Rock Mechanics and Engineering: Prediction and Control of Landslides and Geological Disasters 012822424X, 9780128224243

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Rock Mechanics and Engineering

Rock Mechanics and Engineering Prediction and Control of Landslides and Geological Disasters Helin Fu Wei Chen Jiajun Fu

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2021 Central South University Press. Published by Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-822424-3

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Acknowledgments The studies in this book were supported by the National Natural Science Foundation of China, and as a project of the Railway Corporation (the former Ministry of Railway) in China, the Western Project of the Management Center of the Ministry of Communications in China, the Department of Communications of Guizhou Province in China, the Department of Communications of Hunan Province in China, the Science and Technology Department of Guizhou Province in China, the Committee of the Guangzhou Rail in China, and the Central South University in China. In the process of writing this book, Mr. Jiajun Fu, Mr. Hui Dong, Mr. Wei Chen, Mr. Zhong Zhou, Mr. Ning Zhou, Ms. Qingyan Tian, Ms. Fen Chen, Ms. Cuisong Bu, Mr. Weiguo Zhang, Mr. Jiangfeng Guo, Mr. Chunlong Nie, Mr. Xin Tan, Mr. Yunsi Liu, Mr. Weizhi Hou, Mr. Long Chen, Mr. Qibing Huang, and other team members have contributed a great deal of model verification and other work. The team members as mentioned have made other similar contributions to this book. Prof. Liang Li, Dr. Qiang Luo, Dr. Hanua Tan, Mr. Wang Xue Long, Prof. Jianbing Lv, Ms. Yingwei Liu and other friends, including Wuzhuang Luo, provided their photos and suggestions for use in this book for demonstration. Much appreciation also goes to the School of Civil Engineering Central South University, National Engineering Laboratory of high-speed railway construction. The author expresses his heartfelt thanks to them.

xi

Foreword Under the action of various geological forces or abnormal changes in the geological environment, geological disasters often occur. As the main type of geological hazards, collapses, slips, and debris flows are characterized by sudden occurrence, wide distribution, and certain concealment, these can cause major damage to traffic infrastructure. Geological hazards pose a potential threat to the construction and operation of the high-speed railway infrastructure and restrict the sustainable and stable development of high-speed railway transportation. Although scientific and technological progress has brought improvements to the level of high-speed railway survey, design, construction and operations management, technology still cannot fully adapt to the geological environment of complex areas of railway construction. For example, the Sichuan-Tibet railway and the Yunnan-Tibet railway, which will be built soon, will have more intensive environmental transformations. In order to ensure the smooth completion of these projects, it is necessary to study the environmental and physical laws of geological hazards, the mutual feedback effects between structures and the geological environment, the grading and prediction of geological hazards, and the digital geological hazard prediction systems. Strengthening the monitoring and prediction of geological hazards in order to reduce and prevent geological hazards of high-speed railways are key directions for railway construction. A high-speed railway has its own characteristics, such as a complete maintenance system and long railway line; in addition to roadbed slopes, bridge foundation slopes, tunnel entrance and exit slopes, the geological body is not only subjected to static loads, but also to cyclic train dynamic loads. The damage of the geological body is not only closely related to rock and soil strength and hydrology but is also affected by fatigue load. Therefore, it is complicated to study the physical laws of disaster-causing activities as well as the monitoring and prediction of geological hazards on high-speed railways; it is necessary to study the regularity of disaster-causing activities, the monitoring and forecasting technology of rock (soil) slope collapse, and landslide and debris flow disasters induced by rainfall and human activities on high-speed railways. In the experience of departments of the railway, water conservancy, mining, land and resources, urban construction, etc., based on the characteristics of the high-speed railway project, research on monitoring and forecasting technology suitable for the characteristics of high-speed railway geological hazards has become a research hotspot.

xiii

Foreword This book mainly studies the monitoring and forecasting of geological hazards in the survey and design stage, construction stage, and operations stage of high-speed railways, optimizes the technology of geological hazards remediation, ensures that the design scheme in the survey and design stage is economical and reasonable and that the prevention and control measures in the construction stage are safe and effective, and finally achieves the goal of traffic safety, which has far-reaching significance for railway construction in China. The main content and innovations of this book are as follows: 1. The mechanism of rainfall-type landslide is studied, and the influence of triggering factors on the mechanism of landslide hazards and movement law is revealed through the study of residual slope sediment. This provides a reliable theoretical basis for the time, space, and strength prediction of landslides, and is of great significance for guiding engineering practice. 2. From laboratory tests of landslides, it is found through analysis that the occurrence of collapse of a soil slope is affected by three factors: slope, soil quality, and rainfall, and the interaction relationship among the three factors is revealed. 3. Through the study of the collapse process of a rock slope, the relationship between energy rate (frequency of major events) and collapse probability is obtained. Energy rate is related to the state of rock mass: when energy rate is high, the frequency of major events is high, and the probability of collapse is high. This book puts forward how to use the curve of acoustic emission events and energy rate of rock mass structural plane with time to guide the monitoring work of acoustic emission in a collapse site, and provides a strong basis for collapse prediction. 4. The principle of slope monitoring is established: the slope in a stable state (K > 1.25) doesn’t need to be monitored; the slope in a relatively stable state (1.05 < K < 1.25) should be inspected on the surface; the slope in a potentially unstable state (0.95 < K > > > > > > > > > > > > > > > > > > > > > > > > < Namely:

 1 Wj, i + Wi, j 2 εx ¼ 

∂Wx ∂x

εy ¼ 

∂Wy ∂y

εz ¼ 

∂Wz ∂z



∂Wx ∂Wy > > > γ xy ¼  + > > ∂y ∂x > > > >

> > ∂Wz ∂Wy > > > γ yz ¼  + > > ∂y ∂z > > > >

> > > ∂Wz ∂Wx > > + : γ yz ¼  ∂x ∂z

(2.25)

46

Chapter 2

③

Stress balance equation.

If it is calculated that the head distribution in the slope is H (x, y, z) in some cases, the pore water pressure (seepage pressure) p distribution on a certain action surface can be calculated as: p ¼ γ ðH  zÞ

(2.26)

It is also possible to calculate the distribution of the seepage volume force f in the seepage region, and it is known from the hydraulics principle that the seepage volume force is proportional to the hydraulic gradient: 8 9 ∂H > > > > γ w > > 8 9 > > ∂x 8 9 > > > > f x> > > > J γ < = < = < = x w ∂H γw fy ¼  γ J (2.27) ¼ y ∂y : w ; > > > : > ; > > γ J > >

> w z > fz > > ∂H > > > > : γw ; 1 ∂z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2.28) f ¼ fx2 + fy2 + fz2 In the formula: f is the amount of volume force generated by seepage; γ w is the weight of water; fx, fy, fz is the component of the seepage volume force in the x, y, and z directions; and Jx, Jy, Jz is the hydraulic gradient of the unit in the x, y, and z directions. For the two-dimensional seepage problem, the calculation of the seepage volume force takes the first and third terms shown in Eq. (2.24). The stress balance equation that introduces the seepage volume force into the solid skeleton is: σ ij, j + fi + Ti ¼ 0

(2.29)

In the formula: fi is the seepage volume force and Ti is the volume force. Simultaneous formula (2.12) and formula (2.29):   σ 0ij, j + pδij , j + fi + Ti ¼ 0

(2.30)

Substituting the formula (2.23) into the preceding equation, we can get the equilibrium equation with the basic unknown Wz, Wy, Wz, n, p as a variable: 8 ∂εV ∂p ∂fx > > + μr2 Wx + + ¼0 > ðλ + μÞ > > ∂x ∂x ∂x > > < ∂εV ∂p ∂fy ðλ + μÞ + μr2 Wy + + ¼0 (2.31) > ∂y ∂y ∂y > > > > > ∂ε ∂p ∂fy > : ðλ + μÞ V + μr2 Wz + + + Tz0 ¼ 0 ∂z ∂z ∂z

Similar simulation tests of landslides 47 2 2 ∂ ∂ ∂ z In the formula: Tz0 ¼ ∂F ∂z ¼ ½ð1  nÞρs + nρw   g; r is the Laplacian operator, r ¼ ∂x2 + ∂y2 + ∂z2 . 2

2

2

For an isotropic elastic body: λ¼

Eν E , G¼μ¼ ð1 + νÞð1  2νÞ 2ð1 + νÞ

In the formula: E, v is the elastic modulus and Poisson’s ratio of solid skeleton, respectively. G λ + μ ¼ ð12ν Þ; substituting into Eq. (2.31): 8 G ∂ε ∂p ∂fx V > + Gr2 Wx + + ¼0 > > > ð1  2νÞ ∂x ∂x ∂x > > > < G ∂ε ∂p ∂fy V + Gr2 Wy + + ¼0 > ð1  2νÞ ∂y ∂y ∂y > > > > > G ∂εV ∂p ∂fy > : + + Gr2 Wz + + Tz0 ¼ 0 ð1  2νÞ ∂z ∂z ∂z

(2.32)

Eq. (2.32) is the stress field equation under the influence of a seepage field in the flow-solid coupling analysis of a compressible earth-rock mixture. (4) Seepage field equation under the influence of stress field. As stated earlier, the seepage force generated by the seepage field acts on the solid skeleton and will change the stress field and displacement field of the earth-rock mixture. The variation of stress state of the earth-rock mixture will change its seepage property, that is, the variation of stress field and displacement field will change the porosity ratio and porosity of the earth-rock mixture. At the same time, the permeability coefficient of the earth-rock mixture is closely related to the distribution of the pores. The variation of the pore ratio and porosity will inevitably cause a change in the permeability of the medium, such as the permeability coefficient. The seepage field of the earth-rock mixture is affected and will change accordingly. Therefore the variation of the seepage property of the earth-rock mixture will change the distribution law of seepage flow and will also change the permeability of the earth-rock mixture. The essence of the influence of stress field on the seepage field is that the stress field changes the distribution of pores in the earth-rock mixture, thereby changing the permeability characteristics of the earth-rock mixture. For the fluid-solid coupling problem of a saturated earth-rock mixture, the basic equations of analysis and research must comprise, in addition to the force balance equation, constitutive equation, and geometric equation under the influence of a seepage field: ① the effective stress principle of the fluid and solid skeleton sharing load; ② the pore fluid balance equation; and ③ the pore fluid continuity equation.

48

Chapter 2

Since the seepage occurs in the deformable earth-rock mixture, not only does the fluid have a certain percolation velocity, but the skeleton particles also have a certain moving speed, so the velocity of the fluid particle is: !

!

!

Vf ¼ Vr + Vs

(2.33)

!

! Vs

∂W ¼ ∂t

! Vr

V jD ¼ nSj

!

!

!

In the formula: V f is the absolute speed of fluid motion; V s is the absolute speed of the !

movement of the skeleton particles; V r is the velocity of the fluid relative to the framework !

particles; j is the fluid phase; Sj is the saturation of earth-rock mixture; and V jD is the phase j fluid velocity. According to Darcy’s law: !

V jD ¼ 

 Kj  r pj  ρj gH μj

(2.34)

In the formula: Kj is the permeation coefficient of phase j fluid; μj is the absolute viscosity of phase j fluid; and H is the head height. Because it is assumed to be a single-phase fluid (saturated earth-rock mixture), Sj ¼ 1, and then: !

Vr ¼ 

 K  r p  ρf gH nμ

(2.35)

In the formula: K is the earth-rock mixture permeability coefficient and ρf is the fluid density. The continuity equation of the earth-rock mixture skeleton is: h ! i ∂½ρs ð1  nÞ r  ρs ð1  nÞV s + ¼0 ∂t

(2.36)

In the formula: ρs is the solid skeleton density. The continuity equation for pore fluids (regardless of source and sink terms) is: r




!  ρf nV f +

  ∂ ρf n ¼0 ∂t

(2.37)

Similar simulation tests of landslides 49 Simplify the preceding two formulas to get: !

ρs ð1  nÞr  V s + ð1  nÞ !

!

ρf nr  V r + ρf nr  V r + n

∂ρs ∂n  ρs ¼ 0 ∂t ∂t

(2.38)

∂ρf ∂n + ρf ¼ 0 ∂t ∂t

(2.39)

Divide the preceding two sides by ρs, ρf and add them to obtain the flow-solid coupling seepage field continuity equation of the compressible earth-rock mixture: !

!

nr  V r + r  V s +

ð1  nÞ ∂ρs n ∂ρf + ¼0 ρs ∂t ρf ∂t

(2.40)

In general, the equation of state for a fluid under isothermal conditions can be expressed as: ρf ¼ ρ0 e½ðpp0 Þ=Ef 

(2.41)

In the formula: Ef is the volumetric elastic compressive modulus of the pore fluid. 1 ∂ρf 1 ∂p ¼ ρf ∂t Ef ∂t

(2.42)

1 ∂ρs 1 ∂p ¼ ρs ∂t Es ∂t

(2.43)

Similarly, for solid skeletons:

In the formula: Es is the volume elastic compressive modulus of the solid particles of the earth-rock mixture skeleton.
! !! ∂ r  W ! ∂W ∂εV ¼ And r  V s ¼ r  ; substitute the formulas (2.42), (2.43), and (2.35) ¼ ∂t ∂t ∂t into the formula (2.40) to obtain the flow-solid coupling seepage field differential equation of the compressible earth-rock mixture:

   K ∂εV 1  n n ∂p + rp  ρf grH + ¼0 (2.44) + r  μ Es Ef ∂t ∂t The permeability coefficient K of the earth-rock mixture in the preceding formula is not constant and is specifically described by the formula (2.22). Eqs. (2.19), (2.32), (2.44) form a mathematical model describing the flow-solid coupling seepage of saturated earth-rock mixtures. The model has a total of five equations, the solution variables are Wx, Wv, Wz, n, p, a total of five, and the control equations are closed. (5) Definite condition. For the previous control equations, appropriate boundary conditions and initial conditions should be added to form a solution problem. Since the model contains the seepage field

50

Chapter 2

equation and the stress field equation, it is necessary to provide respective corresponding boundary conditions. For the stress field equation, the stress boundary and the displacement boundary should be provided according to the basic knowledge of elasto-plastic mechanics; for the seepage field equation, the seepage boundary should be provided. The entire flow state or flow control conditions (such as pore pressure distribution, etc.) in the seepage field at the beginning of the test or calculation are referred to as initial conditions. Boundary conditions and initial conditions are collectively referred to as definite conditions. Determining conditions play a decisive role in the flow process and are usually determined by field observations or experiments. The problem is finding a function such as the water head function h (x, y, z) so that it satisfies the differential equation and the condition of the solution, which is called the solution problem. Definite conditions and differential equations constitute a mathematical model for solving seepage and stress fields. ①

Seepage field definite condition. a. Boundary conditions.

The boundary conditions can be divided into the conditions of the geometric boundary traits of the flow field and the dominance of the boundary. According to the mathematical model describing flow, there are usually three kinds of boundary conditions: A. First type of boundary condition (known water head boundary conditions) This refers to the location potential function or head distribution on the boundary, also known as the head boundary condition, which is the most common case. For example, if the seepage boundary in the submerged water in the stable seepage field is the equipotential surface or the equal water head surface, that is, h ¼ constant, the free seepage section and the free surface boundary of the slope, the water head should be equal to the position height, that is, h* ¼z; water head values of drainage ditch and pipe wells are known, etc., all of which belong to this type of boundary condition. For unsteady seepage, the boundary conditions are related to time t, and the change of boundary conditions during the whole process can be expressed by the following formula: HjΓ1 ¼ H1 ðx, y, z, tÞ ðx, y, zÞ Γ1

(2.45)

In the formula: Γ1 is a collection of known boundaries of the water head in the area. B. Second type of boundary condition (flow boundary condition) The normal derivative of a potential function or water head on a boundary is known or can be expressed by a definite number. The flow boundary condition can be expressed mathematically as: ∂h ¼ vn =k ¼ f2 ðx, y, z, tÞ (2.46) ∂n Γ2

Similar simulation tests of landslides 51 When considering the anisotropy of soil, it can also be written as:


 ∂H ∂H ∂H ! ! ! kx cos n , x + ky cos n , y + kz cos n , z ¼ q ðx, y, zÞ Γ2 ∂x ∂y ∂z

(2.47)

In the formula: q is the Boundary flow; Γ2 is a set of boundaries whose normal velocity is known


 ! ! ! in the region; and cos n , x , cos n , y , cos n , z is the direction cosine of the outside normal direction of the boundary. For steady flow, flow on the flow replenishment or outflow boundary q ¼ constant or the corresponding ∂H ∂n ¼ constant. The impervious layer, the symmetrical flow surface, and the free surface of the stable seepage are all boundary conditions of this type, namely ∂H ∂n ¼ 0. In the process of unsteady seepage, the variable free surface boundary should meet the flow replenishment relationship of the second type of boundary condition, in addition to the first type of condition h∗ ¼ z. As shown in Fig. 2.42, the free surface falls to a certain position after passing Δt time, and the microbody q  dΓ  dt is taken. If the outer normal direction is positive, the single-wide flow rate from the boundary can be considered as the free surface decreases: q¼μ

∂h∗ cos θ ∂t

(2.48)

When there is rainfall infiltration on the free surface of the seepage, the preceding formula becomes: q¼μ

∂h∗ cosθ  w ∂t

(2.49)

In the formula: h∗ is the water head on the free boundary; w is the infiltration amount; μ is the water supply within the range of free surface variation; and θ is the angle between the free surface normal and the lead line.

q

q

n dG

0

t t+dt h*(t)

z

x

Fig. 2.42 Schematic diagram of flow replenishment boundary when free surface falls.

