工程科学与技术   2018, Vol. 50 Issue (3): 233-239

1. 石家庄铁道大学 土木工程学院，河北 石家庄 050043;
2. 中交四航局第二工程有限公司，广东 广州 510230

Evaluation Method of Earthquake-induced Bending Collapse of the Upside-down Dangerous Rock-Mass Based on Cantilever Beam Theory
YUAN Wei1, ZHENG Chuanchang2, WANG Wei1, LI Zonghong1, LI Jiaxin1, WEN Lei1, CHANG Jiangfang1
1. School of Civil Eng., Shijiazhuang Tiedao Univ., Shijiazhuang 050043, China;
2. The Second Engineering Company of CCCC Fourth Harbor Engineering Co. Ltd, Guangzhou 510230, China
Abstract: The cantilever beam type is a common form of the upside-down dangerous rock-mass,and it is easily to collapse under earthquake action.Thus,the study on the evaluation method of earthquake-induced bending collapse of the upside-down dangerous rock-mass have great significance on evaluating its security degree.On basis of the calculation theory of the cantilever beam,firstly,the calculating formula for the maximum tensile stress of the cantilever beam type dangerous rock-mass with arbitrary shape under the action of seismic loading and its own gravity has been proposed,and the time-history curve of the maximum tensile stress has been established.At the same time,the tensile failure criterion is considered as the criterion of crack propagation,combined with the relationship between the elastic strain energy and fracture propagation energy,and an approach for calculating the depth of crack has been established to study the crack depth along with the earthquake time.Secondly,according to the shapes of dangerous rocks in real engineering,they are simplified into two special forms:rectangular parallelepiped and cylinder,and has proposed a calculation formula for the critical depth of crack to define the safety factor of dangerous rock under the action of earthquake.At last,a dangerous rock-mass located at the entrance of a tunnel is taken as an example to illustrate the application of the proposed method in this paper.The results show that,under the action of 8.0 magnitude earthquake,the total depth of crack induced by earthquake of this dangerous rock-mass at 14 s is 0.1759 m,which is greater than the critical depth of crack (0.167 m),implying that this dangerous rock-mass would collapse at 14 s.
Key words: upside-down dangerous rock-mass    earthquake-induced collapse    crack growth    cantilever beam theory    maximum tensile-stress criterion

1 立论依据

 ${F_{\rm A}}(t) = m \cdot {a_x}(t) = m \cdot g \cdot \delta (t)$ (1)

 $\begin{array}{*{20}{c}} {{M_{\rm P}} = {F_{\rm G}} \cdot {l_x}}\text{，}&{}&{{M_{\rm J}} = {F_{\rm A}}} \!\!\!\end{array}\cdot \sqrt {l_Y^2 + l_{\textit{z}}^2}$ (2)

 ${M_{\rm T}} = \sqrt {M_{\rm P}^2 + M_{\rm J}^2 + 2{M_{\rm P}} \cdot {M_{\rm J}} \cdot \cos\, \theta }$ (3)

 $\cos \,\theta = \frac{{{l_Y}}}{{\sqrt {l_Y^2 + l_{\textit{z}}^2} }}$ (4)

$O$ 点垂直 $OT$ 作一条直线，假设该直线为 $\varepsilon$ 轴， ${\rm I} - {\rm I}$ 界面绕 $\varepsilon$ 轴的截面抗弯模量为 ${W_\varepsilon }$ ，截面面积为 $S$ ，则 $T$ 点的拉应力 ${\sigma _{\rm T}}$ 可表示如下：

 ${\sigma _{\rm T}} = \frac{{{M_{\rm T}}}}{{{W_\varepsilon }}} + \frac{{{F_{\rm A}}}}{S}$ (5)

 ${\sigma _{\rm T}}(t) = mg(\frac{{\sqrt {l_x^2 + (l_Y^2 + l_{\textit{z}}^2) \cdot {\delta ^2}(t) + 2{l_x} \cdot {l_Y} \cdot \delta (t)} }}{{{W_\varepsilon }}} + \frac{{\delta (t)}}{S})$ (6)

 图1 危岩体受力状态 Fig. 1 Stress state of dangerous rock-mass

 图2 Ⅰ–Ⅰ断面示意图 Fig. 2 Schematic diagram of Ⅰ–Ⅰ section

 图3 Ⅰ–Ⅰ 断面裂缝扩展 Fig. 3 Crack propagation of Ⅰ–Ⅰ section

2 两种特殊形状的倒悬危岩体

 图4 地震波水平加速度解析 Fig. 4 Horizontal component of seismic wave

2.1 等截面长方体危岩体

 ${\sigma _{\rm T}}(t) = \frac{{mg}}{{bh}}[\frac{{6l}}{h} + \delta (t)]$ (7)

