工程科学与技术   2017, Vol. 49 Issue (3): 68-75

Analytical Model for Stand-up Time of Rock Mass Surrounding Tunnel with Consideration of Softening Effect
WU Jianxun, REN Song, FAN Jinyang, CHEN Jie, LIU Wei, DENG Gaoling, YUAN Xi
State Key Lab. of Coal Mine Disaster Dynamics and Control, Chongqing Univ., Chongqing 400044, China
Abstract: The existing analytical solution of stand-up time neglected the softening effect of underground water,which resulted in overestimating the surrounding rock quality.This paper focused on the softening effect of water on stand-up time for the gypsum rocks with typical softening characteristics.The research method combined the softening test results of gypsum with the axisymmetric circular tunnel elastic theory.Firstly,the softening test of gypsum with different soaking time was carried out;thus,the absorption,elasticity modulus and Poisson’s ratio changing over soaking time were obtained,respectively.The integrality index and water content coefficient were introduced into the quantitative relationship between the elasticity modulus and the soaking time and that between the Poisson’s ratio and the soaking time; thus,the softening equation of mechanical parameter of rock mass was obtained.Then,based on the elastic solution of axisymmetric circular tunnel and the influence of tunnel face,the displacement formula of tunnel free face without supporting effects was developed.Then,the softening equation of mechanical parameter of rock mass was embedded into this displacement formula;thus,the displacement solution of free face considering softening effects was obtained.At last,the analytical model for stand-up time of rock mass surrounding tunnel with consideration of softening effect was established by introducing the active span and the critical displacement value into this displacement solution.This model takes into consideration the engineering features of rock mass,the mechanical performance of rock,and the softening behavior of rock.Moreover,it is able to show the relationship between the stand-up time and the active span.This model was further investigated by self-consistent analysis,which showed that the variation rule of each parameter accorded with actual situation and the self-consistency was met.This model was employed for the Lirang tunnel,which indicated that it is practical for in-site application.
Key words: stand-up time    rock mass classification    softening    tunnel stability    active span    gypsum rock

1 软化试验 1.1 试验条件及方法

1）岩样加工成直径50 mm、高100 mm的圆柱形试件（图1）。在40 ℃烘干箱内烘至质量不变，记录烘干后质量M1

2）浸泡烘干后的试件于蒸馏水中，浸水时间方案见表2，每组时间梯度下重复3个试件。

3）完成浸泡后，擦干表面，保证表面无明显水迹，记录试件质量M2。进行单轴压缩试验：采用位移加载方法，加载速率为0.1 mm/min，直至试件完全破坏。

 图1 试件 Fig. 1 Test samples

1.2 试验结果及分析

 ${w_t} = \frac{t}{{\alpha t + \beta }}$ (1)

 图2 石膏岩吸水率随浸水时间变化 Fig. 2 Water absorption of gypsum changes over soaking time

 ${E_t} = E{\text{e}^{ - mt}}$ (2)

 图3 石膏岩弹性模量随浸水时间变化 Fig. 3 Elasticity modulus of gypsum changes over soaking time

 ${\textit{υ} _{\rm{t}}} = \frac{t}{{t + n}} + \textit{υ}$ (3)

 图4 石膏岩泊松比随浸水时间变化曲线 Fig. 4 Poisson ratio of gypsum changes over soaking time

2 自稳时间解析模型

2.1 模型建立

1）实际工程中围岩变形随时间增加，且变形速率及大小受岩体力学性能的影响[17]。研究侧重隧道在时间尺度上的稳定性，因此，假设浸水软化对岩体的影响集中表现为由力学参数下降而导致的围岩变形随时间增加。

2）工程应用中，为简化问题，通常用一个影响系数概括地下水对工程的影响效果[18]。基于此，视地下水对围岩的软化具有一个整体效果，建模中暂不考虑地下水的分布特征。

3）实际施工中，通常用位移管理法判断岩体是否失稳[1920]。基于此方法的原理，认为当围岩变形量达到一个临界值时即失稳。

 $u = \frac{{({\sigma _0} - {p_0}){R^2}}}{{2Gr}}$ (4)