52

Chapter 2

∗ ∂h Because of q ¼ vn ¼ k ∂h ∂n ¼ k ∂z cosθ, the formula (2.48) can be changed to:

μ ∂h∗ ∂h∗ ¼ ∂z k ∂t

(2.50)

C. Third type of boundary condition (mixed boundary condition) Mixed boundary condition means that the internal and external head difference of the aquifer boundary maintains a certain linear relationship with the exchanged flow rate: h+a

∂h ¼β ∂n

(2.51)

In the formula: a, β are the known number of points at the boundary. The calculation needs to be iterated to meet the known relationship between the heads h and ∂h/∂n. When studying the movement of groundwater in large areas, such boundary conditions will be encountered when the water content at the boundary of the aquifer is affected by the change of water level. Such boundary conditions are also included in the clogging of the riverbed surface, the cross-flow of the weak permeable layer in the well wall, and the aquifer. The previous three types of boundary conditions are mathematically referred to as Dirichlet conditions, Neumann conditions, and Fourier conditions. Taking the simplified model of the earth-rock mixture slope field test (Fig. 2.43) as an example, the boundary conditions are as follows: 1-2, 2-3, 5-6 are equal water heads, h ¼ pγ + z ¼ constant,

the first type of boundary condition; 3-4 is the free seepage section, h ¼ z, the first type of boundary condition; 6-7, 7-1 are the impermeable layers, the second type of boundary condition, ∂h ∂n ¼ 0; 4-5 is the groundwater level, the position of the line (wetting line) is the

Fig. 2.43 Slope boundary condition diagram.

Similar simulation tests of landslides 53 p second type of boundary condition in stable seepage, ∂h ∂n ¼ 0, and because of r ¼ 0, it has to meet the condition h∗ ¼z. In the case of unsteady seepage, the free surface is not the second type of boundary condition, and it should meet the flow replenishment boundary condition of Eq. (2.48).

b. Initial conditions. Initial condition usually refers to the original distribution of the head of the seepage field at the initial moment or from a certain moment. The initial condition is the first type of boundary condition, that is, the head distribution of the seepage field, which dominates the entire flow field at the start time t ¼ 0. Therefore in the calculation or test of unsteady seepage, the head distribution of the stable seepage field at the beginning can be obtained as the initial condition (the flow field at the beginning is usually a stable seepage field); any seepage state can also be taken as initial conditions. Only in special cases will the initial conditions be the second and third type of boundary conditions. Hðx, y, z, t0 Þ ¼ H0 ðx, y, z, t0 Þ

(2.52)

In the formula: H0—The known water head. ②

Stress field setting condition.

The stress model boundary conditions are stress boundary conditions and displacement boundary conditions. a. Displacement boundary conditions: ! W

!

boundary

¼ W1

(2.53)

!

In the formula: W 1 is the displacement on the boundary. b. Stress boundary conditions: ! σ 0ij  n j

boundary

¼ Ti

(2.54)

In the formula: Ti is the force on the stress boundary. (6) Finite element algorithm for fluid-solid coupling equation. Formulas (2.19), (2.32), and (2.44) and formula conditions (2.45)–(2.54) constitute a complete single-phase fluid flow-solid coupling seepage mathematical model. In theory, the exact solution of the coupling analysis of the seepage field and the stress field of the earth-rock mixture slope with the pore pressure distribution p(x,y,z) and the stress field distribution σ ij(x,y,z) of the seepage

54

Chapter 2

field can be obtained by solving the previously mentioned various equations. But in fact, for the seepage field or stress field with complicated boundary conditions in engineering practice, it is impossible to solve the analytical solution of the seepage field or the stress field separately. The joint solution is even more difficult. Therefore, coupling analysis can be performed only by numerical solution. For this mathematical model, we can use the finite difference, finite element, and other numerical methods to find the discrete solution, so as to solve the distribution of pore pressure ( p), porosity (n), and displacement field (Wi) of the seepage field. ①

Matrix representation of coupled equations.

The constitutive equations (2.23), geometric equations (2.25), equilibrium equations (2.29) and effective stress principle equations (2.12) of linear elastic isotropic earth-rock mixtures are expressed in matrix form. The equilibrium matrix equation (2.32) expressed by displacement and pore pressure is thus derived as follows: ½∂T ½D½∂½W   ½∂T ½MP  ½G ¼ 0

(2.55)

In the formula: [D] is the elastic matrix; [W] is the displacement matrix; [M] is the pore water pressure distribution matrix; and [G] is a matrix of constant terms (including volume and seepage volume forces caused by gravity). Similarly, the continuity matrix equation of the seepage matrix expressed by displacement and pore pressure is obtained as follows: ∂ ½MT ½∂½K ½MP  ½MT ½∂½W  ¼ 0 ∂t

(2.56)

In the formula: [K] is the permeability coefficient matrix. Eqs. (2.55), (2.56) are differential operator (matrix) equations for coupled model equations. ②

Spatial dispersion of coupled equations.

e pe According to the basic principle of the FEM, we need to establish the approximate solution W, of Eqs. (2.55), (2.56), and discretize with the 8-node isoparametric element, and set the soil region Ω to be divided into a finite number of subregions Ω1, Ω2…Ωn (Ω ¼ ΣΩe). The boundary surface of the subregion Ωe is denoted by Γ e. According to the Galerkin variational principle, Eq. (2.54) becomes: ððð ððð
h i  e  ½MPe Ni dxdydz ¼ ½∂T ½D½∂ W ½GNi dxdydz (2.57) Ωe

Ωe

Eq. (2.57) is embodied, and the spatial discrete algebraic equations of the equilibrium equation expressed by the unit node displacement and the pore pressure are obtained:

Similar simulation tests of landslides 55

e ½Ke fδge + ½Kc fpge ¼ Rf

(2.58)

ÐÐÐ

T In the formula: ½Ke  ¼ Ωe ½B ½D½Bdxdydz is a unit stiffness matrix;

 ÐÐÐ T ½Kc  ¼ Ωe ½B ½M N dxdydz is the unit coupling matrix; and

e Ð Ð Ð ÐÐ e T T Rf ¼ Ωe ½G  dxdydz + Γe ½N fFgds is the unit equivalent node load matrix.

Similarly, the seepage continuity equation (2.56) Galerkin variational equation is: ! ððð ððð e ∂W T T ½M ½∂ ½GNi dxdydz  ½M ½∂½K ½MPe Ni dxdydz½∂ ¼ ∂t Ωe

(2.59)

Ωe

Eq. (2.59) is embodied, and the spatial semidiscrete equations of the seepage continuity equation expressed by the unit junction displacement and the pore pressure are obtained:  e

e  e ∂fδge ½Kc  δ  ½Ks fpge ¼ Rq , δ ¼ (2.60) ∂t

 ÐÐÐ T In the formula: ½Kc  ¼ Ωe ½B ½M N dxdydz is a unit coupling matrix;

e Ð Ð  ÐÐÐ T ½Ks  ¼ Ωe ½Bs  ½K ½Bs dxdydz is the unit flow matrix; and Rq ¼ Γe N Vn ds is the unit equivalent node flow matrix. ③

Time domain dispersion of coupled equations.  e Eq. (2.60) contain the differential term δ of the node displacement with respect to time t, and further need to be discrete on the time t domain. Let tn and tn+1 be two points in the time domain; the element node displacement and pore pressure corresponding to time tn are fδgn e and fpgn e , respectively, and the element node displacement and pore pressure increment in period Δtn ¼ tn+1  tn are fΔδgn e and fΔpgn e , respectively. The element node displacement and pore pressure at time tn+1 are expressed as:  fδge n + 1 ¼ fδgn e + fΔδge (2.61) fpge n + 1 ¼ fpgn e + fΔpge Thus the spatially discrete equilibrium equations (2.58) are written in incremental form:

e ½ke fΔδge + ½kc fΔpge ¼ ΔRf (2.62) Integrate the continuity equations (2.60) on t from tn to tn+1 ð tn + 1 ð tn + 1 ð tn + 1  e

e e ½kc  δ dt  ½ks fpg dt ¼ Rq dt tn

tn

tn

(2.63)

Ð Use the integral approximation to calculate the substitutions of equations tntn+1{p}edt Ð Δt({p}en + θ{Δp}e) and tntn+1[Rq]edt Δt[Rq]e, so that the previous equation becomes:

56

Chapter 2

e ½kc fΔδge  θΔt½ks fΔpge ¼ ΔRp

(2.64)

  e  1

> > < = ΔWyi ½ΔUi  ¼ (2.67) ΔWzi > > > > : ; Δpi 8 9 ΔR > > xi > > < = ΔRyi ½ΔRi  ¼ (2.68) ΔR > > > : zi > ; ΔRpi When solving the discrete equations (2.65) by the FEM, it is necessary to calculate the element flow-solid coupling matrix [k] and the equivalent node load, flow, gravity and seepage increment array [ΔR]e, and then solve Eq. (2.65). 3) Numerical analysis model and calculation conditions. (1) Calculation model. The coordinate system used for the calculation is the Y-axis in the vertical direction and the X-axis in the horizontal direction. The simplified calculation model dimensions are shown in Fig. 2.44. The slope height is 8 m and the horizontal distance is 18 m. The calculation range in the horizontal direction is two times the slope width (36 m), and the calculated depth is three times the slope height (24 m). The soil layer is divided into upper and lower layers. The upper layer is a residual slope with the Quaternary loose deposit(Qdl+el)as the main body, and the thickness is 12–16 m. The lower bedrock is mainly composed of moderately weathered argillaceous siltstone. The second layer is the Longtan (P2l) coal seam formation. On the basis of tests and site survey data, the mechanical parameters of the rock and the soil of the slope are shown in Table 2.4.

Similar simulation tests of landslides 57

Fig. 2.44 Calculate model size (m).

The constitutive model of the material uses the Mohr-Coulomb elasto-plastic model. Due to the large calculation range of the selected slope, the influence of the Saint-Venan effect on the slope stability analysis can be neglected, and the deformation and failure of the slope mainly occur in the shallow part of the slope. The tectonic stress is in the long-term geology and the process has disappeared, so the model boundary does not consider the effect of tectonic stress. Only the self-heavy stress is considered. The ground stress at the left and right boundaries of the slope is calculated according to the static earth pressure. The displacement boundary conditions used in the calculation model are as follows: the base adopts a rigid boundary and constrains the displacement in the horizontal direction and the vertical direction; the left and right boundaries adopt horizontal restraints, constrain the horizontal displacement, and only allow vertical settlement; the surface boundary is a free boundary. Seepage boundary conditions are as follows: both sides and bottom of the model are assumed to be impervious to the boundary; the slope is a freely permeable surface. The initial stress field is the self-heavy stress field, the initial seepage field is selected from the borehole groundwater level data, and the initial groundwater level is taken as the bedrock surface. The calculated grid diagram is shown in Fig. 2.45. The model is divided into 72  48 units and 73  49 nodes. In order to facilitate the analysis of the trend of stress, deformation, and velocity of the slope, I(17,48), II(17,44), III (17,40), and IV(17) are selected from the slope observation points. The vertical and downward section lines of each point are used as observation lines to monitor the displacement, velocity, and principal stress development of the slope. (2) Simulated rainfall process. The monitoring data from the artificial simulated rainfall test showed that about 50% of the rainfall was in the form of surface runoff with a rainfall intensity of 60 mm/h and 2 h of rain and

Table 2.4: Geotechnical parameters.

Test content Index

Test content Index

Water content ω (%) 21

Wet density ρ (g/cm3)

Plasticity index Ip

1.73 Permeability coefficient K (cm/s)

29.7

4.2  103

Dry density ρd (g/cm3)

Saturation Sr (%)

Pore ratio e

1.43 62 0.94 Optimum Maximum dry water content density ρdmax Proportion Gs ωcp (%) (g/cm3) (kN/cm3) 18

1.73

2.78

Liquid limit ωL (%)

Plastic limit ωp (%)

Liquid index IL

58.2 Cohesion c (kPa)

28.5 Internal friction angle φ (°)

0.02 Compression modulus Es (kPa)

6.5

25.5

19,800

Similar simulation tests of landslides 59 I(17,48) II(17,44) III(17,40) IV(17,36)

Fig. 2.45 Slope section finite element meshing.

1 h without rain (actual rainfall intensity of 40 mm/h) and the actual infiltration was 20 mm/h. According to monitoring data in the field test, the simulated rainfall intensity in this numerical analysis is 40 mm/h, the infiltration intensity is 20 mm/h, and the bedrock surface is the transient groundwater level surface, ignoring the permeability of the bedrock. When the natural density and the saturation density are known, it is easy to calculate that it takes 9 h for each 1 m rise of the infiltrated surface. The groundwater level position is referred to the Hoek and Bray [9] slope water level map, which is divided into five working conditions, as shown in Fig. 2.46. The working condition ① is the bedrock surface (impermeable surface), that is, the initial groundwater level surface, as the initial state. The working condition ⑤ is

Fig. 2.46 Groundwater infiltration surface at different times.

60

Chapter 2 Table 2.5: Rainfall intensity and rainfall duration under different working conditions.

Working condition Water level rise (m) Rainfall duration (h) Accumulated rainfall (mm) Cumulative infiltration (mm)

①

②

③

④

⑤

0

6

10

12

14

0

12

20

24

28

0

480

800

960

1120

0

240

400

480

560

fully saturated. The rainfall intensity and rainfall duration for different working conditions are shown in Table 2.5. 4) Analysis of simulation results As the rainfall continues, the groundwater level of the slope increases gradually. When simulating and analyzing different working conditions, the development trend of slope displacement and stress under the interaction of seepage field and stress field in the slope is explored. According to the calculation principle of flow-solid coupling and the preceding calculation model, first the self-weight stress field and initial seepage field of the slope are simulated, the historical displacement is initialized to zero, and the stability of the slope under the action of fluid-solid coupling is calculated and analyzed. The development of stress field, seepage field, and displacement field is shown in the following. (1) Stability analysis. Based on the strength reduction method, the stability coefficient of the slope under different working conditions (water level) was calculated and analyzed, as shown in Table 2.6. The calculation of the role of water in the slope was divided into the conventional calculation method and the fluid-solid coupling calculation method. In the conventional calculation method, only the hydrostatic pressure was calculated (by the method of taking the floating weight of the material), regardless of the hydrodynamic pressure (i.e., the seepage force). The role of the fluid-solid coupling calculation considers the combined effect of hydrostatic Table 2.6: Stability factor at different water levels. Working condition

①

②

③

④

⑤

Conventional

1.72

1.72

1.37

1.07

0.94

Fluid-solid coupling

1.72

1.72

1.31

1.02

0.89

Similar simulation tests of landslides 61 pressure and hydrodynamic pressure. It can be seen from Table 2.6 that with the increase of the groundwater level, the slope stability coefficient gradually decreased. The stability coefficient of the slope decreased to less than 1.0 when the condition was ⑤, indicating that the slope loses stability after 28 h of continuous rainfall. Under the action of fluid-solid coupling, the rise of water level has no effect on the stability of the slope before the groundwater level reaches the toe of the slope. With the continuous elevation of the groundwater level, the slope stability coefficient was reduced by about 5% compared with the conventional calculation method. (2) Stress field analysis. Through the numerical simulation of the stress field of the residual slope under the action of fluid-solid coupling, the maximum unbalanced force history curve, the maximum shear strain increment in the initial state and the fully saturated state, and the plastic zone distribution in the fully saturated state were obtained. The maximum principal stress distribution is shown in Figs. 2.47–2.51. From the evolution curve of the maximum unbalanced force of the slope under flow-solid coupling (see Fig. 2.47), the initial stage of the calculation is to first reach the equilibrium of the initial stress field, and then introduce the seepage field. Each percolation cycle is followed by sufficient mechanical cycles, alternating seepage field and stress field analysis in different calculation time steps, and iterative until the difference between the two adjacent seepage fields and the stress field solution results meet the accuracy requirement (convergence). Comparing the analysis of the maximum shear strain increments in the initial state (case ①) and the fully saturated state (case ⑤) (Figs. 2.48 and 2.49), it can be found that in the initial state, the shear strain concentration formed only near the foot of the slope, and no shear slip zone was

Maximum imbalance (103N)

8

6

4

2

0

0

5

10

15

20

4

Time step (10 )

Fig. 2.47 Maximum unbalanced force history curve of slope under fluid-solid coupling.

62

Chapter 2

Fig. 2.48 Maximum shear strain increment distribution in initial state (m).

Fig. 2.49 Maximum shear strain increment in fully saturated state (mm).

Fig. 2.50 Plastic zone distribution map in fully saturated state.

Similar simulation tests of landslides 63

Fig. 2.51 Maximum principal stress cloud map (Pa) in fully saturated state.

formed, indicating that the slope was in a stable state; when fully saturated, a completely penetrating shear slip zone was formed from the shoulder to the foot of the slope, and the slip surface was approximately arc-shaped. The slope entered an unstable state. Fig. 2.50 shows the plastic zone distribution after the iterative calculation in the fully saturated state. It can be seen that the plastic zone of the slope is mainly distributed in the sliding body, mainly due to shear failure, and there is a tensile stress zone distributed in the middle, and the slope is in an unstable state. From the maximum principal stress distribution diagram (Fig. 2.51) in the fully saturated state, the maximum principal stress gradually increased from top to bottom. Under the influence of the seepage force, the direction of the maximum principal stress near the surface was basically parallel to the slope surface. The direction of the maximum principal stress of the soil was dominated by the vertical direction. (3) Seepage field analysis. Through the numerical simulation of the seepage field under the fluid-solid coupling of the slope, the distribution of pore water pressure at different water levels, the vector of the seepage field and the distribution of the water head at full saturation were obtained (Figs. 2.52–2.54). Fig. 2.52 shows the pore water pressure distribution at different water levels. The pore water pressure in the figure is equal to zero, which is the position of the immersion line. It can be seen that as the rainfall continues, the wetting line gradually increases, and the pore water pressure in the slope gradually increases, which is more detrimental to the stability of the slope. In the fully saturated state, the seepage field vector diagram (Fig. 2.53) shows that the slope shoulder soil has mainly downward movement, and the soil from the shoulder to the toe gradually transforms to the trend of moving outward. The seepage velocity is also gradually

64

Chapter 2 (A)

(B) 0 25 50 75 100 125 150 175

0 25 50 75

(C) 0 25 50 75 100 125 150 175 200

225

Fig. 2.52 Pore water pressure distribution diagram at different water levels (105 Pa). (A) Pore water pressure distribution in initial state. (B) Pore water pressure distribution in condition ③; (C) Pore water pressure distribution in fully saturated condition.

increased. A water outlet is formed at the foot of the slope. It can be seen from the direction of seepage that the seepage dynamic water pressure caused by the seepage increases the sliding force of the slope and reduces the safety factor, which is also verified by the stability analysis of the slope. Fig. 2.54 shows the water head distribution in the fully saturated state (case ⑤). The maximum water head value at the upper left corner of the slope is 1.77 m. The seepage direction of the slope soil is from the high head to the low head.

