平面Rayleigh波入射下饱和土中浅埋隧道复合式衬砌的动力响应

范凯祥; 申玉生; 闻毓民; 黄海峰; 王帅帅; 高波

doi:10.11779/CJGE202203006

平面Rayleigh波入射下饱和土中浅埋隧道复合式衬砌的动力响应 English Version

范凯祥^1,,
申玉生^1, ,,
闻毓民¹,
黄海峰¹,
王帅帅^{1, 2},
高波¹

1.
西南交通大学交通隧道工程教育部重点实验室, 四川成都 610031
2.
中交第二公路工程局有限公司, 陕西西安 710065

基金项目:

国家自然科学基金项目 51778540

国家重点研发计划项目 2019YFC0605104

详细信息

作者简介:
范凯祥(1988—)，男，2011年本科毕业于吉林大学，博士，主要从事隧道抗减震方面的研究。E-mail: fkx_0706@my.swjtu.edu.cn

通讯作者:
申玉生: 许成顺, E-mail: sys1997@163.com

中图分类号: U45
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出版历程
- 收稿日期: 2021-02-22
- 网络出版日期: 2022-09-22
- 刊出日期: 2022-02-28

Dynamic response of composite linings of shallowly buried tunnels in saturated soils subjected to incidence of plane Rayleigh waves

1.
Key Laboratory of Transportation Tunnel Engineering of the Ministry of Education, Southwest Jiaotong University, Chengdu 610031, China
2.
CCCC Second Highway Engineering Co., Ltd., Xi'an 710065, China

摘要

摘要: 基于Biot波动理论，采用Fourier-Bessel级数展开法，建立了平面Rayleigh波入射下，饱和土中浅埋隧道复合式衬砌的散射力学模型，求解了频域内复合式衬砌的动应力集中系数、孔压集中系数的解析解，通过参数化分析，研究了Rayleigh波在不同入射频率作用下，内衬和外衬的刚度比、厚度比、隧道埋深等因素对复合式衬砌动力响应的影响规律。结果表明，入射频率对复合式衬砌的动应力集中系数和孔压集中系数的空间分布和幅值影响显著；增大内衬和外衬的刚度比、厚度比可以显著降低外衬的动应力集中系数和孔压集中系数，最大降幅可达90%以上，但会显著放大内衬的动应力集中系数，而且超过一定幅值后，对外衬的减震效果影响有限，建议内衬和外衬刚度比取值范围为2~4，厚度比取值范围为1~2；随埋深的增大，内衬的动应力集中系数逐渐降低，Rayleigh波对浅埋隧道动力响应的影响更为显著。研究成果可为水下隧道的抗减震设计提供理论支撑。
- 隧道工程 /
- 饱和土 /
- Rayleigh波 /
- 浅埋隧道 /
- 复合式衬砌 /
- 动力响应
Abstract: Based on the Biot wave theory and the Fourier-Bessel series expansion method, a mechanical model for scattering of composite linings of shallowly buried tunnels in saturated soils subjected to incidence of plane Rayleigh waves is established. The analytical solutions of dynamic stress concentration coefficient, pore pressure concentration coefficient and displacement of saturated soils of the composite linings in frequency domain are solved. Through the parameterization analysis, the influences of the stiffness ratio, thickness ratio and tunnel depth on the dynamic response of the composite linings subjected to Rayleigh waves in different frequencies are studied. The results show that the incident frequency has a significant effect on the dynamic stress concentration coefficient and pore pressure concentration coefficient of the composite linings. Increasing the stiffness ratio and thickness ratio of inner linings to outer linings can significantly reduce the dynamic stress concentration factor and pore pressure concentration factor of the outer linings, and the maximum decrease can be more than 90%, but can significantly amplify the dynamic stress concentration factor of the inner linings, and the impact of shock absorption of the outer linings is limited when the amplitude exceeds a certain value. It is suggested that the stiffness ratio of inner linings to outer linings should be 2~4, and the thickness ratio should be 1~2. With the increase of the buried depth, the dynamic stress concentration coefficient of the inner linings decreases gradually, and the influences of Rayleigh waves on the dynamic response of shallowly buried tunnels are more significant. The results may provide theoretical support for the anti-shock design of underwater tunnels.
- tunnel engineering /
- saturated soil /
- Rayleigh wave /
- shallowly buried tunnel /
- composite lining /
- dynamic response

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0. 引言

劲性水泥搅拌桩（SDCM column）也称“混凝土芯搅拌桩”,是一种由水泥搅拌桩(DCM column)壳和中心处较小面积预制混凝土芯桩组成的复合材料桩。按照芯桩与DCM桩壳二者长度间的相对关系,SDCM桩分为3种类型：芯桩长度小于DCM桩壳长度的短芯SDCM桩、芯桩长度等于DCM桩壳长度的等芯SDCM桩和芯桩长度大于DCM桩壳长度的长芯SDCM桩。较之传统的DCM桩,SDCM桩的竖向和水平向承载力高,沉降控制效果好。它既可作为桩基础,也可作为复合地基的竖向增强体,在国内外都得到了大量应用^[1-2]。