 $u = \frac{{\gamma HR}}{{2G}}$ (5)

 $E(t) = KE{\text{e}^{ - mst}}$ (6)
 $\textit{υ} (t) = \frac{{st}}{{t + n}} + \textit{υ}$ (7)

 $s = 1 - {k_{\rm{w}}}$ (8)

 $G(t) = \frac{{E(t)}}{{2[1 + \textit{υ} (t)]}} = \frac{{(t + n)KE{\text{e}^{ - mst}}}}{{2[st + (\textit{υ} + 1)(t + n)]}}$ (9)

 $u = \frac{{\gamma HR}}{{2G}} = \frac{{\gamma HR[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}}$ (10)

 ${u_{\rm{l}}} = u(1 - {\text{e}^{\frac{{ - al}}{R}}})\;\;\left( {l \le {l_0}} \right)$ (11)

 $\left\{ {\begin{array}{*{20}{l}}{u = \displaystyle\frac{{\gamma HR[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}}{\kern 1pt} ,l > {l_0}\text{；}}\\[10pt]{{u_{\rm{l}}} = \displaystyle\frac{{\gamma HR[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}}(1 - {\text{e}^{\frac{{ - al}}{R}}}),l \le {l_0}\;}\end{array}} \right.$ (12)

Lauffer定义了活跃跨度Las图5[7]：当lB时，Las=ll > B时，Las=B=2R。据此，把Las代入式（12）中得：

 $\left\{ {\begin{array}{*{20}{l}}{u = \displaystyle\frac{{{\rm{0}}{\rm{.5}}\gamma {L_{{\rm{as}}}}H[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}} ,l > {l_0};}\\[10pt]{{u_{\rm{l}}} = \displaystyle\frac{{{\rm{0}}{\rm{.5}}\gamma {L_{{\rm{as}}}}H[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}} (1 - {\text{e}^{\frac{{ - al}}{{0.5{L_{{\rm{as}}}}}}}}) ,B < l \le {l_0};}\\[10pt]{{u_{\rm{l}}} = \displaystyle\frac{{\gamma HR[st + (\textit{υ} + 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}}{\kern 1pt} (1 - {\text{e}^{\frac{{ - a{L_{{\rm{as}}}}}}{R}}}),{\rm{0}} < l \le {\rm{B}}}\end{array}} \right.$ (13)
 图5 Las的定义 Fig. 5 Definition of Las

 $\left\{ {\begin{array}{*{20}{l}}{{u_{{\rm{cv}}}} = \displaystyle\frac{{{\rm{0}}{\rm{.5}}\gamma {L_{{\rm{as}}}}H[st + (\textit{υ}+ 1)(t + n)]}}{{(t + n)KE{\text{e}^{ - mst}}}}{\kern 1pt} {\kern 1pt} ,l > {{B;}}}\\[10pt]\! {{u_{{\rm{cv}}}} \! = \! \displaystyle\frac{{\gamma HR[st \! + \! (\textit{υ} \! + \! 1)(t \! + \! n)]}}{{(t \! + \! n)KE{\text{e}^{ \! - \! mst}}}} (1 \! - \! {\text{e}^{\frac{{ - a{L_{{\rm{as}}}}}}{R}}}),0 \! < \! l \! \le \! B}\! \end{array}} \right.\! \!$ (14)

 $(1 - {\text{e}^{\frac{{ - a{L_{{\rm{as}}}}}}{R}}})\quad {\rm{ = }}\frac{{a{L_{{\rm{as}}}}}}{R} - \frac{1}{2}{(\frac{{a{L_{{\rm{as}}}}}}{R})^2} + \cdots$ (15)

$(1 - {\text{e}^{\frac{{ - a{L_{{\rm{as}}}}}}{R}}}) = \displaystyle\frac{{{L_{{\rm{as}}}}}}{R}$ 代入（14）中，化简得：