Fig. 2.53 Seepage vector field in fully saturated state.

Similar simulation tests of landslides 65

Fig. 2.54 Water head distribution map in fully saturated state (m).

(4) Displacement field analysis. The displacement and velocity changes of slope, monitoring points, and observation lines of residual slopes under different working conditions under flow-solid coupling are shown in Figs. 2.55–2.58. It can be seen from the displacement vector diagram of the landslide in full saturation (Fig. 2.55) that the shape of the sliding body is clearly identifiable. The displacement is the largest at the foot of the slope, and the soil at the shoulder is mainly downward from the shoulder to the foot of the slope. Gradually it converts to a tendency to move out of the slope, and the maximum displacement is 56 mm. The character and direction of the velocity vector diagram (Fig. 2.56) are basically consistent with the displacement vector diagram. The maximum velocity is 0.00126 mm/s, and the shape of the plastic zone distribution map is also consistent with the shape of the displacement and velocity vector diagram. In the fully saturated state, the X-direction displacement cloud map (Fig. 2.57) shows that the displacement from the shoulder to the foot is gradually increased, and the direction is horizontal

Fig. 2.55 Displacement vector in fully saturated state.

66

Chapter 2

Fig. 2.56 Rate vector in fully saturated state.

Fig. 2.57 X-direction displacement cloud map in fully saturated state (mm).

Fig. 2.58 Y-direction displacement cloud map in fully saturated state (mm).

Similar simulation tests of landslides 67 to the outside of the slope. The Y-direction displacement cloud map (Fig. 2.58) shows that the displacement is the largest at the shoulder and the direction is vertically downward. In the middle and lower part of the slope, the displacement of the Y direction is gradually reduced due to the bulging effect, and at the foot of the slope forms an upward positive displacement. In case ④, from the X and Y displacement tracking curve (Figs. 2.59 and 2.60), it can be seen when the iteration is calculated to 2200 steps, the horizontal displacement and the vertical displacement approach a certain fixed value. It shows that the slope has reached a steady state after deformation and stress adjustment. The horizontal displacement decreases from the shoulder to the foot of the slope, and the vertical displacement increases from the shoulder to the foot. From the full saturation state (case ⑤) X- and Y-displacement tracking curve (Figs. 2.61 and 2.62), it can be found that the horizontal displacement and vertical displacement of each monitoring point on the slope continuously increase, and there is no stable trend indicating instability of the slope. The stability analysis shows that the stability coefficient is 0.89. Comparing the analysis in Figs. 2.63 and 2.64, it can be seen that the rate of the tracking point I (17, 48) in the X-direction and the Y-direction in the condition oscillated after a period of time, and finally tended to zero, which is related to the horizontal displacement. It tended to a certain fixed value, which indicates that the slope had reached the steady state after deformation and stress adjustment. The rate of the tracking point I (17, 48) in the X-direction and the Y-direction was always changing and constantly increasing in case ⑤, and there was no tendency to zero, indicating that the slope had been unstable.

Horizontal displacement (mm)

0

Time step (103) 2 4

6

–2

–4 –6 –8

–10

I

II III IV

Fig. 2.59 X-displacement tracking curve at condition ④.

68

Chapter 2 Time step (103) 2 4

6

0 IV

Horizontal displacement (mm)

–2

III

–4

–6 II –8

–10 I –12

Fig. 2.60 Y-displacement tracking curve at condition ④.

It can be seen from the horizontal displacement and vertical displacement of each point on the slope at different water levels (Figs. 2.65 and 2.66) that as the groundwater level increases, the displacement of each point on the slope gradually increases. And the horizontal displacement is gradually increased from the shoulder to the foot of the slope. The vertical displacement is

Horizontal displacement (mm)

0

1

Time step (103) 2 3

4

–20

–40

–60

I II III IV

–80

Fig. 2.61 X-displacement tracking curve at condition ⑤.

Similar simulation tests of landslides 69

Vertical displacement (mm)

10 1

0

Time step (103) 2 3

4 IV III

–20 II

–40 I

–60

Fig. 2.62 Y-displacement tracking curve at condition ⑤.

gradually reduced from the shoulder to the foot of the slope. Because the foot of the slope is pushed, the displacement has an upward bulging and upward trend. The displacement from working condition ① to working condition ④ increases slowly. When the water level reaches the working condition ⑤, the horizontal displacement increases sharply. The stability analysis result of the slope also shows that from the working condition ④ to the working condition ⑤ is the turning point of stability to instability of the slope.

0

Rate (10–5 m/s)

–0.2

Time step (103) 2 4

6

Y velocity X velocity

–0.4 –0.6 –0.8 –1.0

Fig. 2.63 Rate history curve of I (17, 48) at condition ④.

Chapter 2

0

Rate (10–5 m/s)

70

1

Time step (103) 2 3

–1

–2

4

X velocity

Y velocity

–3

Fig. 2.64 Rate history curve of I (17, 48) at condition ⑤.

Fig. 2.65 Horizontal displacement of each monitoring point on the slope at different water levels.

Fig. 2.66 Vertical displacement of each monitoring point on the slope at different water levels.

Similar simulation tests of landslides 71 Fig. 2.67 shows the horizontal displacement change of each point in the slope on the line when fully saturated (case ⑤). The position of the inflection point corresponds to the potential sliding surface, and the slip points in the I, II, III, and IV observation points are 6.5, 6.0, 5.0, and 3.5 m below the slope surface, and the depth from the shoulder to the slope is gradually reduced. The maximum horizontal displacement occurs at 1 m from the foot of the slope with a value of 55 mm. The same sliding surface position on each observation line can be obtained from the vertical displacement variation curve (Fig. 2.68) of the observation point on the line at the working condition ⑤. The vertical displacement is the largest at the shoulder with a value of 33 mm. Comparing and analyzing the horizontal displacement and vertical displacement changes, it can be seen that the displacement is dominated by horizontal displacement. Comparing the numerical simulation results with the on-site monitoring test results of artificial rainfall-induced slippage, it is found that the slip surface depth, the slip surface shape, and the variation rule of the horizontal displacement and vertical displacement from the shoulder to the foot of the slope show good consistency. The difference is that the numerical simulation values of the rainfall area and rainfall are greater than the field test value, so the displacement value of the numerical simulation is correspondingly larger than the displacement value of the field test.

Fig. 2.67 The curve of horizontal displacement versus depth at each point on the observation line at condition ⑤.

72

Chapter 2

Fig. 2.68 The curve of vertical displacement versus depth at each point on the observation line at condition ⑤.

2.4 Field simulation test of landslides induced by mechanical excavation 2.4.1 Purpose and implementation content The natural slope is in a relatively stable state due to the transformation of the long-term geological period. The excavation breaks the relative equilibrium state, causing the slope rock mass to unload and rebound in the direction of the air surface, and the deformation is further increased. This is one of the important triggers for railway landslides. The purposes and content of the field simulation test of a landslide induced by mechanical excavation are as follows: (1) To find the Vcr (criteria) value of the slipping slope in the case of excavation. (2) To perform inversion analysis for accurate shear strength of sliding surface: c and φ values. (3) To provide a basis for engineering design. (4) To provide a basis for research of monitoring and forecasting technology of the subject. The test began in April 2012.

2.4.2 Test design The test design parameters are as follows:

Similar simulation tests of landslides 73

Fig. 2.69 Floor plan for monitoring points (m).

Location: 10 m away from the left side of the artificial rainfall test area. Area: 10 m  10 m. Monitoring content: deep displacement of slope, slope crack. Instrument: American-made Sinco inclino-meter (same as rainfall test). Monitoring section: 2 (see Fig. 2.69). Monitoring points: 6 (Fig. 2.70). Monitoring frequency: 6 h/time.

Fig. 2.70 Excavation test point arrangement.

74

Chapter 2

2.4.3 Excavation sequence After setting the inclinometer and the surface displacement monitoring point, in order to balance the inclinometer and the surrounding soil, it needed to be left alone for about 1 week before testing. With regard to the dangers of mechanical excavation, this test used an excavator for excavation (Fig. 2.71). The excavation sequence is as shown in Fig. 2.72, from the ① soil to the ④ soil until the slope collapsed. Excavation was carried out from 13:30 to 14:00 every day, and the excavation depth was 2 m. The monitoring frequency was four times a day, and the data was recorded every 6 h. The reading time was 4:00, 10:00, 16:00, and 22:00 every day, and the number of readings was increased before the slip.

2.4.4 Landslide development process One week after drilling and burying, the measuring tube and the surrounding soil reached equilibrium and stability, and the initial value of each measuring tube was measured. After the staged excavation, the slope body was gradually deformed and destroyed (see Figs. 2.73 and 2.74). The whole process is shown in Table 2.7.

2.4.5 Test results analysis The monitoring data of each hole was analyzed and plotted. The relationship between the cumulative displacement along the inclined direction of slope and the depth of the hole, the relationship between the cumulative displacement along the transverse direction of slope and the depth of the hole, and the relationship between the cumulative displacement and the depth of the hole were obtained. The horizontal displacement curve of each hole is shown in Figs. 2.75–2.92. It can be seen that the displacement of each hole was mainly along the inclined

Fig. 2.71 Excavation sequence of L1 section (m).

Similar simulation tests of landslides 75

Fig. 2.72 Mechanical excavation.

direction of the slope, and each monitoring hole had an obvious sliding surface position. The positions of sliding surfaces at ZK1–ZK6 monitored by the test were, respectively, 3.5, 3.5, 4, 4, 4, and 4 m. It can be seen from the figures that the displacement deformation zone basically occurs within the range of 0–4 m below the surface, the displacement decreased with the

Fig. 2.73 Landslide complete deformation and destruction.

76

Chapter 2

Fig. 2.74 Stagger cracks in the front edge of landslide.

increase of the depth, and the deformation of the slope surface was the largest. The ZK1 nozzle had the maximum displacement of 32.94 mm along the inclined direction of the slope, the maximum displacement of 13.73 mm along the transverse direction of the slope (on the right side of the slope), and the maximum cumulative displacement of 35.69 mm (the direction was the lower right of the slope, 22.63 degrees).

Table 2.7: Landslide history table. Sequence

Time (April 2012)

1 2

16th 17th

13:30–14:00 13:30–14:00 16:30

3

18th

13:30–14:00 22:00

4

19th

10:00 16:00 22:00

5

20th

4:00 10:00

Event description First excavation Continued to excavate Found the trailing edge crack of the landslide, width 1–3 mm, length 6 m Continued to excavate It was found that the front edge of the slope across the air surface had stagger cracks, with a width of 5–8 mm and a length of 9 m, as shown in Figs. 2.73 and 2.74 The ZK6 inclinometer was deformed by extrusion at 4 m; the probe was blocked and could not be monitored ZK3 hole deformation misalignment, unable to monitor ZK2 hole deformation misalignment, unable to monitor ZK4 hole deformation misalignment, unable to monitor ZK1 and ZK5 holes completely slide

Remarks

Similar simulation tests of landslides 77

Fig. 2.75 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK1.

Fig. 2.76 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK1.

Fig. 2.77 Cumulative displacement versus hole depth curve for ZK1.

78

Chapter 2

Fig. 2.78 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK2.

Fig. 2.79 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK2.

Fig. 2.80 Cumulative displacement versus hole depth curve for ZK2.

Similar simulation tests of landslides 79

Fig. 2.81 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK3.

Fig. 2.82 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK3.

Fig. 2.83 Cumulative displacement versus hole depth curve for ZK3.

80

Chapter 2

Fig. 2.84 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK4.

Fig. 2.85 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK4.

Fig. 2.86 Cumulative displacement versus hole depth curve for ZK4.

Similar simulation tests of landslides 81

Fig. 2.87 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK5.

Fig. 2.88 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK5.

Fig. 2.89 Cumulative displacement versus hole depth curve for ZK5.

82

Chapter 2

Fig. 2.90 Cumulative displacement along the inclined direction of slope versus hole depth curve for ZK6.

Fig. 2.91 Cumulative displacement along the transverse direction of slope versus hole depth curve for ZK6.

Fig. 2.92 Cumulative displacement versus hole depth curve for ZK6.

Similar simulation tests of landslides 83

Fig. 2.93 ZK1 feature point displacement-excavation depth-time curve.

Fig. 2.93 shows displacement of feature point-excavation depth-time curve of ZK1. It can be seen that as the depth of excavation increased, the displacement of the inclinometer tube gradually increased. After 4:00 am on the 19th, the displacement development speed increased significantly, and the average displacement rate at 0.5 m reached 0.552 mm/h. It can also be seen that this deformation was a relaxation deformation, gradually decreasing from the surface to the inside. The displacement at 0.5 m was equivalent to twice the displacement at 2.5 m, there was substantially no displacement at 5 m, and the numerical value was small. The change may be caused by measurement errors. Fig. 2.94 shows the cumulative displacement versus hole depth for the last measurement of each measuring point. It can be seen that the deformation zone of each measuring point basically occurred in the range of 0–4 m, and the deformation gradually decreased from the slope surface to the inside. The maximum displacements detected by the inclined measuring tubes were 35.39, 31.01, 35.47, 71.63, 74.75, and 87.02 mm, respectively. Fig. 2.95 shows the cumulative displacement-excavation depth-time curve of each nozzle. It can be seen that the soil movement gradually increased with the increase of the excavation depth, and the closer to the air surface,

Fig. 2.94 The final cumulative displacement versus hole depth curve of each measuring point.

84

Chapter 2

Fig. 2.95 Cumulative displacement-excavation depth-time curve of each nozzle of the measuring point.

the larger the displacement of the soil was. For example, at 4:00 am on the 19th, the displacements at the nozzles of ZK1–ZK6 were 18.83, 22.91, 34.18, 39.95, 44.16, and 87.02 mm, respectively. Fig. 2.96 shows the displacement rate of feature point-excavation depth-time curve of ZK1. The depth of the inclination monitoring was 11 m from the nozzle of the measuring tube to the inside of the slope. The depth of the sliding surface monitored was also the distance from the nozzle to the sliding surface, and the nozzle also had a certain distance from the slope surface. The actual slip surface depth should be subtracted from the exposed portion of the inclinometer tube. The depth of the slip surface monitored by each inclinometer tube is listed in Table 2.8. The location of the sliding surface can be determined by combining the

Fig. 2.96 ZK1 feature point displacement rate-excavation depth-time curve.

Similar simulation tests of landslides 85 Table 2.8: Sliding surface position monitored by the inclinometer. Drill mark

ZK1

ZK2

ZK3

ZK4

ZK5

ZK6

Distance between sliding surface and nozzle (m) Distance between nozzle and slope surface (m) Distance between sliding surface and slope surface (m)

3.5

3.5

4.0

4.0

4.0

4.0

0.5

0.2

0.3

0.2

0.2

0.3

3.0

3.3

3.7

3.8

3.8

3.7

position of the sliding surface detected by the inclinometer with the staggered crack in the leading edge and the tensile crack in the trailing edge of the slope. The position and shape of the sliding surface of L1 and L2 are shown in Figs. 2.97 and 2.98. It can be seen that the sliding surface of the landslide was very shallow, and the depth of the sliding surface was within the

Fig. 2.97 Sliding surface position on the L1 section.

86

Chapter 2

Fig. 2.98 Sliding surface position on the L2 section.

range of 3 to 4 m below the slope surface. It was a shallow landslide. The sliding deformation zone can also be seen from the figure. The longitudinal slope distance of the sliding deformation zone is approximately equal to the transverse width of the airside of the leading edge of the slope. The shape of the sliding deformation zone was approximately semicircular, and the shape of the three-dimensional space of the sliding deformation zone was approximately dustpan. The staggered fracture of the leading edge of the landslide was located at about 3 m below the slope surface, which meant that the height of the excavation free face above 3 m may cause landslides. In future slope excavation similar to such slopes, 3 m should be used as

Fig. 2.99 After collapse (1).

Similar simulation tests of landslides 87

Fig. 2.100 After collapse (2).

the border height of the excavation free face. Appropriate precautions should be taken for slopes having excavation free face exceeding 3 m in height. On the afternoon of April 20th, the monitoring of the inclination was stopped and the test area was no longer covered with colored strips. Heavy rain fell at the test point on April 23 and 24, and on April 25, the slope collapsed. The trench at the leading edge of the landslide was buried, and the inclined tube was broken, as shown in Figs. 2.99 and 2.100.

2.5 Chapter summary In the Zhengning-Shengguan section of the Shanghai-Kunming high-speed railway passing through Guizhou province, the artificial rainfall on the slope and the on-site test of the excavation sliding were successfully carried out, and the processed data was analyzed. The FLAC fluid-solid coupling calculation model was used to analyze the influence of hydrodynamic pressure on slope stability. A predictive model calculation program was used, and the landslide was tested and forecasted. The following results were obtained: (1) Numerical simulation analysis was conducted for five working conditions (water level), and the results obtained were regular. The development of stress field, seepage field, and displacement field were simulated clearly and reasonably, and were basically consistent with the results of on-site monitoring of artificial rainfall-induced slippage. (2) Under the action of hydrodynamic pressure, the stability coefficient of the slope was reduced by about 6% compared with the conventional calculation. (3) The mechanism of a rainfall-type landslide is as follows: rainfall increases the water content of the soil and leads to the slope foot being full of water, which greatly

88

(4) (5) (6)

(7)

Chapter 2 reduces the shear strength of the sliding surface, and triggers the landslide under the action of hydrostatic pressure and hydrodynamic pressure. The mechanism of excavationinduced landslide is that the natural mountain body maintains a certain degree of balance under long-term various forces. Due to the excavation, the balance is broken and the stress is redistributed. If the antisliding index of the rock mass is smaller than the slipping index, a comprehensive effect is produced during the stress adjustment process, and due to the influence of gravity, a landslide is generated. Slip surface strength index: artificial rainfall c ¼ 12.5 kPa, φ ¼ 9.2°, excavation c ¼ 8.3 kPa, φ ¼ 17.7°. Critical sliding speed: artificial rainfall Vcr ¼ 25.6 mm/day; excavation Vcr ¼ 13.3 mm/day. The sliding surface of the artificial rainfall landslide is an arc of R ¼ 367 m, and the slope of the sliding surface is slow. The sliding surface induced by excavation is basically straight and the slope of the sliding surface is steep. Under the same geological conditions, artificial rainfall simulation tests and mechanical excavation simulation tests were carried out, which revealed the significant influence of triggering factors on landslide disaster mechanism and motion laws, and provided reliable time, space, and intensity prediction for landslides. The theoretical basis is of great significance for guiding engineering practice.