 图5 等截面长方体危岩体 Fig. 5  Dangerous rock-mass depicted as uniform section cuboid

 \left\{ \begin{aligned}& W(t) = \frac{1}{6}b \cdot {[1 - \alpha (t)]^2} \cdot {h^2},\\& S(t) = b \cdot [1 - \alpha (t)] \cdot h\end{aligned} \right. (8)

 ${\sigma _{t\max }}(t) = \frac{{mg}}{{[1 - \alpha (t)] \cdot bh}}[\frac{{6l}}{{[1 - \alpha (t)] \cdot h}} + \delta (t)]$ (9)

 ${a_{\rm c}}(t) = \frac{{{f_t} \cdot [1 - \alpha (t)] \cdot bh}}{m} - \frac{{6l}}{{[1 - \alpha (t)] \cdot h}} \cdot g$ (10)

 $\alpha (t) \cdot h = h - \sqrt {\frac{{6mgl}}{{{f_t}b}}}$ (11)

 图6 长方形截面应力分布图 Fig. 6  Stress distribution of rectangle cross profile

$t$ 时刻，危岩体自身重力使重心 $G$ 点的下移值为 ${Y_{\rm G}}$ ，根据悬臂梁计算理论可知：

 ${Y_{\rm G}} = \frac{{5mg{l^3}}}{{Eb \cdot {{[1 - \alpha (t)]}^3} \cdot {h^3}}}$ (12)

 ${X_{\rm G}} = \frac{{mg \cdot \delta (t) \cdot (2l)}}{{E \cdot b \cdot [1 - \alpha (t)] \cdot h}}$ (13)

 $\varOmega = mg[{Y_{\rm G}} + \delta (t) \cdot {X_{\rm G}}]$ (14)

 $\gamma \cdot b \cdot \Delta h = \varOmega$ (15)

 $\Delta h = \frac{\varOmega }{{\gamma \cdot b}}$ (16)

 $\Delta h = \frac{{{m^2}{g^2}l}}{{E\gamma h{b^2}}}(\frac{{5{l^2}}}{{{{[1 - \alpha (t)]}^3} \cdot {h^2}}} + \frac{{2\delta {{(t)}^2}}}{{[1 - \alpha (t)]}})$ (17)

 $\Delta H = \sum {(\Delta {h_i})}$ (18)

$\Delta H$ 大于临界深度 $\Delta {h_{{\rm cri}}}$ ，则表示危岩体在地震作用下会发生崩塌失稳，反之，则表示危岩体在地震作用下仍然处于安全状态。因此，等截面长方体形式的危岩体在地震作用下的安全系数 $FOS$ 可定义为临界深度与裂缝总深度的比值，即：

 $FOS = \frac{{\Delta {h_{{\rm cri}}}}}{{\Delta H}}$ (19)

$FOS$ 大于1.0表示危岩体在地震作用下仍处于安全状态，小于1.0表示危岩体会发生崩塌失稳，等于1.0则表示危岩体处于临界状态。

2.2 等截面圆柱体危岩体

 ${\sigma _{\rm T}}(t) = \frac{{mg}}{{{\text{π}} {r^2}}}[\frac{{4l}}{r} + \delta (t)]$ (20)

 图7 等截面圆柱体危岩体 Fig. 7  Dangerous rock-mass depicted as uniform section cylinder

 ${I_1} = \frac{{{r^4}}}{4} \cdot \psi [\alpha (t)]$ (21)

 $\psi [\alpha ] \!=\! 2(1 \!-\! 2\alpha )(1 \!-\! 8\alpha \!+\! 8{\alpha ^2})\sqrt {\alpha (1 \!-\! \alpha )} \!+\! \arcsin\,\theta$ (22)

 $\theta = \arccos (1 - 2\alpha )$ (23)

 ${I_2} = \frac{{{\text{π}} {r^4}}}{4} - \frac{{{r^4}}}{4} \cdot \psi [\alpha (t)] = \frac{{{r^4}}}{4}[{\text{π}} - \psi (\alpha )]$ (24)

 ${S_1} = {r^2}(\theta - \sin \;\theta \cdot \cos\; \theta )$ (25)

 ${S_2} = {r^2}({\text{π}} - \theta + \sin\; \theta \cdot \cos\; \theta )$ (26)

 ${\sigma _{t\max }} = \frac{{mgl}}{{{I_2}}} \cdot [r - 2\alpha (t) \cdot r] + \frac{{m \cdot \delta (t) \cdot g}}{{{S_2}}}$ (27)

 ${\sigma _{t\max }} = \frac{{mg}}{{{r^2}}}[\frac{{4l(1 - 2\alpha )}}{{r({\text{π}} - \psi (\alpha ))}} + \frac{{\delta (t)}}{{{\text{π}} - \theta + \sin \;\theta \cdot \cos\; \theta }}]$ (28)