很多学者通过室内外试验和数值模拟等方法,从承载力、荷载传递机制和控沉效果等方面对短芯SDCM桩和短芯SDCM桩复合地基进行了较为系统的研究。凌光荣等^[3]、董平等^[4]和吴迈等^[5]通过现场单桩载荷试验研究了SDCM桩的轴向承载特性。试验发现,芯桩的插入使搅拌桩桩侧摩阻力充分发挥,SDCM桩的竖向承载力可高于混凝土灌注桩。Voottipruex等^[6]采用三维有限元法研究了芯桩长度和截面积等因素对SDCM桩竖向和水平向承载力、芯桩轴力和弯矩分布的影响。吴迈等^[7]、丁永君等^[8]和顾士坦等^[9]采用理论分析和现场实测芯桩桩身应力的方法研究了SDCM桩的荷载传递特性。Wonglert等^[10-11]基于室内模型试验成果,提出短芯SDCM桩有土体破坏、芯桩底部破坏和搅拌桩壳顶部破坏3种破坏模式,研究了芯桩长度和搅拌桩壳强度对悬浮SDCM桩竖向承载力和破坏模式的影响。Wang等^[12]通过现场荷载板试验,比较了刚性基础下SDCM桩复合地基与DCM桩复合地基的竖向承载力。Voottipruex等^[13]和Wang等^[14]进行了路堤下SDCM桩复合地基现场试验,比较了柔性基础下SDCM桩复合地基与DCM桩复合地基的沉降控制效果。叶观宝等^[15-16]提出了悬浮SDCM桩复合地基桩-土应力比和沉降的计算方法。杨涛等^[17]建立了端承短芯SDCM桩复合地基的固结计算模型。

近年来,随着长芯SDCM桩在中国的逐渐应用,其承载机理和设计理论的研究开始受到学术界的关注。陈华顺等^[18]和程博华^[19]分别提出了长芯SDCM桩桩侧摩阻力和竖向承载力的计算方法。杨涛等^[20]给出了端承长芯SDCM桩复合地基固结解析解,但该解答无法用于分析悬浮长芯SDCM桩复合地基的固结问题。有鉴于此,本文研究悬浮长芯SDCM桩复合地基的固结计算方法,分析悬浮长芯SDCM桩复合地基的固结特性,进一步完善SDCM桩复合地基的固结计算理论。

1. 固结模型与基本假定

1.1 轴对称固结模型

图1给出悬浮长芯SDCM桩复合地基轴对称固结模型,r和z分别是径向和竖向坐标。r_e为一根SDCM桩的影响区半径,可由桩的间距和布桩方式计算得到。水泥搅拌桩壳外半径和长度分别为r_p和L_p=H₁,上部SDCM桩的置换率为m（m=(r_p/r_e)²）,压缩模量为E_p。芯桩打穿水泥搅拌桩壳,其半径和长度分别为r_sp和L_sp（L_sp=H₁+H₂）,压缩模量为E_sp。芯桩的截面含芯率为ρ（ρ=(r_sp/r_p)²）。根据水泥搅拌桩壳和芯桩的长度将复合地基加固区分为Ⅰ和Ⅱ二部分,加固区Ⅰ内桩间土的厚度、渗透系数、固结系数和压缩模量分别为H₁,k_v1,c_v1,E_s1,加固区Ⅱ内桩间土的厚度、渗透系数、固结系数和压缩模量分别为H₂,k_v2,c_v2,E_s2。下卧层土厚度、渗透系数、固结系数和压缩模量分别为H₃,k_v3,c_v3,E_s3。复合地基总厚度H= H₁+H₂+H₃。

图 1 复合地基固结模型

Figure 1. Consolidation model for composite ground

下载: 全尺寸图片幻灯片

1.2 基本假定

本文公式推导采用如下假定：①地基土完全饱和,水的流动符合Darcy定律；②搅拌桩壳不排水；③芯桩桩端以下的下卧层土和搅拌桩壳以下芯桩桩端以上的土仅发生径向渗流；④芯桩与搅拌桩壳间无相对滑移,加固区中任意深度处的桩和土的竖向应变相等；⑤大面积均布荷载p瞬时施加,在待加固地基中引起的竖向附加应力沿深度均布；⑥土的渗透系数和压缩模量不随固结而变化。