References [1] Biot MA. General theory of three dimension consolidation. J Appl Phys 1956;12:155–65. [2] Detourray E, Cheng AH-D. Poroelastic response of borehole in a non-hydro static stress field. Int J Rock Mech Min Sci Geomech Abstr 1988;25(3):171–82. [3] Lubinski A. The theory of elasticity for porous bodies displaying a strong pore structure, In: Proc second US Natl Congr Appl Mech; 1954. p. 247–56. [4] Kojic M, Cheatham JB. Theory of plasticity of porous media with fluid flow. SPE J 1974;263. [5] Risnes R, et al. Sand stresses around a wellbore. SPE J 1982;22:883–9. [6] Wang Y, Dusseault MB. Borehole yield and hydraulic fracture initiation in poorly consolidated rock strata II perneable media. Int J Rock Mech Min Sci Geomech Abstr 1991;28:247–60. [7] Xu ZH. Fluid-solid coupling problem of seepage and its application. Chin J Rock Mech Eng 1999;05:3–5. [8] Settari A, Walters DA. Advances in coupled geomechanical and reservoir modeling with applications to reservoir compaction. SPE J 2001;334–42. [9] Hoek E, Bray JW. Rock slope engineering. 3rd ed. London: The Institute of Mining and Metallurgy; 1981.

CHAPTER 3

Rockfall mechanisms and block theoretical stability analysis 3.1 Generation mechanism of rockfalls 3.1.1 Definition of a rockfall A rockfall is a sudden and sharply dynamic geological phenomenon that occurs on a steep slope. During a rockfall, the rock mass or soil mass under the control of gravity and external forces suddenly and rapidly topples, tumbles, and jumps from the mother rock (soil) body of the steep slope. After a rockfall, the relative positions of various parts of the deformation bodies are disordered and unrelated to each other. The small blocks tumble a short distance, while the large blocks tumble farther and pile up into inverted stone cones.

3.1.2 Differences between landslides and rockfalls The differences between rockfalls and landslides are mainly reflected in the following four aspects: 1) After a rockfall, the rockfall material often accumulates on the toe of the slope, which is a cone-shaped body with a disorderly structure and no sequence. Landslide deposits often have a certain external shape, and the landslide body has good integrity, reflecting the sequence and structural features. In the landslide deposits, there is no great change in the upper and lower layers of the rock mass (soil) and in the old and the new geological relations, which means that it is still a regular distribution. 2) The rockfall body completely separates from the mother body (mountain body), and the original integrity is completely destroyed. However, landslide bodies are rarely completely separated from the mother body, and most still retain the original relative integrity. 3) After a rockfall, the vertical displacement of the rockfall materials is much larger than the horizontal displacement, and the position of the gravity center of the rockfall materials is much lower than before. Usually, the horizontal displacement of a landslide is greater than the vertical displacement, and the position of the gravity center of most landslides is not Rock Mechanics and Engineering. https://doi.org/10.1016/B978-0-12-822424-3.00003-7 # 2021 Central South University Press. Published by Elsevier Ltd. All rights reserved.

89

90

Chapter 3

greatly reduced. The sliding distance is large and, at the same time, the sliding speed of a landslide is generally slower than that of a rockfall. 4) The surface of a rockfall deposit is substantially free of crack distribution. The landslide body, especially for newly occurring landslides, has many regular and vertical cracks on its surface. For example, arc-shaped tensile cracks are distributed in the upper part of the landslide body; shear cracks are distributed on both sides of the middle part of the landslide body; transverse cracks are distributed in the front part of the landslide body and the direction is perpendicular to the direction of the sliding or sliding pressure; and fan-shaped cracks or radial cracks are distributed in the front and middle part of the landslide body.

3.1.3 Classification of rockfalls Rockfalls, commonly known as “stone blossoming,” are caused by weathering of the rock mass in nature and artificial mountain building, blasting, etc. As a result, the rock is cracked and falls. There are currently four methods used to classify rockfalls. According to location, they can be classified as mountain rockfalls or rive (reservoir) bank rockfalls. According to physical characteristics, they can be divided into rock rockfalls, soil rockfalls, mixture rockfalls, and avalanches. According to the causes, they can be divided into fault rockfalls, joint and fissure rockfalls, weathered gravel rockfalls, and hard-soft rocks contact zone rockfalls. According to their deformation and failure modes, they can be divided into toppling type, sliding type, bulging type, tension type, and staggered type rockfalls. Generally, the rockfall is dominated by toppling and slipping. The specific classifications are shown in Table 3.1.

3.1.4 Factors affecting the occurrence of rockfalls There are many factors influencing the occurrence of rockfalls, which can be roughly divided into internal and external factors. The internal factors are related to the potential rockfall of the slope itself, which can be roughly divided into three major types: geotechnical, geological structure, and topography. The external factors are environmental factors that directly cause rockfalls, such as rock weathering, continuous rainfall, earthquakes, etc. According to the 2005 China Geological Environment Bulletin • Geological Hazards, the induced factors of sudden geological disasters are natural and human factors, with 96.6% of geological disasters caused by natural factors, and 3.4% related to unreasonable human engineering activities. Among the natural factors, rainfall is the most important predisposing factor for sudden geological disasters. 1) Internal factors affecting rockfall (1) Geotechnical materials: rocks and soils are the material conditions of rockfalls. In general, all kinds of rocks and soils can form rockfalls, but rockfall sizes are different for different rocks and soils. Generally, various types of magmatic rocks,

Table 3.1 Rockfall classifications table. Types Toppling type rockfall

Sliding type rockfall

Bulging type rockfall

Lithology

Structural surface

Landform

Loess, limestone Mostly vertical Canyon, upright joints, columnar bank slope, cliff, and other etc. joints, upright upright rock rock layers formations Mostly alternates between soft and hard rock layers, such as limestone with thin layer of shale Upright loess, clay, or hard rock with thick softer layers

Tension type rockfall

More common in alternate soft and hard rock layers

Staggered type rockfall

Hard rock or loess

Rockfall shape

Force state

Plate-shaped, long column

Mainly affected by overturning moment

Sliding surface is Steep slopes are May be Structural mainly usually greater combined into surface with a subjected to than 55° various shapes tendency to free shearing force such as plate, face (possibly wedge, cylinder, flat, wedge, or etc. curved) Steep slope Rock mass is The upper part high is vertical joint, columnar joint, and the lower part is near horizontal structural plane Mostly Upper The upper hard weathered protruding cliff rock layer fissures and protrudes in the gravity tensile form of a cracks cantilever beam Steep slope Mostly plateVertical fissures shaped, long develop, usually greater than 45° column without a structural surface for free face

Starting form of Main factors of motion instability Topple

Slip

Hydrostatic pressure, hydrodynamic pressure, seismic force, gravity Gravity, hydrostatic pressure, hydrodynamic pressure

The lower soft rock is vertically squeezed

Inflated with sinking, slipping, tilting

Gravity, softening by water

Tension

Tension crack

Gravity

Shear force caused by selfweight

Staggered

Gravity

92

Chapter 3 metamorphic rocks and sedimentary rocks, quartz sandstones, glutenits, diagenetic rocky loess, and densely structured loess form large-scale rockfalls. Shale, marl and other interbedded rocks and loose soil layers are often dominated by small falls and spalling. (2) Geological structures: structures such as faults, folds, and rupture surfaces and layers also have a considerable impact on rockfalls. Generally, the soil near the fault zone is not water permeable and often blocks the groundwater flow, increases the local water pressure, reduces the shear strength of the fault zone, and causes easy slipping. In addition, the fault zone is mostly composed of soft and broken materials, and can be compressed easily due to weak strength. The folds mainly affect the development of the terrain, causing the slope of the rock formation and the fracture surface of the rock mass. The existence of various discontinuities in the rock mass is the basic condition for the rockfall. When the occurrence and combination of various discontinuous surfaces are favorable for rockfall, they become a decisive factor. There are many reasons for the occurrence of a rupture surface. Regardless of its cause, the influence of the development of the rupture surface on the stability of the slope is mostly negative, such as reducing the strength of the rock mass, causing subsidence, providing a sliding surface, and causing the rock mass to break. In addition, its fissure makes it easy for water to enter the rock mass, which not only increases the water pressure in the gap, but also accelerates the weathering. The more development of fractures in the slope, the more likely a rockfall. A steeply inclined structural plane parallel to the direction of the slope extension is most conducive to the formation of rockfall. (3) Topography and geomorphology: river, lake (reservoir), bank slopes and hillsides, railway and road slopes, engineering building slopes, and various types of artificial slopes are the landforms that are most favorable for rockfalls. High-steep slopes with gradient greater than 45 degrees, erosion bank slopes, isolated mountain mouths, and concave steep slopes are terrains favorable for rockfall formation.

These three factors, geotechnical type (rock and soil type), geological structure, and topography, are collectively referred to as geological conditions, which are the basic internal conditions for the formation of rockfall. 2) External factors affecting rockfall A potential rockfall body will not fall without external forces or other factors. Virtually any potential rockfall is subject to the long-term effects of various complex factors. Among these factors, atmospheric rainfall, surface water, weathering, earthquakes, and human factors are the most important. It is undoubtedly important to understand these factors and their impact on potential rockfall. (1) Earthquakes cause the slope to sway, destroying the slope balance and inducing rockfall. Earthquakes with a general intensity greater than 7 degrees will induce a large number of rockfalls.

Rockfall mechanisms and block theoretical stability analysis 93 (2) Heavy rain is a catalyst for inducing geological disasters. The scouring, dripping, and infiltration of heavy rains reduce the shear strength of the rock and soil, especially the antisliding strength of the structural surface of the geological body, which causes it to develop into a sliding surface and a rockfall interface. In addition, the self-weights of the rocks and soils are added to due to rain, which increases the dynamic pressure and static pressure of the groundwater, further reducing the stability of the slope and inducing rockfall. In addition, the rockfall body and the landslide body form a debris flow under the action of high-intensity water flow. (3) Erosion and immersion of surface water is another major factor, as rivers and other surface waters continuously wash the slopes or soak the toe of the slope, weakening the slope support or softening the rocks and soils, reducing the slope strength, and possibly inducing rockfalls. (4) Human activities, such as the excavation of the toe of a slope, underground goaf, reservoir water storage, water discharge, and other activities that change the original balance state of the slope, will induce rockfall. Other factors, such as frost heave, temperature difference between day and night, etc., can also cause rockfall. 3) Relationship between slope shape and rockfall There are many factors affecting rockfall on a slope, which can be roughly divided into internal and external factors. The internal factors are related to the potential of the slope itself to cause rockfall, and the external factors are the environmental factors that directly cause the rockfall. According to the specific actual conditions, we focus on the impact of the shape of the slope on the rockfall. The shape of the slope is the most basic condition for the occurrence of slope rockfall. The main indicators for characterizing the terrain are slope gradient, slope height, and surface morphology. To this end, studying the impact of slope shape on rockfall is helpful in understanding the conditions and mechanisms of rockfall formation as well as the dangers of overexploiting steep slopes. (1) Form of slope rockfall The slope formed by nature presents various forms. According to the presence or absence of soil in the upper part of the slope, it can be divided into semiinfinite slope and finite slope. There are three common forms of rockfall related to semiinfinite slopes. The first is constrained by parallel bedrock surfaces and the failure surface is the plane along the bedrock, as shown in Fig. 3.1A; the second is limited by the upper and lower bedrock faces, and the failure surface is arcshaped, as shown in Fig. 3.1B; the third is an arc-shaped failure occurring on a homogeneous slope, as shown in Fig. 3.1C. Of course, there is also a composite failure surface located between the preceding rockfall forms. There are also three common forms of failure for finite slopes. The first is planar rockfall on the homogeneous slope, as shown in Fig. 3.2A; the second is arc-like rockfall on monoclinic

94

Chapter 3 Failure surface Failure surface Bedrock surfaces

Bedrock surfaces

(B) (A)

Failure surface

Bedrock surfaces

(C) Fig. 3.1 Rockfall form on a semiinfinite slope. (A) Uniform thickness. (B) Interlayer. (C) Different thickness.

Failure surface Failure surface Failure surface

Bedrock surfaces

Bedrock surfaces

(A)

(B)

(C)

Fig. 3.2 Rockfall form on a finite slope. (A) Homogeneity. (B) Single bedrock surface. (C) Single bedrock surface.

bedrock, as shown in Fig. 3.2B; the third is the arc-shaped rockfall on a composite bedrock surface, as shown in Fig. 3.2C. Usually the rockfall part is on a valley or hillside, as shown in Fig. 3.3. (2) Influence of slope gradient on rockfall Mountain disasters involve the loss of plastic balance of the slope, causing damage, displacement, and accumulation, which affect human activities and land use. According to the survey data in Panxi, Sichuan, there have been 816 landslides and rockfalls with volume above 10
104 m3. According to the average slope gradient, the gradient value is between 36 and 45 degrees. The type of slope failure is mostly rockfall. Most of the general rockfalls occur on slopes with gradient greater than 30 degrees. Showing the actual situation regarding rockfalls, the literature records

Rockfall mechanisms and block theoretical stability analysis 95

Fig. 3.3 Schematic diagram of the location of the rockfall.

Collapse 600

400

200

0 20

40

60

80

Slope (°) Fig 3.4 Probability distribution of different rockfalls.

2238 instances of rockfall data over the past 30 years. About 80% of the rockfall slopes have a 30- to 50-degree gradient, and most of the gradient is about 40 degrees, as shown in Fig. 3.4. From the analysis of the properties of soil and rock that constitute slopes, the gradient of general soil slopes is smaller than that of rock slopes, and when soil slopes fall, most of the gradient is between 30 and 40 degrees, while the gradient of the rock slope is between 30 and 50 degrees. (3) Influence of slope height on rockfall According to statistics, most rockfalls occurred on slopes with slope heights greater than 20 m. The higher the slope height, the greater the probability of rockfall. Table 3.2 shows the

96

Chapter 3 Table 3.2 The relationship between the number of rockfalls and the slope height. Slope height (m)

Rockfall (times)

Proportion (%)

20 >30 >40 >50

2 11 10 11 23

3.5 19.3 17.5 19.3 40.4

relationship between the number of rockfalls in the Fengzhou Works Section and the slope height. Japan also carried out statistical analysis on the impact of slope height on rockfall: the slope with height of 10–20 m has the highest probability of rockfall, accounting for 32.9% of the total, followed by 20–30 m, accounting for 15.8%, and the average slope height is 35.5 m, as shown in Fig. 3.5. It can be seen from Table 3.2 and Fig. 3.5 that the slope heights of rockfalls in different areas are different, which is mainly related to the soil and the rock condition. The rock slope is high and steep, and the soil slope is low and slow. (4) Influence of slope morphology on rockfall The movement of a rockfall block is different from that of a landslide. The rockfall body is cracked from the ground ! toppled to the free surface ! to instantaneously tear away from the

Fig. 3.5 Probability distribution of different slope height rockfall.

Rockfall mechanisms and block theoretical stability analysis 97 mother body with high-speed movement. The whole movement process shows free fall, rolling, jumping, collision, pushing, and coexisting compound processes. During the movement, large rock blocks are broken and disintegrate into small pieces due to jumping and collision. Generally, the shape of the mountain slope is irregular, but the complex terrain can be divided into three basic conditions: planar, concave, and convex. Due to the variation of the longitudinal and transverse sections and the combination of different radii of curvature, various shapes can be formed. However, the basic slope morphology can be divided into nine types, and three of them have little or no probability of appearing. According to the literature, the six common types are the planar type, ascending type, descending type, gully type, ridge type, and water collecting type. Japan’s observation of slope morphology is very detailed, and the probability of rockfall in longitudinal and transverse sections for various forms was investigated, as shown in

0.9% Other 4.7%

9%

2.

%

2.7 4.4%

23.6%

4.6% 4.9% 5.9%

20.6%

7.7

%

12.5%

Fig. 3.6 Probability of rockfall in various forms. Table 3.3 Probability of rockfall of section shape (%). Longitudinal section

Cross section

Descending Compound type Planar type type

Ascending type

Planar type

Gully type

Ridge type

Other

32.6

7.0

60.1

18.1

17.8

4.0

36.1

24.3

98

Chapter 3 Probability 80

I II III

60 40 20 0 Catchment Divide water

Flat

Fig. 3.7 Effect of cross-sectional shape on rockfall.

