地铁区间隧道震害特点、震害分析方法及减震措施的探讨[J]. 振动与冲击, 2003(1): 72-73, 65, 76, 111. https://www.cnki.com.cn/Article/CJFDTOTAL-ZDCJ200301018.htm ZHANG Yu-e, BAI Bao-hong, ZHANG Yao-hui, et al. Study on subway tunnel's behavior due to seismic damage, methods of analyzing seismic response and earthquake-proof means[J]. Journal of Vibration and Shock, 2003(1): 72-73, 65, 76, 111. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZDCJ200301018.htm
[2] 袁勇, 包蓁, 禹海涛, 等. 考虑行波效应的盾构隧道多点振动台试验[J]. 中国公路学报, 2017, 30(8): 174-182. doi: 10.3969/j.issn.1001-7372.2017.08.020 YUAN Yong, BAO Zhen, YU Hai-tao, et al. Multi-point shaking table test on shield tunnels in consideration of wave-passage effect[J]. China Journal of Highway and Transport, 2017, 30(8): 174-182. (in Chinese) doi: 10.3969/j.issn.1001-7372.2017.08.020
[3] 袁勇, 禹海涛, 燕晓, 等. 超长沉管隧道多点振动台试验模拟与分析[J]. 中国公路学报, 2016, 29(12): 157-165. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL201612023.htm YUAN Yong, YU Hai-tao, YAN Xiao, et al. Multi-point shaking table test simulation and analysis of a super-long immersed tunnel[J]. China Journal of Highway and Transport, 2016, 29(12): 157-165. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL201612023.htm
[4] 禹海涛, 袁勇, 顾玉亮, 等. 非一致激励下长距离输水隧道地震响应分析[J]. 水利学报, 2013, 44(6): 718-725. https://www.cnki.com.cn/Article/CJFDTOTAL-SLXB201306016.htm YU Hai-tao, YUAN Yong, GU Yu-liang, et al. Effect of non-uniform excitation on seismic response of long-distance water-conveyance tunnel[J]. Journal of Hydraulic Engineering, 2013, 44(6): 718-725. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SLXB201306016.htm
[5] St JOHN C M S, ZAHRAH T F. Aseismic design of underground structures[J]. Tunnelling and Underground Space Technology, 1987, 2(2): 165-197. doi: 10.1016/0886-7798(87)90011-3
[6] HASHASH Y M A, HOOK J J, SCHMIDT B, et al. Seismic design and analysis of underground structures[J]. Tunnelling and Underground Space Technology, 2001, 16(4): 247-293. doi: 10.1016/S0886-7798(01)00051-7
[7] 禹海涛, 吴胤翔, 涂新斌, 等. 盾构隧道纵向地震响应的多尺度分析方法[J]. 中国公路学报, 2019, 33(1): 139-144, 152. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL202001014.htm YU Hai-tao, WU Yin-xiang, TU Xin-bin, et al. Multi-scale method for longitudinal seismic response analysis of shield tunnels[J]. China Journal of Highway and Transport, 2019, 33(1): 139-144, 152. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL202001014.htm
[8] 周彦良. 曲线隧道的地震响应特性及抗震减震方法研究[D]. 西安: 西安科技大学, 2013. ZHOU Yan-liang. Study on Seismic Response Properties and Anti-Seismic and Seismic-Relieving Measures of Curved Tunnel[D]. Xi'an: Xi'an University of Science and Technology, 2013. (in Chinese)
[9] 穆嘉豪, 王国波. 小半径平面曲线隧道地震响应分析[J]. 岩土工程学报, 2019, 41(增刊2): 197-200. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC2019S2051.htm MU Jia-hao, WANG Guo-bo. Seismic response of small-radius planar curved tunnels[J]. Chinese Journal of Geotechnical Engineering, 2019, 41(S2): 197-200. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC2019S2051.htm
[10] FABOZZI S, BILOTTA E, YU H, et al. Effects of the asynchronism of ground motion on the longitudinal behavior of a circular tunnel[J]. Tunnelling and Underground Space Technology, 2018, 82: 529-541.
[11] 禹海涛, 张正伟, 李攀, 等. 土岩变化地层长隧道纵向地震响应解析解[J]. 岩土工程学报, 2019, 41(7): 1244-1250. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201907010.htm YU Hai-tao, ZHANG Zheng-wei, LI Pan, et al. Analytical solution for longitudinal seismic responses of long tunnels crossing soil-rock stratum[J]. Chinese Journal of Geotechnical Engineering, 2019, 41(7): 1244-1250. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201907010.htm
[12] 禹海涛, 袁勇. 长大隧道地震响应分析与试验方法新进展[J]. 中国公路学报, 2018, 31(10): 19-35. https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL201810002.htm YU Hai-tao, YUAN Yong. Review on seismic response analysis and test methods for long and large tunnels[J]. China Journal of Highway and Transport, 2018, 31(10): 19-35. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZGGL201810002.htm
[13] 禹海涛, 蔡创, 张正伟. 任意动载作用下长隧道纵向响应解析解[J]. 同济大学学报(自然科学版), 2018, 46(1): 1-6. https://www.cnki.com.cn/Article/CJFDTOTAL-TJDZ201801002.htm YU Hai-tao, CAI Chuang, ZHANG Zheng-wei. Analytical solutions for long tunnels under arbitrary dynamic loadings[J]. Journal of Tongji University (Natural Science), 2018, 46(1): 1-6. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-TJDZ201801002.htm
[14] CLOUGH R W, PENZIEN J. Dynamics of Structures[M]. the United: States of America Computers and Structures, Inc, 2003.