2. 固结方程与定解条件

2.1 固结方程

基于前述基本假定,参考杨涛等^[20-21]的研究,可得到荷载p瞬时施加情况下悬浮长芯SDCM桩复合地基得固结方程如下：

。

${\begin{cases} \frac{\partial {\bar{u}}_{s 1}}{\partial t} = c_{v 1 e} \frac{\partial^{2} {\bar{u}}_{s 1}}{\partial z^{2}} \begin{matrix} \end{matrix} (0 \leq z < H_{1}), \\ \frac{\partial {\bar{u}}_{s 2}}{\partial t} = c_{v 2 e} \frac{\partial^{2} {\bar{u}}_{s 2}}{\partial z^{2}} \begin{matrix} \end{matrix} (H_{1} \leq z < H_{1} + H_{2}), \\ \frac{\partial {\bar{u}}_{s 3}}{\partial t} = c_{v 3 e} \frac{\partial^{2} {\bar{u}}_{s 3}}{\partial z^{2}} \begin{matrix} \end{matrix} (H_{1} + H_{2} \leq z \leq H) 。 \end{cases}$

(1)

$c_{v 1 e} = (1 + \frac{m [ρ E_{s p} + (1 - ρ) E_{p}]}{(1 - m) E_{s 1}}) c_{v 1}$ ,

(2)

$c_{v 2 e} = (1 + \frac{m ρ E_{s p}}{(1 - m ρ) E_{s 2}}) (\frac{1 - m}{1 - m ρ}) c_{v 2}$ ,

(3)

$c_{v 3 e} = (1 - m ρ) c_{v3} 。$

(4)

式中 ${\bar{u}}_{s 1}$ , ${\bar{u}}_{s 2}$ , ${\bar{u}}_{s 3}$ 分别为加固区Ⅰ、Ⅱ中桩间土和下卧层土中任意深度z处的平均超静孔隙水压力；c_v1e,c_v2e,c_v3e分别为考虑劲性桩和芯桩影响的加固区Ⅰ、Ⅱ中桩间土和下卧层土的等效竖向固结系数。

2.2 定解条件

（1）边界条件

考虑复合地基上边界排水、下边界不排水的单面排水情况,边界条件如下：

$z = 0, {\bar{u}}_{s 1} = 0$ ,

(5)

$z = H, \frac{\partial {\bar{u}}_{s 3}}{\partial z} = 0$ 。

(6)

（2）孔压和水流连续性条件

考虑到 $(1 - m) {\bar{u}}_{s 1}$ 和 $(1 - m ρ) {\bar{u}}_{s 2}$ 分别为加固区Ⅰ和加固区Ⅱ内任意深度处的平均孔压,可得加固区Ⅰ与加固区Ⅱ分界面处、加固区Ⅱ与下卧层分界面处孔压和水流连续性条件如下：

z=H₁时,

$(1 - m) {\bar{u}}_{s 1} = (1 - m ρ) {\bar{u}}_{s 2}$ ,

(7)

$(1 - m) k_{v 1} \frac{\partial {\bar{u}}_{s 1}}{\partial z} = (1 - m ρ) k_{v 2} \frac{\partial {\bar{u}}_{s 2}}{\partial z}$ 。

(8)

z=H₁+H₂时,

$(1 - m ρ) {\bar{u}}_{s 2} = {\bar{u}}_{s 3}$ ,

(9)

$(1 - m ρ) k_{v 2} \frac{\partial {\bar{u}}_{s 2}}{\partial z} = k_{v 3} \frac{\partial {\bar{u}}_{s 3}}{\partial z}$ 。

(10)

（3）初始条件

荷载施加的瞬时,桩间土和下卧层土中没有竖向变形。参照杨涛等^[20-21]的研究,容易写出如下初始条件：

${\bar{u}}_{s 1} (z, 0) = \frac{p}{1 - m}$ ,

(11)

${\bar{u}}_{s 2} (z, 0) = \frac{p}{1 - m ρ}$ ,

(12)

${\bar{u}}_{s 3} (z, 0) = p$ 。

(13)

3. 固结解析解

3.1 方程与定解条件的函数变换

为便于求解,进行如下函数变换：

${\begin{cases} {\hat{u}}_{s 1} = (1 - m) {\bar{u}}_{s 1}, \\ {\hat{u}}_{s 2} = (1 - m ρ) {\bar{u}}_{s 2}, \\ {\hat{u}}_{s 3} = {\bar{u}}_{s 3} 。 \end{cases}$

(14)

显然, ${\hat{u}}_{s 1}$ 和 ${\hat{u}}_{s 2}$ 分别是复合地基上、下加固区任意深度处的平均超静孔隙水压力。将式（14）代入固结控制方程式（1）和定解条件式（5）～（13）,则变换后的复合地基固结控制方程和求解条件如下：