Fig. 3.6 and Table 3.3. However, it should be noted that the statistical data includes both natural slopes and artificially rectified slopes. It can be seen from Table 3.3 that the probability of rockfall when the cross-sectional shape is the planar type is 60.1%, which is larger than that of the gully and ridge types. This is due to the fact that there are many planar type slopes that have been artificially rectified in the statistical data. For example, if human factors are removed, statistics are also used in the literature to compare the effects of cross-sectional morphology on slope rockfall from three different regions, as shown in Fig. 3.7. The results show that the gully type has the highest probability of rockfall, and the probability of rockfall for the ridge type is slightly more than that of the planar type. There are many kinds of slopes in nature. The longitudinal shape of the slope can be categorized as convex slope (ridge type), concave slope (gully type), and straight slope (planar type). Among them, the convex slope is steepest, which is conducive to rockfall. The development of a concave slope is mostly as the residual back wall of an ancient rockfall, which is conducive to the accumulation of surface water and groundwater, and thus prone to inducing slope rockfall. Many gully sources also exhibit a gentle slope to steep slope, which is classified as the descending type. Due to the strong erosion of the gully head, the gully is prone to slip, and then exhibits a steep slope to gentle slope, which belongs to the ascending type in wide river valleys. As a typical slope morphology, the ascending type generally does not cause a large number of rockfalls. The timing and scale of the rockfalls of different types of slopes are different. Similarly, the probabilities of a variety of slope shapes having the capability of rockfall are also different. According to the statistical analysis of the data, there are three kinds of slopes whose probability of rockfall is very small, as shown in Table 3.4. Table 3.4 Basic shape of the slope. Transverse Basic shape Longitudinal

Plane Probability Concave shape Probability Convex shape Probability

Plane

Concave shape

Convex shape

I Common II Common III Common

IV Common V Common VI Rarely

VII Common VIII Rarely IX Almost none

Rockfall mechanisms and block theoretical stability analysis 99 As can be seen from Table 3.4, there are six types of slopes that are prone to rockfall: (I) Planar type slope: Both the slope and the bedrock are flat and parallel. (II) Ascending type slope: The longitudinal section of the slope and the bedrock surface are both concave, and the slope gradient becomes larger and larger from the lowest point to highest point. (III) Descending type slope: The longitudinal section of the slope and the bedrock surface are convex, and the slope gradient becomes larger and larger from the highest point to the lowest point to. (IV) Gully type slope: The cross section of the slope surface and the bedrock surface is concave, which is similar to the shape of the gully. (V) Ridge type slope: The cross section of the slope surface and the bedrock surface is convex, which is similar to the ridge. (VI) Water collecting type slope: The longitudinal section and the cross section of the slope surface and the bedrock surface are both concave, and this shape forms a water collecting pit. 4) Rockfall simulation (1) Design of rockfall simulation test In order to study the possibility of rockfall of a slope, a simulation was carried out using a variable slope test tank (cited from XU Yongnian). The designed slope test sink was 6-m long, 2-m wide, and 1.2-m high. There was a rain device at the top, the maximum rainfall intensity was set to 100 mm/h, and the rainfall could be adjusted. The artificial rainfall simulation test tank is shown in Fig. 3.8. The test sand was a mixture of natural sand with diameter smaller than 2 mm and powder clay (clay content of about 24%) in a certain ratio. According to the clay contents of 4%, 8%, and 12%, a test grading curve of the test sand, river sand, silty clay, and clay content of 8% is shown in Fig. 3.9. Nozzle

Rainfall device

Slope groove 3.0 m 1.2 m

Elevator Safety equipment Tap water

Fig 3.8 Artificial rainfall simulation test sink.

100 Chapter 3 Lass than a certain particle size (%) 100 80 Power clay

60 River sand

40

Test sand 8%

20 0 10

1

0.1

0.01

IE-3

Particle diameter (mm)

Fig 3.9 Gradation curve of test sand.

Because the soil slope is the most likely to rockfall, and the slope gradient is 30–40 degrees when the rockfall occurs, in order to investigate the slope gradient influence on the rockfall, the slope was set at 30, 35, and 40 degrees. Six different slope morphology tests were carried out with a gradient of 35 degrees. Three gradients and six types of morphology slopes, due to one type of coincidence, were tested in eight groups. Other conditions are shown in Table 3.5. During the test, the variation of pore water pressure and soil movement with cumulative rainfall was measured. The pore water pressure gauge probe was placed at the center line of the slope groove, 1.5 m above the end and 5 cm away from the bottom of the groove. The displacement meter was placed at the center line of the slope groove, 4.0 m above the end, and 40 cm away from the bottom of the groove. The results are shown in Figs. 3.10 and 3.11. (2) Rockfall test results ① Rockfall part of slope The type of slope morphology is related to rockfall occurrence, especially the location of slope rockfall. The ascending type slope falls in the upper part, the descending type slope falls in the lower part, the gully type slope falls on both sides of the upper half, and the ridge type slope falls on both sides of the lower part. ② Forms of slope rockfall It was found that the rockfall process for the descending type slope begins at the bottom and then gradually develops upward. From morphological analysis after the rockfall, it could be considered as a step-shape rockfall. The middle of the ridge type slope is higher, with the effect of scattered water, and the two sides first partially fall. Then, it falls again in the middle, which can be considered a separate rockfall. The gully type slope tends to collect water due to its low middle, and partially falls in the middle, and then rockfall occurs on both sides, which can be called a dendritic-type rockfall. The water collecting type of slope was high in the surrounding

Table 3.5 Test condition summary. Test group No. No. No. No. No. No. No. No.

1 2 3 4 5 6 7 8

Slope shape Parallel Parallel Parallel Descending Ascending Gully Ridge Water collecting

Vertical shape

Vertical curvature

Transverse shape

Flat Flat Flat Convex Concave Flat Flat Concave

R¼∞ R¼∞ R¼∞ 15 m 15 m R¼∞ R¼∞ 5m

Flat Flat Flat Flat Flat Concave Convex Concave

Transverse curvature R R R R R

¼∞ ¼∞ ¼∞ ¼∞ ¼∞ 2m 2m 5m

Slope gradient (°)

Dry density (g/cm3)

Clay content (%)

Rainfall (mm/h)

30 35 40 30 35 35 35 35

1.55 1.65 1.75 1.65 1.65 1.75 1.55 1.55

4 8 12 12 4 12 8 12

40 60 80 60 80 40 80 60

102 Chapter 3 Pressure (kpa) 3

No. 4

No. 5

No. 7

No. 8

No. 1

2 No. 2

1

No. 3 No. 6

0 0

100

200

300

500 600 400 Cumulative rainfall (mm)

Fig 3.10 The pore water pressure of each group varies with the accumulated rainfall. Displacement (mm) 500

No. 2

No. 3

400

No. 4

No. 5 No. 8

300 200

No. 6

0

0

No. 7 No. 1

100 100

200

300

400

500

600

Cumulative rainfall (mm)

Fig 3.11 Soil displacement in each group varies with accumulated rainfall.

area, the middle was low, and the water flowed to the middle. The rockfall shape was similar to a scoop shape. The upper part of the gradient of the ascending slope was larger than that of the lower part, and the rockfall started from the upper part. The shape after the rockfall resembled the shape of a horseshoe. Since the planar type slope was flat, the shape after the fall was close to a planar shape. ③ Slope rockfall speed and volume The timing of the rockfall on a slope is not only closely related to the material that constitutes the slope but is also related to the accumulated rainfall. When rainwater infiltrates, pore water pressure is generated in the soil. When the pore water pressure reaches a limit, the soil loses balance and is destroyed. The scale of the rockfall is due to the thickness and location of the rockfall layer, but the scale of the rockfall directly affects the treatment plan. In the test, a camera was used to observe the seepage condition of the rainwater, the rockfall process, and the rockfall position. The timing and scale of the rockfall of each group are shown in Table 3.6. The peak pore water pressure in the table was the maximum value from the probe set at the center line of the slope groove, 1.5 m above the end and 5 cm from the bottom of the groove. The rockfall time was recorded according to the data from a displacement meter set at the center line of the slope groove, 4.0 m above the end and 40 cm away from the bottom of the groove. The rockfall scale was estimated according to the rockfall range and average depth at the end of

Rockfall mechanisms and block theoretical stability analysis

103

Table 3.6 Slope rockfall timing and scale statistics. Peak water pressure Test Rainfall count (mm)

Water level (cm)

Rockfall time

Seepage Rainfall rate (cm/s) (mm)

Rockfall scale

Speed (cm/ min)

Range (m)

Width (m)

Depth (m)

Volume (m)

Rockfall type Partial rockfall Overall rockfall Surface rockfall Overall rockfall Surface rockfall Surface rockfall Overall rockfall Partial rockfall

No. 1

256.4

21.4

2.5
103

–

–

1.5–3.0

2.0

0.2

0.6

No. 2

230.0

15.7

4.3
103

225.0

37.8

0.0–4.0

2.0

0.4

3.2

No. 3

–

–

–

207.0

32.5

0.0–5.0

1.0

0.3

1.5

No. 4

320.0

28.9

2.7
103

342.0

38.8

0.0–4.0

2.0

0.5

4.0

No. 5

387.0

29.0

6.3
103

389.3

35.0

0.0–4.0

2.0

0.2

1.6

No. 6

319.7

14.5

2.9
103

206.0

33.0

0.0–4.5

2.0

0.3

2.7

No. 7

500.0

28.2

1.0
103

468.0

12.2

0.0–6.0

2.0

0.6

7.2

No. 8

530.0

28.6

1.1
103

536.0

9.8

0.0–6.0

2.0

0.1

0.7

the test. The rockfall type was classified according to the rockfall volume. The partial rockfall volume was less than 1.0 m3, the surface rockfall volume was less than 3.0 m3, and the overall rockfall volume was greater than 3.0 m3. From the results of the rockfall simulation test and the analysis of the field observation data, it can be concluded: ① No. 1 had a partial rockfall during the test with a slope gradient of 30 degrees, while the slope of No. 2 had a total rockfall with a gradient of 35 degrees, which indicated that the larger the gradient, the more likely the occurrence of a rockfall. The slope of No. 3 had a gradient of 40 degrees and, due to the large content of clay and large dry density, it was not easy for the rainwater to infiltrate. Therefore, no pore water pressure was generated and, due to the steep slope, only surface rockfall occurred. ② The test results of various slope morphologies affecting rockfall showed that, for different slopes with the same size, an equal radius of curvature (except for the plane type) and the same soil condition, the risk was: No. 4 descending type > No. 5 ascending type, No. 2 plane type > No. 8 water collecting type, No. 6 gully type and No. 7 ridge type were not very different. ③ Rockfall on slopes was mostly caused by the maximum pore water pressure in the soil. For example, for No. 2 and Nos. 4–8, the pore water pressure increased sharply and was the intrinsic force causing a rockfall. However, No. 1 had pore water pressure, but because the slope gradient

104 Chapter 3 was 30 degrees, there was no rockfall. No. 3 had no pore water pressure, but because the slope gradient was 40 degrees, it also had rockfall. It can thus be seen that the slope gradient was also a key factor affecting slope stability. ④ The displacement of the soil was related to the accumulated rainfall. When the accumulated rainfall reached or exceeded a certain limit, the soil lost stability and rockfall occurred. ⑤ For gully and ridge type slopes, the gully slope tended to collect water due to the lower middle part, and the ridge slope tended to scatter water due to the higher middle part. According to the observed form of rockfall, the middle part of the gully type slope first partially falls and then it falls on both sides, while the sides of the ridge type slope first partially fall and then it falls in the middle.

3.1.5 Regularity of the time of rockfalls The time of rockfall occurred roughly as follows: 1) Rockfall generally occurred in rainy seasons with heavy rainfall. Fig. 3.12 shows the time of major geological disasters in 2005 (43.2% of rockfalls in 2005) and the relationship between geological disasters and monthly rainfall from January to September 2007. 2) Rockfall tended to occur during or after each rain. The rainfall process mentioned here mainly refers to torrential rains, heavy rains, and continuous rainfall for a long time. This was the time when the rockfalls occurred the most frequently. 3) Rockfall tended to occur during a strong earthquake. This mainly refers to the occurrence of rockfalls in the epicentral area (mountain area) during a strong earthquake with magnitude 6 or above. There was little rockfall after earthquakes. 4) Rockfall tended to occur during or after the process of excavating the toe of the slope. Due to the excavation of the toe of the slope in a project (or construction site), the stability of the upper rock mass (soil) was destroyed and rockfall often occurred. Some of the rockfalls occurred during construction, which mostly caused small rockfalls. More rockfalls occurred during the period after construction. 5) Rockfall occurred during the initial stage of reservoir water storage and the flood peak period of the river. At the beginning of a reservoir impoundment or the first peak period of the reservoir water level, the reservoir rock soil was first immersed (softened), and the upper rock and soil body was easily made unstable and rockfalls occurred. 6) Rockfalls occurred after strong mechanical construction and major blasting. It can be concluded that the occurrence of rockfalls was mainly affected by factors such as stratum lithology, structural distribution, vegetation cover, topography, and atmospheric

Rockfall mechanisms and block theoretical stability analysis

105

Time distribution of major sudden geological disasters in 2005 30.0% 26.1% 25.0% 21.2% 20.8% 20.0% 15.0% 9.8% 10.0%

7.9%

5.0% 1.0% 0.9% 1.0%

7.9%

2.2% 0.4% 0.7%

0.0%

Month 1

2

3

4

5

6

7

8

9

10

11

12

(A) 3500

140 mm

3000

120

2500

100

2000

80

1500

60

1000

40

500

20

0 Month 1

(B)

0 2

3

4

5

6

7

8

9

Number of disasters Monthly precipitation

Fig 3.12 Relationship between the occurrence time of geological disasters and monthly rainfall. (A) Time distribution of major outbreaks of geological disasters in 2005. (B) Relationship between geological disasters and monthly rainfall in 2007.

precipitation intensity. Under normal circumstances, lithology fragility, tectonic development, sparse vegetation, and steep terrain contributed to rockfall during heavy precipitation.

3.2 Stability analysis of sliding rockfalls using block theory A rock slope mainly exhibits toppling and slipping damage. This section focuses on slipping rockfall as the research object, and uses block theory to analyze its stability.

106 Chapter 3

3.2.1 Sliding rockfall stability analysis principle The direction, extent, and volume of a slip rockfall are often affected by joint crack cutting. The qualitative study of the stability of rock slope mass is an indispensable step in the whole stability analysis. Compared with quantitative calculation, it is the basis and has a guiding significance for quantitative calculation. Quantitative calculations without qualitative evaluation guidance are often detached from reality and meaningless. Qualitative evaluation methods include the engineering geological conditions analysis method, engineering geological comparison method, and graphic method. This section mainly uses the block theory analysis method based on the graphic method and the vector analysis method developed in recent years. In a specific possible rockfall site, in order to analyze the stability of the potential rockfall, it is necessary to first find the engineering geological conditions of the site and the factors affecting the rockfall, and finally analyze and evaluate the development trends and stability of the site. The basis of stability analysis is to find the engineering geological conditions of the supporting project; any adverse physical geological phenomena are based on certain engineering geological conditions. Among the many conditions for rockfall formation, the most important ones are: ① high and steep topographical conditions and ② specific rock structure conditions (potential rockfalls are cut by various unstable structural faces). With these two conditions of the potential rockfall site, there is a possibility of rockfall. On the contrary, if these two are not present, the rock mass cannot fall. These two are thus necessary conditions for judging potential rockfall. Based on the engineering geological conditions and influencing factors, considering the existing state of the potential rockfall body, one can judge its development trend. For example, if rain is found to be the main factor affecting stability of the potential rockfall on a road, it is necessary to study the climatic phenomena, rainfall distribution, and rainfall in the area. Combined with the engineering geological conditions, we can then judge its development trend and its stability. The essence of the slope stability problem is mainly the contradiction between sliding and antisliding in the stress-strain relationship. The analysis of rock mechanics involves quantitatively examining the relationship between the two opposites and their transformation conditions. The rockfall of a rock slope is the result of a portion of an unstable structure that is pulled along certain structural planes and moves along a certain structure toward a certain space. Therefore, the main boundary conditions for the partial structure to lose balance and slide or fall are as follows: 1) Sliding surface When the rock mass is broken, the structural surface along which large shear stress and frictional resistance are generated is called the sliding surface. The intact rock slope is

Rockfall mechanisms and block theoretical stability analysis

107

deformed due to the force exceeding the strength, resulting in sliding or falling, which rarely happen. If there is no possible sliding surface in the rock mass, it is basically stable; if there is a sliding surface in the rock mass, whether the rock mass can slide or fall depends mainly on the characteristics of the sliding surface. 2) Cutting surface The structural surface between the unstable structure and the rock mass (not subject to large normal stress in sliding failure, does not produce significant frictional resistance, and only pulls along this surface) is called the cutting surface. If there is only a sliding surface in the rock mass and there is no corresponding cutting surface, the sliding or rockfall cannot be formed. However, as a general matter, since the rock mass is often cut by a plurality of sets of structural faces, the cutting surface forms easily. Sometimes the cutting surface also produces a certain frictional resistance, but it is still not the main sliding surface. In this condition, the cutting surface is called the cutting sliding surface. The rock mass of nature is cut into structural bodies of different shapes by the structural plane, but only the structural body cut by the cut surface and the sliding surface is an unstable sliding body or potential rockfall body. Due to these factors, the block theory can be used to analyze the stability of a block rockfall.

3.2.2 Basic concepts of block theory In hard and semihard strata, the rock mass is cut into various types of spatial inlays by the structural plane. In the natural state, these spatial blocks are in static equilibrium. When the rock mass is excavated, some rock blocks will first slip along the structural surface, and then a chain reaction will occur, causing damage to the entire project. Block theory is a method for analyzing the stability of blocks in the fractured rock mass by means of topology, set theory, geometry, and vector algebra. 1) The basic assumptions of block theory are: (1) The structural surface is planar and runs through the concerned rock mass. (2) The structural body is a rigid body. (3) The instability of a rock mass is due to the rock mass first produceing shear displacement along the structural plane under various loads. Based on these assumptions, block theory first considers the structural plane and the excavation free surface as a spatial plane, the structural body as a convex body, and the various acting loads as a space vector, and then applies geometric methods to study the number of block types for rock mass and their mobility in detail, and to give a strict mathematical proof under the given conditions of space and planes.