 图8 圆形截面应力分布图 Fig. 8  Stress distribution of circle cross profile

 ${a_{\rm c}}(t) = [\frac{{{f_t} \cdot {r^2}}}{{mg}} - \frac{{4l(1 - 2\alpha )}}{{r({\text{π}} - \psi (\alpha ))}}] \cdot ({\text{π}} - \theta + \sin \;\theta \cdot \cos \;\theta ) \cdot g$ (29)

${a_{\rm c}}(t) = 0$ ，可得到关于 $\alpha (t)$ 的超越方程，假设 $\alpha (t)$ ${\alpha _{\rm c}}$ 时，该超越方程成立，因此，把 ${\alpha _{\rm c}}$ 代入式（23）和式（25）便可得到危岩体裂缝的临界面积 ${S_{{\rm cri}}}$ ，当裂缝面积（区域1的面积）大于该临界值时，危岩体在自重作用下就会发生崩塌失稳。 ${S_{{\rm cri}}}$ 可表示如下：

 ${S_{{\rm cri}}} = {r^2}({\theta _{{\rm cri}}} - \sin {\theta _{{\rm cri}}} \cdot \cos {\theta _{{\rm cri}}})$ (30)

 $\left\{ {\begin{array}{*{20}{l}} {{X_{\rm G}} = \displaystyle\frac{{mg \cdot \delta (t) \cdot (2l)}}{{E \cdot {r^2}({\text{π}} - \theta + \sin\,\theta \cdot \cos\,\theta )}}} \text{，}\\ {{Y_{\rm G}} =\displaystyle \frac{{5mg{l^3}}}{{3E{r^4}[{\text{π}} - \psi (\alpha )]}}} \end{array}} \right.$ (31)

 $\Delta {S_i} = \frac{\varOmega }{\gamma }$ (32)

 $\Delta S = \sum {(\Delta {S_i})}$ (33)

 $FOS = \frac{{{S_{{\rm cri}}}}}{{\Delta S}}$ (34)

$FOS$ 的物理意义与第2.1节相同。

3 工程应用

 图9 危岩体示意图 Fig. 9 Schematic diagram of dangerous rock-mass

 图10 水平加速度时程曲线 Fig. 10 Time history curve of horizontal accelerated speed

$\alpha = 0.012\,5$ 代入式（10）计算 $t = 11\;{\rm s}$ 时刻的临界加速度 ${a_{\rm c}} = 0.086g$ 。地震波水平加速度在11～14 s之间皆小于 $0.086g$ ，说明在此时间段内，裂缝不会继续扩展。当 $t$ =14 s时，地震波水平加速度为 $0.1g$ ，超过了临界加速度 $0.086g$ ，因此，在该时刻裂缝会继续扩展，把 $\alpha = 0.012\,5$ $\delta = 0.1$ 等代入式（17）得到此时间点裂缝扩展量 $\Delta {h_2} = 0.088\,2\,{\rm m}$ 。两次裂缝扩展的总深度 $\Delta H$ 为0.175 9 m，超过了该危岩体的临界深度 $0.167\,{\rm m}$ ，因此，危岩体在地震第14 s时发生崩塌失稳。

4 结　论

1）地震水平加速度超过临界加速度时，其裂缝才能往深部扩展。临界加速度不是一个定值，不仅与岩体抗拉强度、危岩体尺寸等存在关联，亦与当前裂缝深度有关。当前的裂缝深度越大，则临界加速度越小。因此，随着裂缝的继续扩展，岩体裂缝更容易往深部延伸。

2）危岩体裂缝存在一个临界深度（或临界面积），即当裂缝的实际深度（或面积）等于临界深度（或临界面积）时，危岩体仅在自身重力作用下崩塌失稳的临界状态。

3）裂缝的单次扩展量不仅与危岩体的弹性模量、几何尺寸、单位面积裂缝扩展所克服的岩体表面能等相关，亦与地震水平加速度和当前裂缝深度相关。地震水平加速度越大，单次裂缝扩展量越大；当前裂缝深度越大，单次裂缝扩展量亦越大。

4）当裂缝扩展的累积量超过临界深度（或临界面积）时，危岩体就会发生崩塌失稳。因此，临界深度（或临界面积）可作为危岩体崩塌失稳的判据。

5）危岩体的安全系数可由预测计算的裂缝深度（或面积）除以临界深度（或临界面积）表示。当安全系数大于1.0时，表示危岩体不会在地震作用下发生崩塌失稳；当安全系数小于1.0时，则表示危岩体会发生地震崩塌失稳；当安全系数等于1.0时，则表示危岩体地震后处于临界状态。

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