${\begin{cases} \frac{\partial {\hat{u}}_{s 1}}{\partial t} = c_{v 1 e} \frac{\partial^{2} {\hat{u}}_{s 1}}{\partial z^{2}} \begin{matrix} \end{matrix} (0 \leq z < H_{1}), \\ \frac{\partial {\hat{u}}_{s 2}}{\partial t} = c_{v 2 e} \frac{\partial^{2} {\hat{u}}_{s 2}}{\partial z^{2}} \begin{matrix} \end{matrix} (H_{1} \leq z < H_{1} + H_{2}), \\ \frac{\partial {\hat{u}}_{s 3}}{\partial t} = c_{v 3 e} \frac{\partial^{2} {\hat{u}}_{s 3}}{\partial z^{2}} \begin{matrix} \end{matrix} (H_{1} + H_{2} \leq z \leq H) 。 \end{cases}$

(15)

$z = 0 ： {\hat{u}}_{s 1} = 0$ ,

(16)

$z = H ： \frac{\partial {\hat{u}}_{s 3}}{\partial z} = 0$ ,

(17)

z=H₁,

${\begin{cases} {\hat{u}}_{s 1} = {\hat{u}}_{s 2} \\ k_{v 1} \frac{\partial {\hat{u}}_{s 1}}{\partial z} = k_{v 2} \frac{\partial {\hat{u}}_{s 2}}{\partial z} \end{cases}$ ,

(18)

z=H₁+H₂,

${\begin{cases} {\hat{u}}_{s 2} = {\hat{u}}_{s 3} \\ k_{v 2} \frac{\partial {\hat{u}}_{s 2}}{\partial z} = k_{v 3} \frac{\partial {\hat{u}}_{s 3}}{\partial z} \end{cases}$ ,

(19)

${\hat{u}}_{s 1} (z, 0) = {\hat{u}}_{s 2} (z, 0) = {\hat{u}}_{s 3} (z, 0) = p$ 。

(20)

3.2 固结解析解

显然,式（15）,（16）～（20）分别是瞬时荷载p作用在由复合地基上、下加固区复合土和下卧层土组成的三层土系统固结问题的固结方程和定解条件。与瞬时荷载p作用下的三层天然地基固结问题相比,只是各层天然地基土的固结系数c_v1,c_v2和c_v3分别被c_v1e,c_v2e和c_v3e代替而已。设三层土系统中土层i的渗透系数为k_vi(i=1,2,3),则其压缩模量为E_sie=c_vier_w/k_vi (i=1,2,3）。

为使计算公式得以简化,定义5个无量纲参数：

$\begin{array}{l} a_{i} = \frac{k_{v i}}{k_{v 1}}; b_{i} = \frac{E_{s 1 e}}{E_{s i e}}; c_{i} = \frac{H_{i}}{H_{1}}; μ_{i} = \sqrt{\frac{b_{i}}{a_{i}}} (i = 1, 2, 3) ； \\ d_{i} = \sqrt{\frac{a_{i - 1} b_{i - 1}}{a_{i} b_{i}}} \begin{matrix} \end{matrix} (i = 2, 3) 。 \end{array}}$

(21)

根据谢康和^[22]的研究,容易得到：

（1）各加固区和下卧层的平均超静孔压

${\begin{cases} {\hat{u}}_{s 1} (z, t) = p \sum_{m = 1}^{\infty} C_{m} s i n (λ_{m} \frac{z}{H_{1}}) e^{- λ_{m}^{2} T_{v1e}}, \\ {\hat{u}}_{s 2} (z, t) = p \sum_{m = 1}^{\infty} C_{m} [s i n (λ_{m}) c o s (μ_{2} λ_{m} \frac{z - H_{1}}{H_{1}}) + \\ d_{2} c o s (λ_{m}) s i n (μ_{2} λ_{m} \frac{z - H_{1}}{H_{1}})] e^{- λ_{m}^{2} T_{v1e}}, \\ {\hat{u}}_{s 3} (z, t) = p \sum_{m = 1}^{\infty} \frac{C_{m} A_{m}}{c o s (μ_{3} c_{3} λ_{m})} c o s (μ_{3} λ_{m} \frac{H - z}{H_{1}}) e^{- λ_{m}^{2} T_{v1e}}, \end{cases}$

(22)

$T_{v1e} = c_{v1e} t / H_{1}^{2}$ 。

(23)

（2）桩间土和下卧层土的平均孔压

将式（14）代入式（22）,可得悬浮长芯SDCM桩复合地基上、下部加固区中桩间土和下卧层土平均超静孔隙水压力：

${\begin{cases} {\bar{u}}_{s 1} (z, t) = \frac{p}{1 - m} \sum_{m = 1}^{\infty} C_{m} s i n (λ_{m} \frac{z}{H_{1}}) e^{- λ_{m}^{2} T_{v 1 e}}, \\ {\bar{u}}_{s 2} (z, t) = \frac{p}{1 - m ρ} \sum_{m = 1}^{\infty} C_{m} [s i n (λ_{m}) c o s (μ_{2} λ_{m} \frac{z - H_{1}}{H_{1}}) + \\ d_{2} c o s (λ_{m}) s i n (μ_{2} λ_{m} \frac{z - H_{1}}{H_{1}})] e^{- λ_{m}^{2} T_{v1e}}, \\ {\bar{u}}_{s 3} (z, t) = p \sum_{m = 1}^{\infty} \frac{C_{m} A_{m}}{c o s (μ_{3} c_{3} λ_{m})} c o s (μ_{3} λ_{m} \frac{H - z}{H_{1}}) e^{- λ_{m}^{2} T_{v1e}}, \end{cases}$