108 Chapter 3 2) Characteristics of block theory are: (1) Block theory is completely three-dimensional. (2) The core of block theory is to find the key blocks on an excavation surface. For rock mass engineering that needs to maintain stable excavation, it is necessary to carry out engineering treatment before the key block is completely exposed during the excavation process. (3) The research object of block theory has spatial movement with an obvious sliding surface, considering only the shear strength of the structural plane, regardless of the strength, damage, and deformation of the rock mass itself. (4) As with other analytical methods, the reliability of the analysis results of block theory depends on the accuracy of the analysis parameters, first depending on the accuracy of the mechanical parameters c and φ of the structure, and, second, on the structural surface. 3) Basic types of blocks The block can be divided into two categories: finite block and infinite block. The finite block can be further categorized into nonmovable block and movable block. The movable block can then be divided into stable block, possibly unstable block, and key block, as shown in Fig. 3.13. The block generally refers to the rock mass cut by various structural planes and free surfaces. The infinite block is not completely cut into isolated blocks by the structural planes and the free surfaces, as shown in Fig. 3.14, whereas the finite block is completely cut into isolated blocks by the structural planes and the free surfaces; see Fig. 3.15. The nonmovable block, which moves in any direction, is restricted by adjacent blocks. If adjacent blocks do not move, such a block will not be able to move. The stable block remains stable, even when the shear strength of the sliding surface is equal to zero. The potentially unstable block is possibly unstable under the action of the engineering force and the self-weight due to the reduction of the shear strength on the sliding surface. The key block is theoretically unstable without reinforcement measures, because the shear strength on the sliding surface is insufficient to resist the sliding force under the action of the engineering force and self-weight. The block theory research method involves the following steps: first eliminate all infinite blocks and nonmovable blocks by geometric analysis; then through kinematics analysis, identify the blocks that may be unstable under engineering force and self-gravity; then according to the physical and mechanical properties of the sliding surface, determine the key blocks on the excavation face of the project, and provide corresponding engineering reinforcement measures.

Rockfall mechanisms and block theoretical stability analysis

Structural surface

109

Airfront

Airfront

Airfront

Structural surface

(C)

Structural surface

Airfront

(B)

Airfront

(A)

Structural surface

Structural surface

(D)

(E)

Fig. 3.13 2D schematic diagram of block type. (A) Infinite block. (B) Inverted wedge block. (C) Stable block. (D) Potentially unstable block. (E) Key block.

P3 U3

P3 L3

U2 P2 L2

0 U3 0

L3 O

U1L2L3

0

U1

UI

L1

(A)

0

U2 P2 0 L2

0

P1

L1

(B)

P1

Fig 3.14 Infinite block two-dimensional diagram. (A) Infinite block before moving. (B) Infinite block after moving.

110 Chapter 3 P3 U3

P1

P3

0

UI

0

U3

0

L1

L3

0

L3

O 0

U2 L1U2L3

U2 L1

U1 L2

P2 0

P2

L2

O

P1

(A)

(B)

Fig 3.15 Finite block two-dimensional diagram. (A) Finite block before moving. (B) Finite block after moving.

4) Types of pyramids Definition of pyramid: If the structural planes and the free planes in the space are translated to pass through the origin point of the coordinates, the spatial planes will constitute a series of pyramids with the origin of the coordinates as the vertices. There are several types of pyramids: Fracture pyramid: a pyramid in a half-space of rock mass bounded only by structural planes, symbol JP. Excavation pyramid: a pyramid consisting of a half-space of rock mass bounded by free surfaces, symbol EP. Space pyramid: a pyramid in the space outside the space of the excavation pyramid, symbol SP. Block pyramid: a pyramid consisting of several structural planes and more than one free surface in a half-space of rock mass, symbol BP. 5) Block finiteness and mobility principle After the rock mass is cut by the structural planes, infinite blocks and finite blocks are formed. Finite blocks are the blocks of interest. The finite block theorem is: JP \ EP ¼ φ, where φ is an empty set. The mathematical calculation of the finite block can be expressed as follows: let the interfaces of the block pass through the origin point to form a pyramid, which can be expressed by the following inequalities:

Rockfall mechanisms and block theoretical stability analysis A1 x + B1 y + C1 z  0 A2 x + B2 y + C2 z  0 … An x + Bn y + Cn z  0

111

(3.1)

If there is a unique solution (0,0,0) in Eq. (3.1), the block pyramid is an empty set, and the corresponding block is finite. On the contrary, if (3.1) has a nonzero solution, the block pyramid is a nonempty set, and the corresponding block is infinite. 6) Movable block discriminant principle If the block cut by the structural surfaces and the free surfaces is finite and the fracture block cut only by the structural surfaces is infinite, the block is movable. If the block cut by the structural surfaces and the free surfaces is finite and the fracture block cut only by the structural surfaces is also finite, the block is immovable. The mathematical expression is: JP 6¼ φ EP \ JP ¼ φ

(3.2)

7) Labeling method of block or pyramid There are three methods for labeling blocks or pyramids: visual annotation, numerical numbering, and symbolic numbering. (1) Visual annotation Let Ui and Li denote the upper and the lower half space of plane Pi, respectively. There are a group of structural planes P1, P2, P3, P4 and free plane P5 in the rock mass. If block B1 consists of the lower half of P1, P2, and P3 and the upper half space of P4, then B1 can be marked as L1L2L3U4. If block B2 consists of the upper half space of P1 and the lower half space of P3, P5, then B2 can be marked as U1L3L5. This method of labeling is intuitive but is not suitable for analysis and calculation. (2) Numerical numbering The relationship between each structural plane, the free surface, and the block in the rock mass is represented by four numbers 0, 1, 2, and 3, where “0” indicates that the block is in the upper half space of the plane; “1” indicates that the block is in the lower half of the plane; “2” indicates that the plane is not the interface of the block; “3” indicates that a pair of interfaces parallel to each other formed in the block. By using the numerical numbering method, all possible combinations of the structural and free surfaces of the rock mass can be clearly marked for analysis.

112 Chapter 3 Table 3.7 The formula for the total number of fractured blocks, the number of infinite fractured blocks, and the number of finite blocks. Number of parallel structure planes

Total number of fractured blocks

Each set of structural planes is not parallel A set of parallel structural planes is determined Any set of parallel structural planes Two sets of parallel structural planes are determined Any two sets of parallel structural planes m sets of parallel structural planes are determined Any m sets of parallel structural planes

2

Total number of infinite blocks 2

Total number of finite blocks 2

Condition

n n+2

2  (n + n  2)

n 1

2n1

2(n  1)

2n-1  2(n  1)

n 2

n2n1

2n(n  1)

n[2n-1  2(n  1)]

n 2

2n1

2

2n2  2

n 3

n(n  1)2n3

n(n  1)

n(n  1)(2n3  1)

n 3

2nm

0

2nm

n m 3

nm Cm n2

0

nm Cm n2

n m 3

n

n

(3) Symbolic numbering In order to facilitate logical operations on the computer, the upper half space is represented by “+1”, the lower half space is represented by “1”, and the pair of parallel interfaces of the block are represented by “1”. 8) Determination of the total number of fractured blocks After the rock mass is cut by the structural planes, fractured blocks are formed. Most of them are infinite blocks and a few are finite blocks. Table 3.7 shows the calculation formulas for the total number of fractured blocks, the number of infinite fractured blocks, and the number of finite blocks in several cases.

3.2.3 Vector discriminating steps in block theory The basic steps in using vector analysis to identify fractured blocks are as follows:

Rockfall mechanisms and block theoretical stability analysis

113

(1) Considering n groups of structural planes, the normal vector n^i in the upward direction of Pi can be calculated based on the inclination angle αi and the tendency βi of Pi: n^i ¼ ðAi , Bi , Ci Þ ¼ ð sin αi sinβi , sinαi cos βi , cos αi Þ

(3.3) !

(2) Find the intersection line of structural planes, represented by the edge vector I ij . ! I ij




x y z


     ! ! ¼ n i
n j ¼

Ai Bi Ci

¼ Bi Cj  Bj Ci , Aj Ci  Ai Cj , Ai Bj  Aj Bi
Aj Bj Cj


(3.4)

In the formula: n^i ¼ ðAi , Bi , Ci Þ n^j ¼ Aj , Bj , Cj

(3.5)

The total number of edge vectors is C2n, which is n ! /[(n  2) ! 2!] (Table 3.7). !

For each possible intersection of the structural planes, calculate whether each edge e ij is the true edge of the pyramid. When the dot product of a vector and an upward normal vector of a plane is greater than zero, the vector is in the upper half space; conversely, if the dot product is less than zero, the vector is in the lower half space. To facilitate the analysis, the concept of "direction parameter" Iijk is introduced:    Ikij ¼ sign n^i
n^j  n^k ði 6¼ jÞ

(3.6)

In the formula: when the function

8 8 0 < +1 L ¼ ¼ 0 , signðLÞ ¼ 0 : :

< +1, Explain that the edge vector Ik is in the upper half of the interface Pk ij Ik 0, Explain that the edge vector Ikij is just on the interface Pk > : 1, Explain that the edge vector Iij is in the lower half of the interface P k k Each direction parameter constitutes a “direction parameter matrix” [Iijk ]C2n
n, and the symbol numbers Ds ¼ [I(a1),I(a2),…,I(an)] of a block are compiled into the following “block symbol number matrix” [D]:


I ða1 Þ 0



(3.8) ½Dn
n ¼

0 I ða2 Þ

I ðan Þ


114 Chapter 3 Therefore the block “discrimination matrix” [T] can be established: ½T C2n
n ¼ ½IC2n
n  ½DC2n
n

(3.9)

According to Eq. (3.8): h ! i Tkij ¼ Ikij  Iðak Þ ¼ sign I ij  n^  I ðak Þ h! i ¼ sign I ij  ðIðak Þ  n^k Þ

(3.10)

Then the discriminant matrix [T] of a certain BP is obtained, as shown in Table 3.8. Here n^k is the upward normal vector of interface Pk, and v^k ¼ Iðak Þ  n^k is the normal vector of interface Pk pointing to interior of BP. (3) Vector operation discrimination method for block mobility The necessary and sufficient conditions for block mobility are JP 6¼ φ and EP \ JP ¼ φ, or JP  SP. The force acting on the block is shown in Fig. 3.16. !

① The resultant force of main forces r , composed of the block’s own weight, external water pressure, inertial force, and anchor reinforcement force. !

② Normal reaction force on the sliding surface N : ! X Nl v^l N¼

(3.11)

l

In the formula: Nl is the normal reaction force acting on the sliding surface l; it is assumed that the structural surface does not have tensile strength, so Nl 0; and v^l is the unit normal vector of the structural plane l, which points to inside the block. !

③ The tangential frictional resistance on the sliding surface T : X ! T¼ Nl tgϕl S^

(3.12)

l

Table 3.8 Discriminant matrix [T] for BP. A line element Iij in [T]

Judge of the edge Iij

Both are “0” or both “+1” and “21” Both are “0” and “+1” Both are “0” and “21”

Iij is not the edge of BP, the block is movable Iij is the true edge of BP, the block is immovable Iij is the true edge of BP, the block is immovable

Rockfall mechanisms and block theoretical stability analysis

115

Fig 3.16 Force acting on the movable block.

In the formula: ϕl is the internal friction angle of the structural face l and S^ is the block movement direction. ! To judge the state of the block: setting the tangential force F on the sliding surface, the equilibrium equation of the block is: X ! Nl v^l  T S^ (3.13) FS^ ¼ r + l

Therefore, from Eq. (3.13), it can be determined that if F > 0, the block is a key block; if F ¼ 0 it is in a limit equilibrium state; F < 0 indicates that the block is in a stable state. There are three forms of motion of the block: movement away from the rock mass (drop or lift), movement along one side, and movement along both sides, as shown in Fig. 3.17. ①

Detachment movement from rock mass

If the moving direction S^ of the movable block is not parallel to each structural surface, the necessary and sufficient condition for the block to satisfy the equilibrium equation (3.13) is that the moving direction is consistent with the direction of the resultant force of the main forces, as shown in the following: r^ ¼ S^ ②

(3.14)

Sliding along one side

If the moving direction S^ of the movable block is parallel to a certain structural plane i, the necessary and sufficient conditions for the block to satisfy the equilibrium equation (3.13) are: ! S^i ¼ S^ and v^i  r 0

(3.15)

116 Chapter 3

S

(A)

(B)

S=n

S

(C) Fig 3.17 Block motion diagram. (A) Detaching rock mass movement. (B) Sliding along one surface of the block. (C) Sliding along two surfaces of the block. ! where S^i is the projection of r on plane i.

③

Sliding on both sides

If the movable block moves along planes i and j at the same time, the necessary and sufficient conditions for the block to satisfy the equilibrium equation (3.13) are: v^i  S^j 0, v^j  S^i 0

(3.16)

h i ni
^n^j ! sign n^i
n^j Þ: r S^ ¼ |ni
nj |

(3.17)

! where S^i and S^j are the projections of r on planes i and j, respectively.

3.2.4 Calculation of stability coefficient for movable blocks 1) Calculation of movable block weight After determining that the block is a movable block, it is necessary to calculate the stability coefficient to determine whether it is an unstable block or a stable block. The movable block

Rockfall mechanisms and block theoretical stability analysis

117

can be divided into several tetrahedrons. As long as the volume of the tetrahedron can be calculated, the volume of the movable block can be obtained. Further, the weight of the movable block is obtained. Set the coordinates (xi, yi, zi) of the four corners of the tetrahedron; then its volume is:


x2  x1 y2  y1 z2  z1

1
(3.18) V ¼

x3  x1 y3  y1 z3  z1

6
x4  x1 y4  y1 z4  z1
The vertex (xi, yi, zi) of the tetrahedron can be solved by the equation of the structural plane constituting the tetrahedron. Let the structural plane be Pi, Pj, Pk, and the equation is: 8 < A i x + B i y + Ci z ¼ Di A x + B j y + Cj z ¼ Dj : j Ak x + Bk y + Ck z ¼ Dk

(3.19)

From this formula, (xi, yi, zi) can be found, so the weight of the block can be solved: W ¼ ΣVi γ

(3.20)

In the formula: γ is the rock mass density 2) Block stability coefficient calculation After determining the weight of the movable block and determining the possible slip mode, the stability factor calculation can be performed. (1) When the movable block rockfalls directly Stability factor: K¼0

(3.21)

(2) When sliding on one side K¼

W cos αtan ϕ + cΔ W sin α

(3.22)

In the formula: W is the Movable block weight; α is the sliding surface inclination angle; c, ϕ is the sliding surface cohesion and internal friction angle; Δ is the sliding surface area. (3) When sliding on both sides

K¼

W cos αð sin α2 tan ϕ1 + sin α1 tan ϕ2 Þ + ðc1 Δ1 + c2 Δ2 Þ sin ð180°  α1  α2 Þ W sin α

(3.23)

118 Chapter 3 In the formula: α is the inclination angle of the intersection of two sliding surfaces. α1 is the angle between the normal of intersection and the sliding surface P1. α2 is the angle between the normal of intersection and the sliding surface P2. The rest of the symbols have the same meaning as before.

3.2.5 Correction of block theory In the general block theory, assuming that the joint passing through the investigated rock mass is infinitely long is far from reality, because the length of the joint exposed in the project is limited. Since the length of the joint is finite, the block is not completely cut, and sometimes the fractured block cannot be formed. Therefore, a portion of the finite blocks determined by the block theory will become infinite blocks, which will greatly reduce the number of key blocks of the excavated rock mass in actual engineering. In addition, the key blocks group is composed of parallel structure planes. The volume of each key block is different, and the stability coefficient is different. During the excavation process, some key blocks slip and some are not destroyed. Block theory does not consider all aspects of the structural plane. Therefore, it is necessary to revise the block theory. There have been many studies on the joints of rock masses. According to the research results of Chowdhury, it can be found that: (1) The joint trace is subject to a negative exponential distribution. (2) The position of the joint is the same as the position where the block edge is located. The length of the joint follows a negative exponential distribution, and the density function of its probability distribution is: f ðxÞ ¼ μeμx

(3.24)

In the formula: μ is the joint trace endpoint density, which is the reciprocal of the trace length. The cumulative probability is: FðxÞ ¼ 1  eμx

(3.25)

Taking the triangular pyramid as an example, the probability of the key block slipping is analyzed. Assuming that the edge (I) length of the triangular pyramid is a, the length of the structural surface containing the edge I in the pyramid must be greater than a; otherwise the pyramid will not be formed. Therefore, the probability of occurrence of the pyramid can be regarded as the probability that each joint surface is longer than a and appears at the same time. This means that

Rockfall mechanisms and block theoretical stability analysis

119

the probability of sliding for the key block should be equal to the product of the probability of the length of each joint in the joint surfaces constituting the block being larger than the length of the block edge. The calculated result using formula (3.25) is the cumulative probability value. It is the probability value that the joint length is less than x. The probability F1(x) whose trace length is greater than x can be calculated by Eq. (3.26): F1 ðxÞ ¼ 1  FðxÞ ¼ eμx

(3.26)

For the three sets of joints, if the density of the endpoints of each set of joints is respectively μ1, μ2, μ3, then the lengths of three edges formed by the three sets of joint surfaces are respectively x1, x2, x3, and the probability that the three joint lengths are greater than x1, x2, x3 at the same time is: F2 ðxÞ ¼ eðμ1 x1 + μ2 x2 + μ3 x3 Þ

(3.27)

From Eq. (3.27), the probability of the falling key block of the triangular pyramid can be found. Extend Eq. (3.27) to quadrangular pyramids or pyramids with more joint faces. Let the density of the endpoints of each set of jointed traces be μ1, μ2, μ3, …, μn; then the probability of the n faces pyramid whose edge lengths are respectively x1, x2, x3, …, xn becoming a slip block is: F2 ðxÞ ¼ eðμ1 x1 + μ2 x2 + ⋯ + μn xn Þ

(3.28)

From Eq. (3.28), it can be found that the longer the structural plane is, the smaller μ is and the greater the probability F2(x) that the key block becomes a sliding block. If μ is constant and the length of the structural plane is constant, the smaller the edge length x of the block is, the greater the probability F2(x) that the key block becomes a sliding block. This means that the smaller the key block volume, the easier it can become a sliding block. The fewer the surfaces that make up the pyramid, the greater the probability of sliding of the key block.