[15] 姚玲森. 曲线梁[M]. 北京: 人民交通出版社, 1989. YAO Ling-sen. Curved Beam[M]. Beijing: China Communications Press, 1989. (in Chinese)
[16] STANTON S C, MANN B P. On the dynamic response of beams with multiple geometric or material discontinuities[J]. Mechanical Systems and Signal Processing, 2010, 24(5): 1409-1419.
[17] 志波由紀夫, 川島一彦, 大日方尚巳, et al. 応答変位法によるシールドトンネルの地震時断面力の算定法[C]//土木学会論文集, 1989: 385-394. YUKIS S, KAZUHIKO K, NAOMI O, et al. Calculation of internal force of structure during earthquake using response displacement method[C]//Proceedings of the Civil Academy, 1989: 385-394. (in Japanese)
[18] ZHANG J, YUAN Y, YU H. Shaking table tests on discrepant responses of shaft-tunnel junction in soft soil under transverse excitations[J]. Soil Dynamics and Earthquake Engineering, 2019, 120: 345-359.
[19] 李心熙. 曲线隧道纵向动力响应解析解及其影响因素分析[D]. 上海: 同济大学, 2019. LI Xin-xi. Analytical Solution for Longitudinal Dynamic Responses of Curved Tunnels and Parametric Analyses[D]. Shanghai: Tongji University, 2019. (in Chinese)
[20] 公路隧道设计规范 第一册 土建工程：JTG 3370.1—2018[S]. 北京: 人民交通出版社, 2018. Specifications for Design of Highway Tunnels Section 1 Civil Engineering: JTG 3370.1—2018[S]. Beijing: China Communications Press, 2018. (in Chinese)
[21] 范立础, 王君杰, 陈玮. 非一致地震激励下大跨度斜拉桥的响应特征[J]. 计算力学学报, 2001, 18(3): 358-363. https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG200103021.htm FAN Li-chu, WANG Jun-jie, CHEN Wei. Response characteristics of long-span cable-stayed bridges under non-uniform seismic action[J]. Chinese Journal of Computational Mechanics, 2001, 18(3): 358-363. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-JSJG200103021.htm
-
期刊类型引用(15)
1. 牛广利，李天旸，薛广文，崔朋，秦朋，方豪文. 长距离引调水工程智能安全监控预警系统研发及应用. 长江科学院院报. 2025(02): 204-210 . 
百度学术
2. 张中昊，李赛，汪可欣. 蠕滑-地震动联合作用下跨断层输水隧洞衬砌结构损伤分析. 振动与冲击. 2025(04): 275-285 . 百度学术
3. 司才龙，贾治元，贡力. 冰-水耦合作用下流冰对闸墩的撞击破坏力学特性响应. 科学技术与工程. 2025(07): 2943-2950 . 百度学术
4. 李艳丽，章志明，桂单明. 基于组合赋权-云模型的城市配水工程质量评价研究. 中国农村水利水电. 2025(03): 14-22 . 百度学术
5. 邓子昂，张继勋，张玉贤，孙艳鹏. 代理模型驱动的隧洞支护方案多目标优化方法. 水力发电学报. 2025(04): 59-71 . 百度学术
6. 张书豪，艾亚鹏，陈健，靳春玲，姬照泰. 物元可拓法在引水隧洞围岩稳定性评价中的应用. 安全与环境学报. 2024(01): 10-18 . 百度学术
7. 方腾卫，杨孟，张建伟，陈磊，曹克磊. TBM引水隧洞组合结构联合承载特性及荷载分担率研究. 广东水利水电. 2024(01): 31-38 . 百度学术
8. 徐文韬，徐红超，石长征，伍鹤皋，张超，李端正. 有压引水隧洞多层柔性叠合衬砌对断层蠕滑错动的适应性研究. 岩土工程学报. 2024(05): 998-1007 . 本站查看
9. 夏求林，吕兴栋，李平刚，周世华，高志扬. PVA纤维、减缩剂和轻烧氧化镁对水工衬砌混凝土性能影响对比研究. 水利水电技术(中英文). 2024(S1): 429-433 . 百度学术
10. 黄邦辉，李志荣，左澍琼，邓开来，辜文兰，洪彧. 导水屏障调谐减震渡槽振动台试验. 清华大学学报(自然科学版). 2024(07): 1147-1156 . 百度学术
11. 胡江，叶伟，马福恒. 长距离调水工程安全风险综合评估. 南水北调与水利科技(中英文). 2024(03): 417-426 . 百度学术
12. 郑书闽，颜建国，郭鹏程，徐燕，李江，刘振兴. 基于VMD及深度学习的供水管道小尺度泄漏检测研究. 水利学报. 2024(08): 999-1008 . 百度学术
13. 李颂章，李星霖，韦凡秋，罗亮明，夏裕栋. 基于数字图像修正HC法的水工隧洞围岩精细化分级研究. 中国农村水利水电. 2024(11): 227-232+246 . 百度学术
14. 张剑，邹德兵，傅兴安，鲁伟. 超大城市水网深隧绿色建造与智慧工程设计. 水利水电技术(中英文). 2024(S2): 250-254 . 百度学术
15. 包娟，钱波，段正刚，庄锦亮. 长距离输水管道水头损失计算可视化程序设计与应用. 云南水力发电. 2023(07): 124-127 . 百度学术
其他类型引用(8)
下载:
计量
- 文章访问数:
- HTML全文浏览量: 0
- PDF下载量:
- 被引次数: 23