(24)

$C_{m} = \frac{2 d_{2} s i n (2 μ_{3} c_{3} λ_{m})}{λ_{m} [s i n (2 μ_{3} c_{3} λ_{m}) (d_{2} + μ_{2} c_{2} F_{m}) - μ_{3} c_{3} D_{m}]}$ ,

(25)

$A_{m} = s i n (λ_{m}) c o s (μ_{2} c_{2} λ_{m}) + d_{2} c o s (λ_{m}) s i n (μ_{2} c_{2} λ_{m})$ ,

(26)

$D_{m} = s i n (2 μ_{2} c_{2} λ_{m}) E_{m} - d_{2} s i n (2 λ_{m}) c o s (μ_{2} c_{2} λ_{m})$ ,

(27)

$E_{m} = s i n^{2} (λ_{m}) - d_{2}^{2} c o s^{2} (λ_{m})$ ,

(28)

$F_{m} = s i n^{2} (λ_{m}) + d_{2}^{2} c o s^{2} (λ_{m})$ 。

(29)

按下面方程求解出特征值 $λ_{m}$ ：

$A_{m} + d_{3} c o t (μ_{3} c_{3} λ_{m}) B_{m} = 0$ ,

(30)

$B_{m} = s i n (λ_{m}) s i n (μ_{2} c_{2} λ_{m}) - d_{2} c o s (λ_{m}) c o s (μ_{2} c_{2} λ_{m})$ 。

(31)

（3）复合地基整体平均固结度

参考谢康和^[22]的研究,容易得到悬浮长芯SDCM桩复合地基按沉降定义和按孔压定义的整体平均固结度U_s和U_p的计算公式：

$U_{s} = 1 - \sum_{m = 1}^{\infty} \frac{C_{m}}{λ_{m} (1 + b_{2} c_{2} + b_{3} c_{3})} e^{- λ_{m}^{2} T_{v1e}}$ ,

(32)

$\begin{matrix} U_{p} = 1 - \sum_{m = 1}^{\infty} \frac{\sqrt{a_{2} b_{2}} B_{m} (b_{3} - b_{2}) + b_{2} b_{3} + (b_{3} - b_{2} b_{3}) c o s (λ_{m})}{λ_{m}^{} b_{2} c_{3} (1 + c_{2} + c_{3})} \cdot \\ C_{m} e^{- λ_{m}^{2} T_{v1e}} 。 \end{matrix}$

(33)

4. 算例验证

算例中,H=20 m,r_e=1.1 m；芯桩：r_sp=0.175 m,L_sp= H₁+ H₂=12 m,E_sp=20 GPa,泊松比μ_sp=0.17。搅拌桩壳：r_p=0.35 m,L_p=H₁=8 m,E_p=150 MPa,泊松比μ_p=0.25。地基土：H₃=8 m,E_s1=E_s2=3 MPa,E_s3=9 MPa,k_v1=k_v2=k_v3=10^-8 m/s,泊松比μ_s=0.35。为了在数值计算中近似模拟等应变条件,在复合地基表面铺设0.5 m厚的混凝土板,板上荷载p=68 kPa瞬时施加。混凝土板压缩模量和泊松比与芯桩相同。地基土、芯桩、搅拌桩壳和混凝土板均采用线弹性模型。在各材料的弹性模量E和压缩模量E₁之间按E= (1+ μ)(1-2μ)E₁/(1-μ)近似换算,μ为泊松比。图2为悬浮长芯SDCM桩复合地基轴对称有限元模型。模型左、右侧边界上约束径向位移,不排水。模型底边界上径向和竖向均约束,不排水。复合地基表面自由,排水。

图 2 有限元模型

Figure 2. FEM model

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采用ABAQUS有限元软件进行算例固结分析。混凝土板、芯桩和搅拌桩壳采用4结点四边形单元（CAX4）剖分,桩间土采用应力-孔压耦合4结点四边形单元（CAX4P）剖分。芯桩-土交界处设置摩擦接触对,摩擦系数取0.42。模型共剖分2466个单元,结点总数2742个。

图3给出本文解析解计算的悬浮长芯SDCM桩复合地基整体平均固结度（U_s）曲线与有限元计算结果的比较,时间轴采用无量纲时间因数T_u=c_v1t/H²。图3表明,本文解析解与有限元计算结果较为接近,解析解数值略大于数值解,二者差值最大不超过3.0%。计算表明,解析解有较高的计算精度。