3.2.6 Stereographic projection method of block theory As mentioned previously, the failure mode of a rockfall is mainly controlled by the structural plane. Grasping the geometric features of the structural plane is the key to correctly determining the possible instability mode of the slope. In engineering geology, the inclination angle and the tendency of the structural planes are commonly used to represent the spatial morphology of the structural planes. With the stereographic projection method, it is reasonable to simultaneously display the parameters of the inclination angle and the tendency on one plane.

120 Chapter 3 Using the stereographic projection method, it is easy to visually identify the movable block in the slope. The mapping and discrimination methods are as follows: (1) Select the reference circle radius R and draw a reference circle. (2) According to the inclination angle αi and the tendency βi of the structural plane Pi, the corresponding projection large circle is drawn, and the drawing method is as follows: the origin point of the rectangular coordinate system is the same as the reference circle center, east is the x-axis, and north is the y-axis. The following equations can be obtained. r ¼ R=cos α

(3.29)

x ¼ Rtan α sinβ

(3.30)

y ¼ Rtan α cosβ

(3.31)

In these equations, R is the radius of the projection circle, and x and y are the coordinates of the center of the projection circle. (3) Given the JP number, each projection circle is divided into a number of small areas, and each small area corresponds to a nonempty JP. The specific numbering method refers to the relevant literature. (4) According to step (2), the projection large circle of each free surface is given to find the corresponding SP region, and each JP completely included in the SP region is a movable block.

3.2.7 Compilation of program for block theory The block theory calculation process involves a large number of matrix operations, and all the numerical calculation functions of MATLAB are based on the matrix. MATLAB has the most complete and strong operation functions on matrices, and has the following advantages: conciseness, compactness, ease of use, and flexibility, and its library functions are rich and reliable. MATLAB also has the following abilities: provides almost as many operators as the C language; includes structured control statements; features of object-oriented programming; fewer grammatical restrictions; large freedom of programming; good portability of programming; powerful graphics functions; and a powerful toolbox. Based on these factors, MATLAB was selected for the block theory analysis program.

3.3 Stability calculation for potential rockfalls 3.3.1 Basic assumptions (1) Consider the movement of the rockfall body as a whole during the development of the rockfall, especially before the sudden rockfall movement.

Rockfall mechanisms and block theoretical stability analysis

121

(2) Simplify the complex spatial motion problem of the rockfall body into a plane problem, which means taking the unit width of the rockfall body for checking. (3) There is no friction between the two sides of the rockfall body and the stable rock mass, and between the various rockfall bodies.

3.3.2 Basic patterns, force analysis and stability check Force analysis and stability check are performed according to the five types of rockfall in the classification of Table 3.1. K is the stability factor. 1) Toppling type rockfall The basic pattern of a toppling rockfall is shown in Fig. 3.18. It can be seen from Fig. 3.18A that there are cracks between the upper and lower parts of the unstable rock mass and the stable rock mass. Once the toppling occurs, it will rotate with point A as the turning point. In the stability check, the most unfavorable combination of possible additional forces should be considered. In the rainy season, open cracks may be filled with heavy rain, and hydrostatic pressure should be considered. In earthquake areas above VII degrees, seismic problems may also be considered. See Fig. 3.18B for the force pattern. If other forces are not considered, the antioverturn stability coefficient K of the rockfall body can be calculated as follows: K¼

6aW 10h20 + 3Ph

(3.32)

In the formula: W is the rockfall weight (kN/m2); P is the horizontal seismic force (kN); a is the vertical distance from the turning point to the extension line of gravity, and it is half the width of the rockfall body here (m); h0 is the height of water level (m); and h is the height of rock mass (m). 2) Sliding type rockfall The sliding rockfall can be checked according to the sliding stability, which can be calculated according to the method in block theory.

h

2a

f A

(A)

p w

(B)

Fig 3.18 Toppling type rockfall. (A) Basic diagram of toppling rockfall. (B) Calculation diagram of toppling rockfall.

122 Chapter 3 3) Bulging type rockfall The thick and weak rock layers in the lower part of this type of rockfall body are often fault fracture zones, and weathered fractured rock masses. Under the action of water, these weak rock layers are softened first. Under the action of the upper rock mass, if the compressive stress is greater than the unconfined compressive strength of the weak rock stratum, the soft rock layer will be squeezed out, resulting in bulging. The upper rock mass may sink, slip, or topple until a sudden rockfall occurs, as shown in Fig. 3.19. Therefore bulging is the key to this type of rockfall. The stability factor can be calculated from the ratio of the unconfined compressive strength of the lower soft rock layer (the saturated water compressive strength in the rainy season) to the compressive stress generated by the upper rock mass on the soft rock top surface: K¼

Rn A  Rn ¼ W=A W

(3.33)

In the formula: W is the upper rock mass weight (kN); A is the bottom area of the upper rock mass (m2); and Rn is the unconfined compressive strength of the lower soft rock in the natural state (it is saturated in the rainy season). 4) Tension type rockfall A typical case of a tension rockfall is shown in Fig. 3.20. The rock mass existing in the form of a cantilever beam bears the maximum bending moment and shearing force on the AC surface, and the top of the rock layer is pulled and the bottom is pressed. The tensile stress near point A is

Fig. 3.19 Bulging type rockfall.

Rockfall mechanisms and block theoretical stability analysis

123

Fig 3.20 Tension type rockfall.

the largest. Under the action of long-term gravity and long-term weathering, the crack near the point A gradually expands and develops into a deep crack. The pulling force will be more and more concentrated in the part that has not been cracked. Once the tensile stress exceeds the tensile strength of the rock, the upper rock mass falls. Therefore, the key to this type of rockfall is whether the tensile stress on the maximum bending moment section AC exceeds the tensile strength of the rock. The stability can be checked by the ratio of tensile stress to the allowable tensile strength of the rock. If the length of the prominent rock mass is l, the rock mass is equal in thickness, the thickness is h, the width is 1 m (take the unit width), and the rock bulk density is γ, when there is no crack on the AC section, the tensile stress at point A is: σA ¼

My I

(3.34)

In the formula: M is the moment of the AC surface, M ¼ 2l γh, y ¼ h2; I is the moment of inertia of h the AC section, I ¼ 12 ; and γ is the rock density. 3

Then, it can be deduced: σA ¼

3l2 γ h

(3.35)

The stability factor K value can be obtained by the ratio of the allowable tensile strength of the rock to the tensile stress at point A: K¼

½σ  σA

(3.36)

124 Chapter 3 If there is a crack at point A, the crack depth is a, and the lowest point of the crack is B, then the 3

Þ l γh ha moment of inertia I ¼ ðha 12 , y ¼ 2 , and bending moment M ¼ 2 on the BC section, then the tensile stress at point B is: 2

l2 γh h  a  2 2 ¼ 3l γh σ¼ 2 ðh  aÞ3 ðh  aÞ2 12

(3.37)

The stability factor is: K¼

½σ  σB

(3.38)

5) Staggered type rockfall Fig. 3.21 shows the situation of a staggered rockfall. The rock mass ABCD, which may become a rockfall in the figure, is analyzed. As shown, the potential rockfall block is directly connected to the stabilized rock mass. If the additional force such as water pressure and seismic force is not considered, the maximum shear stress will be generated in the EC direction at an angle of 45 degrees to the vertical direction under the action of the rock mass. If the CD height is h, the AD   width is a, and the rock mass density is γ, then the rock mass AECD weight is W ¼ a h  a2 γ.   The normal stress on the rock mass cross-section FOG is γ h  a2 , so the maximum shear stress   τmax on the EC surface is 2γ h  a2 . Therefore, the rock mass stability coefficient K value can be calculated by the ratio of the allowable shear strength [τ] of the rock to τmax: K¼

½τ  4½τ ¼ τmax γ ð2h  aÞ

(3.39)

a A

D

A D

E E B

F C

B

O G C

Fig. 3.21 Staggered type rockfall. (A) Basic diagram of staggered rockfall. (B) Calculation diagram of staggered rockfall.

Rockfall mechanisms and block theoretical stability analysis

125

In the formula: [τ] is the rock allowable shear strength (kPa); γ is the rock density (kN/m2); h is the height of rock mass (m); and a is the width of rock mass (m).

3.4 Chapter summary (1) Through the analysis of the internal and external causes of rockfall on a slope, the mechanism of rockfall geological disasters is discussed, and the time law of the rockfall is summarized. (2) The premise, basic theory, and method of analyzing the stability of a rock slope by general block theory are introduced. The stability of a potential rockfall block on a slope is analyzed by block theory methods. The analysis program can make it easier to find the movable block that affects the stability of the slope, and the stereographic projection map of each structural plane can be used to determine the potential rockfall area of the slope. (3) Block theory is an effective method to analyze the stability of a potential collapsing rock mass, but due to the basic assumption that the joint runs through the researched rock mass, the block study can be considered only in the qualitative analysis stage. In actual engineering, the length of joints is finite. Therefore, considering the finiteness of the length of joints, the theoretical analysis of rock slope stability is more accurate and more targeted. (4) According to the analysis of the trace-length probability distribution model of the joint surface, the key block with larger volume is less prone to rockfall. When the structural surface is longer, and the value of μ is smaller, the probability F2(x) of the key block becoming a sliding block is greater. If μ is constant, and the length of the structural plane is constant, the smaller the edge length x of the block is, the greater the probability F2(x) of the key block becoming a sliding block. The smaller the volume of the key block, the easier it is for the key block to slip. The fewer faces that make up the pyramid, the greater the probability of the key block slipping. (5) The stability calculation formula for single-sided and double-sided rockfall modes are derived. The rockfall is divided into five types of deformation failure methods: toppling type, sliding type, bulging type, tension type, and staggered type rockfalls. The stability calculation methods for various rockfall forms are introduced.

CHAPTER 4

Potential hazard prediction of rockfalls 4.1 Introduction After a rockfall from a slope, the collapsed rock rolls or falls along the slope, and finally accumulates on the slope, seriously affecting the safety of driving and buildings. Hazard studies of rockfall bodies are an important type of slope engineering problem. However, research on slope problems has been mainly based on the stability of the slope as a whole, but the problem of rockfalls on slopes has been less frequently studied. With the rapid development of national economies around the world, the problem of rockfall has become more and more prominent. The study of potential hazard prediction of rockfalls is of great significance for further study of their destructive power along with prevention and control measures. The key to the analysis of rock movement is to determine the speed of the rockfall movement along with its trajectory and destructive force, and to provide a reliable basis for protective designs.

4.2 Research methods for predicting potential hazards of high-speed railway landslides The prediction of the potential hazards of rockfall involves the study of rockfall movement problems on a slope. At present, research and analysis methods for rockfall movement problems on a slope can be summarized into two categories: the empirical method based on experimental research, and the theoretical method based on theoretical derivation. The empirical method mainly includes field test research and indoor simulation model test research. The experimental research method determines the basic physical and mechanical parameters, and aims at understanding the important methods of studying the slope rockfall problem. The data from experimental research has the characteristics of accuracy, objectivity, and comprehensiveness. Undoubtedly, a certain amount of test data is indispensable for studying the problem of rockfalls, and it is also the basis for correctly understanding the rockfall problem. However, experimental research data is not systematic and is prone to strong regional limitations, which causes the results of the test methods to be of less engineering significance. The theoretical derivation method is based on kinematics and dynamics theory to establish a reasonable mathematical calculation model. With the development of computer technology, computer-aided analysis methods in rockfall research have been developed. Some countries Rock Mechanics and Engineering. https://doi.org/10.1016/B978-0-12-822424-3.00004-9 # 2021 Central South University Press. Published by Elsevier Ltd. All rights reserved.

127

128 Chapter 4 began research in this field in the 1980s, and the work is still evolving. For example, CRSP (Colorado Rockfall Simulation Program) is a well-developed computer-aided analysis system. Understanding the causes of rockfall and mastering its possible motion characteristics are prerequisites for successful design of prevention and control measures. The most important design parameters are the impact energy and motion trajectory of a collapsed rockfall, which can be determined by theoretical calculation, numerical simulation, or field test. Rock kinetic energy and bounce height are used to select control measures; this is better for construction safety, lowering project cost, and improving line quality. Generally, engineering measures such as avoidance, protection, roofing, anchoring, and blocking are adopted, supplemented by drainage, vegetation greening, etc. to prevent rockfalls and to ensure safe driving. For the sake of simplification in modeling, the following assumptions are made for the slope rockfall problem: (1) Simplify the rockfall movement as a two-dimensional motion problem, ignoring the interaction between the rockfall. (2) Assuming the shape of the slope is known, the slope equation is f(x, y) ¼ 0; the initial condition of the rockfall movement is determined by field investigation and is considered a known condition in the derivation. (3) Consider only the case where the rockfall is a sphere, and the sphere degenerates into a circle in two-dimensional coordinates. It is also assumed that the mass of the rockfall is evenly distributed, and the rockfall rotates around its center. (4) The movement of the rockfall is divided into several main forms of movement. Each form of movement has its own characteristics and assumptions. The actual form of movement can be summarized into one or a combination of several forms of movement.

4.3 Kinematic equation of a rockfall and determination of its parameters There are five main types of rockfall movements on the slope: (1) falling of rock; (2) sliding of rock; (3) free fall of rockfall; (4) bounce and collision of rockfall; and (5) rolling of rock (Fig. 4.1). For a specific slope, these types of rockfall movements may be combined, depending on the shape of the slope, the geomechanical characteristics of the slope, and the mechanical properties of the rock. The five forms of movement are explained separately.

4.3.1 The falling of a rock The fall is caused by the weathering and erosion of the rock formation and decomposition into smaller rock fragments or cuttings. As rain continues to wash and erode, the rock or debris will collapse and cause the rock to fall. Rocks or earthy stones of different sizes are affected by gravity and fall from steep slopes or cliffs in a free fall. Rockfall often occurs on steep slopes or

Potential hazard prediction of rockfalls 129

Fig. 4.1 Various forms of rockfall movement on a slope.

narrow, high road slopes. Due to the large impact of rockfalls, they often destroy structures on the side of the road, burying houses, blocking traffic, and causing casualties. As mentioned earlier, rockfall movement is characterized by a vertical component in the particle displacement vector that greatly exceeds the horizontal component. Moreover, the fall body is completely detached from the mother rock. In the case of a cliff, the block displacement obeys the law of free fall motion, so its falling speed depends on the height at which the rockfall begins, and the velocity expression is: pffiffiffiffiffiffiffiffi V ¼ 2gh (4.1) In the formula: V is the straight drop speed of the rock block; g is the gravitational acceleration; h is the height of the rockfall when it begins. If it does not collide with protruding rock during the caving process, the rock will fall straight. Its destructive power can be measured by its kinetic energy. According to the knowledge of kinematics, the kinetic energy of its landing is: 1 P ¼ mV 2 2

(4.2)

In the formula: P is the destructive power of the rock mass (i.e., the energy at the time of landing) and m is the quality of the straight rock mass.

130 Chapter 4

4.3.2 Sliding of rockfalls At the beginning and end of the rockfall movement, especially on slopes where the slope is gentle and smooth, sliding rockfalls may occur. This kind of slope is generally flat and not very steep. The sliding distance varies from a few centimeters to several meters. Due to the existence of sliding friction, the speed of a rockfall in the sliding stage is often low and the energy loss is large. For the sliding of the initial stage of the rockfall movement, the design is concerned with the speed at the end of the sliding (that is, the initial speed V0 of the rockfall after the free fall). Assuming that the rockfall starts to slide under the action of gravity, the speed at the end of the sliding is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ 2gð sin α  μ cosαÞs (4.3) In the formula: g is the gravitational acceleration; μ is the coefficient of sliding friction can be determined by a friction test or experience in the field; α is the slope of the sliding section of the slope; and s is the sliding distance. For sliding at the end of the rockfall movement, the design is concerned with the final sliding distance s and assessing the impact of the rockfall disaster. Assuming that the initial velocity of the rockfall before sliding is V0, then the rock stops sliding, and the total sliding distance s can be estimated by the following formula: s¼

V02 2g  ð sin α  μcos αÞ

(4.4)

The parameters in the formula are the same as before.

4.3.3 Free falling of a rock Free falling of rock is related to the shape of the slope section and the initial movement form of the rockfall. In places where the slope angle changes, and after the collision (as shown in Fig. 4.2), the rockfall will often form a flying fall. Under the action of gravity, the gravitational potential energy of rockfall is converted into kinetic energy. Ignoring the influence of air resistance during the flight of a rockfall, the free falling of a rockfall can be described as a simple oblique (flat) throwing motion, and the trajectory is a parabola between a series of collision points. For the free falling of rockfall, the design concerns the location of the collision point, the incident velocity of the collision, and the impact velocity and height of the rockfall at the protection structure setting. As shown in Fig. 4.2, at t ¼ t0, it is assumed that the rockfall is located at point O (x0,y0) in the figure, and the velocity is V0.

Potential hazard prediction of rockfalls 131

Fig 4.2 Rockfall free flying model.

Then at point t ¼ t0 + Δt, the coordinates (x, y) of the rockfall are: x ¼ V0x  Δt + x0

(4.5a)

1 y ¼  gΔt2  V0y  Δt + y0 2

(4.5b)

By eliminating the Δt in the preceding two equations, the equation of motion of the rockfall can be obtained: 2 2V0x V0y 2V0x ðy  y0 Þ + ðx  x0 Þ2 + ðx  x0 Þ ¼ 0 g g

(4.6)

f ðx, yÞ ¼ 0

(4.7)

Slope equation:

The coordinates (x*, y*) of the collision point C can be obtained by solving the simultaneous Eqs. (4.6), (4.7). At the same time, it is easy to obtain the incident velocity Vi of the rockfall at the collision point C: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi ∗ 2 + V + g x  x0 Vi ¼ V0x (4.8) 0y V0x Once the position of the collision point and the incident velocity of the collision are determined, the collision of the rockfall and the subsequent motion trajectory can be further analyzed. In terms of design, in addition to knowing the location of the collision point and the incident velocity, it is also necessary to know the impact velocity and height of the rockfall at the protective structure. Knowing the impact velocity of the rockfall can help in further analyzing the impact energy of the rockfall on the protective structure, and based on this, rationally

132 Chapter 4 designing the protective structure. Knowing the height of the impact, the height of the protective structure can be reasonably designed to avoid the failure of the rockfall to fly over the protective structure and invalidate the protective structure. As shown in Fig. 4.2, assuming that the protective structure is set at point D on the slope, the coordinates are (xD, yD), and the flight path is controlled by the starting point at the collision point C, which will be x0 ¼ x*, y0 ¼ y*, V0x ¼ V1x, V0y ¼ Vy, x ¼ xD. Substituting these into Eqs. (4.5a), (4.5b), it is simple to obtain:     1 xD  x∗ 2 xD  x∗  V1y  (4.9) + y∗  yD h¼ g 2 V1x V1x sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   xD  x∗ 2 2 V ¼ V1x + V1y + g (4.10) V1x In the formula: h is the impact height of the rockfall at the protective structure setting and V is the impact velocity of the rockfall at the protective structure.