 $\left\{ {\begin{array}{*{20}{l}}{{L_{{\rm{as}}}} = \displaystyle\frac{{{u_{{\rm{cv}}}}(t + n)KE{\text{e}^{ - mst}}}}{{{\rm{0}}{\rm{.5}}\gamma H[st + (\textit{υ} + 1)(t + n)]}} ,l > {{B;}}}\\[10pt]{{L_{{\rm{as}}}} = \displaystyle\frac{{{u_{{\rm{cv}}}}(t + n)KE{\text{e}^{ - mst}}}}{{a\gamma H[st + (\textit{υ} + 1)(t + n)]}},0 < l \le B}\end{array}} \right.$ (16)

2.2 模型应用

1）地勘资料显示：石膏岩段上部为平均厚度120 m的灰岩，H=120 m，γ=25.5 kN/m3；石膏段岩体较破碎，完整性系数K=0.4。

2）石膏段最大单位涌水量0.028 6 L/(s·m)，地下水状态属于经常渗水级别，水压介于0.2～0.71 MPa，石膏岩的单轴抗压强度介于11～25 MPa。对照表5，确定kw=0.7，s=0.3。

3）根据文献[20]中的软弱围岩隧道稳定位移管理基准，取ucv=100 mm。

4）Eυmn的值通过软化试验获得，列于表8。隧道在施工过程中及时支护，隧洞开挖长度小于隧洞宽度，取a=0.77。

Lauffer根据活跃跨度与时间关系对围岩分级[5]，是一种较为合理的分级方法，见图8图8中A级围岩质量最好，G级围岩质量最差。是否考虑软化效应会影响分级结果：隧道活跃跨度为10 m；图6中，其对应自稳时间达到375 d；图7中，相同跨度对应的自稳时间能达到1×108 d。依据图8中的分级方法，文中模型所得结果属B级，Nguyen结果属A级。两者定量结果相差之大，以至于导致定性的差别。

1）用上下台阶法，台阶长度为8 m。

2）洞身开挖后，立即施作初期支护锚杆、钢筋网、钢架、喷射混凝土，及时封闭围岩。初期支护采用C30高性能混凝土。

3）初期支护基本稳定后及时施作二衬，二衬距掌子面70 m。

 图6 本文模型计算出的自稳时间与活跃跨度关系曲线 Fig. 6 Curve of stand-up time and active span predicted by model developed in this paper

 图7 由Nguyen模型计算出的自稳时间与活跃跨度关系曲线 Fig. 7 Curve of stand-up time and active span predicted by Nguyen model

 图8 不同等级岩体的活跃跨度与自稳时间关系（Lauffer，1958） Fig. 8 Active span versus stand-up time for different classes of rock mass (Lauffer,1958)

2.3 模型参数自洽分析

 图9 隧道埋深对自稳时间的影响 Fig. 9 Influence of tunnel depth on stand-up time

 图10 岩体完整性系数对自稳时间的影响 Fig. 10 Influence of rock-mass integrity index on stand-up time

 图11 系数s对自稳时间的影响 Fig. 11 Influence of coefficient s on stand-up time

 图12 弹性模量对自稳时间的影响 Fig. 12 Influence of elasticity modulus on stand-up time

3 讨 论

 ${W_{\rm{n}}} = w(r,\theta ,t)$ (17)

 ${E_{\rm{n}}} = g({W_{\rm{n}}})$ (18)
 ${\textit{υ} _{\rm{n}}} = h({W_{\rm{n}}})$ (19)

 $E(t) = \frac{{\iint\limits_D {g[w(r,\theta ,t)]} r{\rm{d}}r{\rm{d}}\theta }}{D}$ (20)
 $\textit{υ} (t) = \frac{{\iint\limits_D {h[w(r,\theta ,t)]} r{\rm{d}}r{\rm{d}}\theta }}{D}$ (21)
 $D = \left\{ {r,\theta |{R_{\rm{0}}} < r < {R_{\rm{m}}},0 \le \theta \le 2{\rm{\text{π}}}} \right\}$ (22)

 图13 轴对称圆形隧道模型 Fig. 13 Model of axisymmetric circular tunnel

4 结 论

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