图 3 解析解和有限元解的比较

Figure 3. Comparison between analytical results and FEM results

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5. 复合地基结性状分析

采用的几何和力学参数基准值如下：H=20 m,H₁=10 m,H₂=5 m,H₃=5 m。m=0.1, $ρ$ =0.25。E_p=150 MPa,E_sp=20 GPa。E_s1=E_s2=3 MPa,E_s3=9 MPa。k_v1=k_v2=k_v3=10^-8 m/s。

（1）桩的贯入比的影响

图4给出长芯SDCM桩贯入比 $β$ =L_sp/H的变化对复合地基固结速率的影响,计算时地基土为均质土,E_s3=3 MPa,搅拌桩壳长度与芯桩长度的比值 $β_{1}$ =L_p/ L_sp=0.67保持不变。图4表明,复合地基的固结速率随着长芯SDCM桩贯入比的增加逐渐增大,当 $β$ >0.75以后,复合地基固结速率的增加率显著增大。

图 4 β对固结速率的影响

Figure 4. Influences of β on consolidation rate

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（2）搅拌桩壳的刚度和几何尺寸的影响

图5～7分别给出上部SDCM桩置换率m、搅拌桩壳长度与芯桩长度之比 $β_{1}$ =L_p/L_sp和搅拌桩壳压缩模量E_p的变化对复合地基固结速率的影响。在图5固结度曲线计算中,芯桩截面积保持不变,即m_iρ_i=mρ=0.025,m越大表示搅拌桩壳的厚度越大。图5～7计算结果表明,搅拌桩壳的厚度、长度和压缩模量的变化对悬浮长芯SDCM桩复合地基的固结速率没有影响。

图 5 　m对固结速率的影响

Figure 5. Influences of m on consolidation rate

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图 6 β₁对固结速率的影响

Figure 6. Influences of β₁ on consolidation rate

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图 7 E_p对固结速率的影响

Figure 7. Influences of E_p on consolidation rate

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（3）芯桩刚度和含心率的影响

图8,9分别给出芯桩的截面含芯率 $ρ$ 和压缩模量E_sp的变化对复合地基固结速率的影响。图8表明,芯桩含芯率 $ρ$ 的变化对悬浮长芯SDCM桩复合地基固结速率近乎没有影响。当含芯率 $ρ$ 增大时,仅在固结前期复合地基的固结速率会略微减小,但降幅非常小,可以忽略不计。从图9中可见,芯桩压缩模量E_sp的变化对悬浮长芯SDCM桩复合地基前期的固结速率有一定影响。随着E_sp数值的增加,前期复合地基固结速率随之减小,但降幅并不大。总的来看,芯桩刚度的变化对复合地基固结速率的影响不大。

图 8 ρ对固结速率的影响

Figure 8. Influence of ρ on the consolidation rate

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图 9 E_sp对固结速率的影响

Figure 9. Influences of E_sp on consolidation rate

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（4）下卧层土体刚度的影响

图10给出下卧层土压缩模量E_s3取不同数值情况下悬浮长芯SDCM桩复合地基的U_s-T_u曲线。图10表明,复合地基的固结速率随下卧层土刚度的增加而增大。这说明,将芯桩的桩端置于承载力较大的持力层上可加速复合地基的固结。

图 10 E_s3对固结速率的影响

Figure 10. Influences of E_s3 on consolidation rate

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6. 结论

（1）本文固结解析解是基于加固区等竖向应变假定获得的,因此,它更适用于刚性基础下悬浮长芯SDCM桩复合地基的固结分析。

（2）桩的贯入比和下卧层土的刚度是影响悬浮长芯SDCM桩复合地基固结快慢的主要因素。桩的贯入比和下卧层土的压缩模量越大,悬浮长芯SDCM桩复合地基的固结越快。

（3）芯桩截面含芯率的变化不会影响悬浮长芯SDCM桩复合地基的固结速率。芯桩刚度的增加会略微减小固结前期复合地基的固结速率,对后期复合地基的固结速率没有影响。

（4）搅拌桩壳的厚度、长度和刚度的变化对悬浮长芯SDCM桩复合地基的固结速率没有影响。

图 1 力学模型

Figure 1. Mechanical model

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图 2 地表正规化位移

Figure 2. Normalized displacements of surface

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图 3 变刚度复合式衬砌动应力集中系数和孔压集中系数

Figure 3. DSCF and PPCF of composite linings with variable stiffnesses

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图 4 汶川地震龙溪隧道开裂

Figure 4. Cracking of Longxi tunnel in Wenchuan earthquake

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图 5 内衬和外衬刚度比对复合式衬砌动应力集中系数和孔压集中系数的影响

Figure 5. Influences of stiffness ratio on DSCF and PPCF of composite lining

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图 6 变厚度复合式衬砌动应力和孔压集中系数

Figure 6. DSCF and PPCF of composite linings with variable thicknesses

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图 7 内衬和外衬厚度比对复合式衬砌动应力集中系数和孔压集中系数的影响

Figure 7. Influences of thickness ratio on DSCF and PPCF of composite lininges

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图 8 隧道埋深对复合式衬砌动应力集中系数和孔压集中系数的影响

Figure 8. Influences of buried depth on DSCF and PPCF of composite linings

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参考文献(25)

[1]	徐长节, 丁海滨, 童立红, 等. 基于非局部Biot理论下饱和土中深埋圆柱形衬砌对平面弹性波的散射[J]. 岩土工程学报, 2018, 40(9): 1563–1570. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201809002.htm XU Chang-jie, DING Hai-bin, TONG Li-hong, et al. Scattering waves generated by cylindrical lining in saturated soil based on nonlocal Biot theory[J]. Chinese Journal of Geotechnical Engineering, 2018, 40(9): 1563–1570. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201809002.htm
[2]	朱赛男, 李伟华, Vincent W Lee, 等. 平面P波入射下海底衬砌隧道地震响应解析分析[J]. 岩土工程学报, 2020, 42(8): 1418–1427. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC202008010.htm ZHU Sai-nan, LI Wei-hua, VINCENT W L, et al. Seismic response of undersea lining tunnels under incident plane P waves[J]. Chinese Journal of Geotechnical Engineering, 2020, 42(8): 1418–1427. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC202008010.htm
[3]	刘中宪, 琚鑫, 梁建文. 饱和半空间中隧道衬砌对平面SV波的散射IBIEM求解[J]. 岩土工程学报, 2015, 37(9): 1599–1612. https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201509008.htm LIU Zhong-xian, JU Xing, LIANG Jian-wen. IBIEM solution to scattering of plane SV waves by tunnel lining in saturated poroelastic half-space[J]. Chinese Journal of Geotechnical Engineering, 2015, 37(9): 1599–1612. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTGC201509008.htm
[4]	刘中宪, 梁建文, 张贺. 弹性半空间中衬砌隧道对瑞利波的散射[J]. 岩石力学与工程学报, 2011, 30(8): 1627–1637. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201108016.htm LIU Zhong-xian, LIANG Jian-wen, ZHANG He. Scattering of rayleigh wave by a lined tunnel in elastic half-space[J]. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(8): 1627–1637. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201108016.htm
[5]	李彬. 地铁地下结构抗震理论分析与应用研究[D]. 北京: 清华大学, 2005. LI Bin. Theoretical Analysis of Seismic Response of Underground Subway Structures and its Application[D]. Beijing: Tsinghua University, 2005. (in Chinese)
[6]	WU D, GAO B, SHEN Y S, et al. Damage evolution of tunnel portal during the longitudinal propagation of Rayleigh waves[J]. Natural Hazards, 2015, 75(3): 2519–2543. doi: 10.1007/s11069-014-1447-2
[7]	GREGORY R D. The propagation of waves in an elastic half-space containing a cylindrical cavity[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 1970, 67(3): 689–710. doi: 10.1017/S0305004100046016
[8]	HÖLLINGER F, ZIEGLER F. Scattering of pulsed rayleigh surface waves by a cylindrical cavity[J]. Wave Motion, 1979, 1(3): 225–238. doi: 10.1016/0165-2125(79)90035-0
[9]	LUCO J E, DE BARROS F C P. Dynamic displacements and stresses in the vicinity of a cylindrical cavity embedded in a half-space[J]. Earthquake Engineering & Structural Dynamics, 1994, 23(3): 321–340.
[10]	梁建文, 纪晓东. 地下衬砌洞室对Rayleigh波的放大作用[J]. 地震工程与工程振动, 2006, 26(4): 24–31. doi: 10.3969/j.issn.1000-1301.2006.04.004 LIANG J, JI X. Amplification of Rayleigh waves due to underground lined cavities[J]. Earthquake Engineering and Engineering Vibration, 2006, 26(4): 24–31. (in Chinese) doi: 10.3969/j.issn.1000-1301.2006.04.004
[11]	LIU Q J, ZHAO M J, WANG L H. Scattering of plane P, SV or Rayleigh waves by a shallow lined tunnel in an elastic half space[J]. Soil Dynamics and Earthquake Engineering, 2013, 49: 52–63. doi: 10.1016/j.soildyn.2013.02.007
[12]	张志军. 饱和半空间中Rayleigh波的传播[D]. 天津: 天津大学, 2007. ZHANG Zhi-jun. The Propagation of Rayleigh Wave in a Half-space Saturated[D]. Tianjin: Tianjin University, 2007. (in Chinese)
[13]	LIN C. Wave Propagation in A Poroelastic Half-Space Saturated With Inviscid Fluid[D]. Southern California: University of Southern California, 2002.
[14]	DERESIEWICZ H. The effect of boundaries on wave propagation in a liquid-filled porous solid: IV Surface waves in a half-space[J]. Bulletin of the Seismological Society of America, 1962, 52(3): 627–638. doi: 10.1785/BSSA0520030627
[15]	刘优平, 龚敏, 徐斌. 半空间饱和土中输水管道对瑞利波的散射[J]. 四川大学学报(工程科学版), 2012, 44(5): 71–77. https://www.cnki.com.cn/Article/CJFDTOTAL-SCLH201205012.htm LIU You-ping, GONG M, XU Bin. Scattering of underground water-filled pipe in half saturated space impacted by rayleigh waves[J]. Journal of Sichuan University (Engineering Science Edition), 2012, 44(5): 71–77. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-SCLH201205012.htm
[16]	徐颖, 梁建文, 刘中宪. Rayleigh波在饱和半空间中圆形洞室周围的散射[J]. 岩土力学, 2017(8): 2411–2424. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201708034.htm XU Ying, LIANG Jian-wen, LIU Zhong-xian. Diffraction of Rayleigh waves around a circular cavity in poroelastic half-space[J]. Rock and Soil Mechanics, 2017(8): 2411–2424. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201708034.htm
[17]	DING H B, TONG L H, XU C J, et al. Dynamic responses of shallow buried composite cylindrical lining embedded in saturated soil under incident P wave based on nonlocal-Biot theory[J]. Soil Dynamics and Earthquake Engineering, 2019, 121: 40–56. doi: 10.1016/j.soildyn.2019.02.018
[18]	ZHANG C, LIU Q, DENG P. Surface Motion of a Half‐Space with a semicylindrical canyon under P, SV, and rayleigh waves[J]. Bulletin of the Seismological Society of America, 2017, 107(2): 1–12.
[19]	LIANG J, BA Z, LEE V W. Diffraction of plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space[J]. Soil Dynamics and Earthquake Engineering, 2006, 26(6/7): 582–610.
[20]	梁建文, 纪晓东. 地下衬砌洞室对Rayleigh波的放大作用[J]. 地震工程与工程振动, 2006, 26(4): 24–31. doi: 10.3969/j.issn.1000-1301.2006.04.004 LIANG Jian-wen, JI Xiao-dong. Amplification of Rayleigh waves due to underground lined cavities[J]. Earthquake Engineering and Engineering Vibration, 2006, 26(4): 24–31. (in Chinese) doi: 10.3969/j.issn.1000-1301.2006.04.004
[21]	李伟华. 含饱和土的复杂局部场地波动散射问题的解析解和显式有限元数值模拟[D]. 北京: 北京交通大学, 2004. LI Wei-hua. Analytical Solution and Explicit Finite Element Numerical Simulation of Complex Local Field Wave Scattering Problem with Saturated Soil[D]. Beijing: Beijing Jiaotong University, 2004. (in Chinese)
[22]	MILTON A, IRENE A S. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[M]. London: Dover Publications, 1965.
[23]	BIOT M A. Theory of propagation of elastic waves in a fluid-saturated porous solid: Ⅰ low-frequency range[J]. The Journal of the Acoustical Society of America, 1956, 28(2): 168–178. doi: 10.1121/1.1908239
[24]	邓永锋, 刘松玉, 章定文, 等. 几种孔隙比与渗透系数关系的对比[J]. 西北地震学报, 2011, 33(增刊1): 64–66. https://www.cnki.com.cn/Article/CJFDTOTAL-ZBDZ2011S1015.htm DENG Yong-feng, LIU Song-yu, ZHANG Ding-yi, et al. Comparison among some relationships between permeability and void ratio[J]. Northwestern Seismological Journal, 2011, 33(S1): 64–66. (in Chinese) https://www.cnki.com.cn/Article/CJFDTOTAL-ZBDZ2011S1015.htm
[25]	吴冬. 山岭隧道洞口段地震损伤反应特性与损伤评价方法研究[D]. 成都: 西南交通大学, 2016. WU Dong. Study on Seismic Response Characteristic and Damage Evaluation Method of Tunnel Portal[D]. Chengdu: Southwest Jiaotong University, 2016. (in Chinese)

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0. 引言
1. 固结模型与基本假定
1.1 轴对称固结模型
1.2 基本假定
2. 固结方程与定解条件
2.1 固结方程
2.2 定解条件
3. 固结解析解
3.1 方程与定解条件的函数变换
3.2 固结解析解
4. 算例验证
5. 复合地基结性状分析
6. 结论

平面Rayleigh波入射下饱和土中浅埋隧道复合式衬砌的动力响应 English Version

作者简介: 范凯祥(1988—)，男，2011年本科毕业于吉林大学，博士，主要从事隧道抗减震方面的研究。E-mail: fkx_0706@my.swjtu.edu.cn

通讯作者: 申玉生: 许成顺, E-mail: sys1997@163.com

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