4.3.4 Bounce and collision of rockfalls The bounce and collision of rockfall occurs when the rock encounters the slope during a free fall. This is the most complicated and uncertain part of the rockfall movement. The collision may be a nearly completely elastic collision or a completely inelastic collision, depending on the rockfall and slope rock, the physical and mechanical properties of the soil, the angle of incidence at the time of collision, the mass of the rockfall, and the incident velocity. We use the recovery factor method to describe the problem of falling rock collisions. Taking the collision problem of rockfall as a rigid body collision, the energy loss in the collision process is considered by the recovery coefficient, and direct discussion of the nonlinear deformation and friction problem during the rockfall collision is avoided. As early as the Newton era, the recovery factor was used to describe the collision of objects. The recovery factor indicates the degree of velocity recovery after an object collides, and also indicates the degree of deformation recovery, and reflects the degree of mechanical energy loss during the collision. The most commonly used recovery coefficients are the normal and tangential recovery coefficients: Rn ¼

Vm Vin

(4.11)

Rt ¼

Vrt Vit

(4.12)

In the formula: Rn, Rt are the normal and tangential velocity recovery coefficients; Vin, Vit represent the normal and tangential velocity before collision; Vrn, Vrt represent the normal and tangential velocity after collision.

Potential hazard prediction of rockfalls 133

Fig 4.3 The model of the bounce and collision of rockfall.

When Rn and Rt are 1, there is no friction damping during the collision, and the collision is a completely elastic collision; when Rn and Rt are 0, it is a completely viscous damping state, and the collision is a completely inelastic collision. Many scholars currently adopt this definition. For rockfall collisions, the design is most concerned with the speed after the collision and the rolling problem after the collision. As shown in Fig. 4.3, the incident velocity Vi of the rockfall collision has been determined by the previous calculations and is used here as a known condition. According to the decomposition of the velocity vector  Vit ¼ Vx∗  cos α + Vy∗  sin α Vin ¼ Vx∗  sin α + Vy∗  cos α  Vrt ¼ V1x  sinα + V1y  cosα Vrn ¼ V1x  cos α  V1y  sin α

(4.13) (4.14)

where Vit, Vin are the tangential and normal components of the rockfall velocity before collision; Vrt, Vrn are the tangential and normal components of the rockfall velocity after collision respectively; V1x, V1y are the levels of rockfall speed after collision, respectively. Vertical component; the angle between the collision tangent plane and the x-axis. The size  0  α is  ∂f ∂f 1 fx  of α is tan and fy0 ¼ ∂y , and f is the slope equation. f 0 ðx∗ , y∗ Þ  , where fx0 ¼ ∂x y

When combined, Eqs. (4.11)–(4.14) can be used to obtain the horizontal and vertical components V1x, V1y, and Vr of the rockfall speed after collision.

(4.15a) V1x ¼ Rt cos 2 α  Rn sin 2 α  Vx∗ + ðRt + Rn Þ  sinα cos α  Vy∗

134 Chapter 4

V1y ¼ ðRt + Rn Þ  sin α cos α  Vx∗ + Rt sin 2 α + Rn cos 2 α  Vy∗ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 + V2 Vr ¼ V1x 1y

(4.15b) (4.16)

The rockfall collision recovery coefficients Rn and Rt are two important parameters for correctly estimating the trajectory of the rockfall. These two parameters are related to the slope, the geomechanical properties of the slope cover, and the size of the rockfall itself. Laboratory tests show that the normal recovery coefficient increases slightly with the increase of slope, but the effect on the tangential recovery coefficient is not obvious; the more the slope is covered with soil, the more complete the collision becomes. For inelastic collisions, the corresponding normal and tangential recovery coefficients are smaller; on the other hand, the harder the exposed bedrock on the slope, the more the collision tends to an elastic collision, and the corresponding normal and tangential recovery coefficients are larger; a smaller mass of rockfall has a larger collision recovery coefficient than massive rock falling on the same slope. The value of the recovery coefficient is directly related to the reliability of the rockfall trajectory estimated on this basis. Due to the complexity of the rockfall problem, the recovery coefficient of the rockfall collision is very discrete and unlike the collision between ordinary objects. A more accurate reference value of the recovery coefficient can be obtained through experiments. At present, there is no specification to clearly indicate the range of values of Rn and Rt. Practical experience tells us that the normal recovery coefficient of rockfall collision is between 0.2 and 0.5, and the tangential recovery coefficient is between 0.4 and 0.9. Generally, when the bedrock is exposed, the larger value is taken; when the slope is conglomerate or hard soil without vegetation cover or a small amount of vegetation, the median value is taken; when the slope is covered with loose residual soil or clay, the smaller value is taken. Tables 4.1 and 4.2 show the values of the normal and tangential recovery coefficients recommended by The Former Transportation Bureau of the Ministry of Railways. Another important issue related to collisions is the estimation of the kinetic energy of a rockfall after a collision. In β ¼ EEvr , where Er is the kinetic energy of the rockfall and Ev is the kinetic energy of the rockfall, β is called the scale factor of rock kinetic energy and translational kinetic Table 4.1 Normal recovery coefficient. Slope feature

Normal recovery coefficient Rn

Smooth and hard surface and paving surface Mostly bevels in bedrock and conglomerate areas Hard soil slope Soft soil slope

0.37–0.42 0.33–0.37 0.30–0.33 0.28–0.30

Potential hazard prediction of rockfalls 135 Table 4.2 Tangential recovery coefficient. Slope feature

Tangential recovery coefficient Rt

Smooth and hard surface and paving surface Mostly bedrock and slopes without vegetation Slope with a small amount of vegetation Vegetation-covered slopes and soil slopes with sparse vegetation cover Shrub-covered soil slope

0.87–0.92 0.83–0.87 0.82–0.85 0.80–0.83 0.78–0.82

energy. In theory, it is difficult to give an accurate solution for β, which is generally based on experience. The Japan Railway Association (JRA), based on nearly 60 years of field tests, suggested that the kinetic energy of the rockfall should be taken as 0.1 of the translational kinetic energy, that is, β ¼ 0.1.

4.3.5 Rock rolling The scroll mentioned here refers to the rolling of the rockfall against the slope. In nature, this kind of scrolling is rarely seen. More commonly, a short-range bounce mode is seen, often forming a series of continuous paraboloids with a small bounce distance and a low bounce height. The state of rockfall rolling is related to the degree of irregularity of the slope, the size of the rockfall, the moving speed of the rockfall, and the friction between the rockfall and the slope. According to experimental observation, generally when the size of the falling rock is smaller than the irregularity of the slope, the rockfall mainly exhibits a small bounce and slip movement. When the size of the falling rock is larger than the irregularity of the slope, the rockfall maintains a sliding movement. Generally, only the lower-speed spherical, columnar, and pie-shaped rockfalls will exhibit rolling on slopes with a gentle and smooth gradient. In order to avoid complicated analysis, we simplified the rolling model of the rock into a frictional rolling of a circular rigid body on an inclined surface. All complex control factors are attributed to the rolling friction coefficient, for generalization. The rolling of the rock mainly occurs at the beginning and ending stages of the movement. On some slopes with steep gradients, there may also be sliding rolling (Fig. 4.4). What we should be concerned with is the time when the rockfall enters the rolling state during the movement and the final distance of the rolling. Here, tanγ ¼ vvrnrt is defined, where Vrn and Vrt are the normal and tangential components of the rockfall bounce speed after collision, respectively, and γ is the bounce and collision rockfall angle. After the rockfall collision, when tan γ < ξγ (where ξγ is an infinitesimal number with an arbitrary value greater than 0, the size can be determined according to the calculation accuracy), the rockfall can be considered to enter a rolling state without rebound. Otherwise, continue to follow the bounce to carry out the calculation.

136 Chapter 4

Fig 4.4 Rockfall rolling model.

For the rolling analysis of a rockfall, as shown in Fig. 4.4, it is assumed that the rockfall enters the rolling state at point O; at this time t ¼ t0, V ¼ V0, and s ¼ 0. Then at any time t, the dynamic equilibrium equation can be obtained: N  mg  cosα ¼ 0

(4.17a)

m  s€¼ mg sin α  f

(4.17b)

s€ ¼f RN  d R

(4.17c)

I

In the formula: N is the slope facing the support of rockfall; f is the friction of the slope facing the rockfall; m is the Rockfall quality; R is the falling radius; and s is the rockfall displacement vector. Available from Eqs. (4.17a)–(4.17c): m



d s€¼  g  sinα  cosα  I R m+ 2 R

 (4.18)

m

I , where B is the constant related to rockfall mass and shape; definition R2 d ur ¼ R ¼ tan βr , which is called the rolling friction coefficient, and βr is called the rolling friction angle. Definition B ¼

m+

Also, a ¼ s€¼ B  g  cos α  ð tanα  tanβr Þ. Thus the relationship between speed V and displacement s can be obtained by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ V02 + 2B  g  cosα  ð tan α  tan βr Þ  s

(4.19)

Potential hazard prediction of rockfalls 137 If a < 0, that is, tan α < tan βr, the rockfall is decelerated and scrolled, and finally stops under the rolling friction. The displacement s at the stop is: s¼

V02 2B  g  cos α  ð tan α  tan βr Þ

(4.20)

This formula can be used to assess the extent of the impact of a collapsed rockfall disaster. At the same time:  2 V  V02 (4.21) μr ¼ tan βr ¼ tanα  2  B  g  cos α  s Like the recovery coefficients Rt and Rn, the rolling friction coefficient μr is also an important parameter for correctly estimating the trajectory of the rockfall. The rolling friction coefficient is related to the size, shape, and speed of the rockfall, the gradient, and the geomechanical properties of the slope. In the case of sufficient test data, the value of μr can be inversely calculated by Eq. (4.21). The results of the field test show that the friction coefficient μr of rockfall rolling is between 0.3 and 1.0 (βr is between 16 and 45 degrees).

4.4 Chapter summary Rockfalls can be divided into five types of movements: falling, sliding, free falling, collision bounce, and rolling. A kinematics analysis is carried out to calculate the movement speed, movement trajectory, and destructive force of the falling rock, and the movable block is determined. Once the body has obtained the destroyed energy and the extent of damage caused by the falling, an effective research method to assess the impact of the rockfall disaster has been established.

CHAPTER 5

Laboratory tests 5.1 Conclusion of indoor rockfall simulation test Based on the rockfall experiments described in Chapter 3, the following conclusions can be drawn: (1) The slope of No. 1 has a partial rockfall during the test with a gradient of 30 degrees, while the slope of No. 2 with a gradient of 35 degrees has a total rockfall, which indicates that the larger the gradient, the more likely the occurrence of a rockfall. For the slope of No. 3 with a 40-degree gradient, both the density and the clay content are high, and rainwater does not easily infiltrate. Although no pore water pressure is generated, a surface rockfall occurs due to the steep gradient. (2) The test results on the impact of various slope morphologies on rockfall show that the order of risk of different slopes having the same size, the same curvature radius (except the plane type), and the same soil condition is: No. 4 descending type > No. 5 ascending type, No. 2 plane type > No. 8 collecting type, No. 6 gully type, and No. 7 spine type are not much different. (3) Most of the slope rockfalls occur when the pore water pressure in the soil reaches a maximum. For example, for No. 2 and Nos. 4–8, the pore water pressure increases sharply and is the intrinsic power causing rockfalls. For No. 1, there is pore water pressure, but because the gradient is 30 degrees, there is no rockfall. No. 3 has no pore water pressure, but the slope is 40 degrees, so it also exhibits rockfalls. It can be seen that the gradient is also a key factor in slope stability. (4) The displacement of the soil is related to the accumulated rainfall. When the accumulated rainfall reaches or exceeds a certain limit, the soil loses stability and rockfalls occur. (5) The gully-type and the ridge-type slopes show different rockfall characteristics based on morphology: the gully-type slope has a tendency to collect water due to its low middle, while the middle of the ridge-type slope is high and is prone to scatter water, and these characteristics affect the form of rockfall. The middle of the gully-type slope is first to exhibit rockfall, and then rockfalls occur on both sides. The sides of the ridge-type slope exhibit a first partial rockfall, and then rockfalls occur in the middle.

Rock Mechanics and Engineering. https://doi.org/10.1016/B978-0-12-822424-3.00005-0 # 2021 Central South University Press. Published by Elsevier Ltd. All rights reserved.

139

140 Chapter 5

5.2 Indoor acoustic emission test This section describes indoor acoustic emission tests carried out as part of the experimental investigations for a rockfall monitoring network and monitoring system for the K79+380
+500 slope in the Guibi section near the Shanghai-Kunming high-speed railway. In order to ensure better monitoring results, rock samples containing the structural surface were first taken from the site. After processing, structural surface mechanical properties tests and the acoustic emission characteristics tests were carried out indoors, to obtain data on the rock mass structural surface to guide the scene. Acoustic emission monitoring is considered to be a sound basis for on-site rockfall prediction, combined with theoretical analysis of the block.

5.2.1 Test procedure (1) Sampling and processing of rock samples In the rockfalls near the K79+380
+500 in the Guibi section near the Shanghai-Kunming highspeed railway, the joints and fissures of the rock mass are particularly developed, and the rock mass of the slope is heavily broken. The rock masses with structural planes used in the laboratory were taken from the rockfall section near the K79+380
500. In order to obtain the internal friction angle and cohesive force of the structural surface, direct shearing and tilting compression molding tests were carried out indoors. In the direct shear test, the size of the sample was 10  10  10 cm3, and concrete mortar mixed with No. 525 cement was poured into a square test piece measuring 15  5  5 cm3. The cement river sand ratio was 1:1; the structural surface was located in the middle part, parallel to the upper and lower sides; the box was divided into upper and lower boxes, and the height was 7.5 cm; the upper and lower boxes (parallel to the structural surface) were in contact with the mortar. A gap of 6–8 mm was left at the surface, so that the actual shear plane was along a structural plane. In the inclined compression molding test, the sample size was 5  5  5 cm3, and the sample was kept at 10–110°C for 24 h in an incubator, and then was taken out for testing. The on-site monitoring uses the YSS geophone, which only has three indicators: large event, total number of events, and energy rate. The indoor SFS-4B acoustic emission meter was used in the tests, so three indicators were also recorded in the indoor test for on-site application. (2) Direct shear test During the test, the vertical load was loaded from small to large, step by step. Each time when the vertical preset load was applied, a horizontal load was added until cutting occurred; after that, another sample was changed to another specified vertical load value, and horizontal load was added until reaching shear failure. After the test data of the group of samples were analyzed and processed, the value for each structural plane was obtained.

Laboratory tests 141 (3) Tilt compression test While performing the tilt compression test, the acoustic emission transducer was mounted on the test piece, and as the pressure increased, parameters including the mold angle and the structural surface acoustic emission event, the total number of events, and the energy rate were recorded. According to different loads and angles, the normal stress and shear stress on the structural plane were calculated, and finally a regression test was carried out to obtain the Cj, φj value of each structural plane. The direct shear and the inclined compression tests were compared, and the Cj, φj values of the structural planes were finally determined, as shown in Table 5.1. (4) Indoor acoustic emission test A block diagram of the indoor acoustic emission test system is shown in Fig. 5.1. The acoustic emission information analyzer used in the test room was the SFS-4B four-channel information analyzer produced by Shenyang Electronics Research Institute. The instrument is a product of the 1990s. The ZF unit has four independent main channels and a main amplifier. The gain is 0–60 dB, which is adjustable every 1 dB, and the output amplitude is 4 V. The instrument can measure seven parameters, including acoustic emission events, energy, ring length, and ringing. (Considering that the geophone used in the field is a YSS type, only three parameters, large events, total number of events, and energy rate, are discussed in this test.)

Table 5.1 Cj, φj values of various structural planes. Structural face number

Cohesion (Cj)

Internal friction (ϕj, 0)

1 2 3 Average

0 0 0 0

32.7 28.5 31.6 30.9

Fig 5.1 Block diagram of indoor acoustic emission test system.

142 Chapter 5 In addition, linear accumulation, linear increment, logarithmic accumulation, and logarithmic increment are the types of output forms. The instrument was also equipped with a preamplifier with a gain of 40 dB; the measured frequency response was flat in the range of 0.7–1200 kHz, and the frequency response was 100 m

Toppling rockfall Falling rockfall Volume of dangerous rock mass

The height of the top of the dangerous rock mass from the steep cliff (slope)

Small dangerous rock Medium dangerous rock Large dangerous rock Extra-large dangerous rock Low-lying rocks Middle-positioned rocks High-positioned rocks Extra-high positioned rocks

7.2 Stability evaluation of high-speed railway landslides and rockfalls 7.2.1 Railway landslide stability evaluation Landslide stability evaluation is a general field defined by qualitative discrimination, limit equilibrium analysis, and fuzzy neural network methods of comprehensive evaluation. The wild qualitative discriminant criteria are described in the following paragraphs. (1) Stable state: the appearance features of the landslide are greatly modified in the later stage, the landslide depression is basically difficult to identify, and the slope of the sliding body ground is gentle (
10 degrees). The leading edge is low and slow (generally 50 m The starting position is located at the back edge of the landslide, and the main power comes from the deadweight or load at the back of the landslide. The initial sliding position is at the leading edge of the landslide, which is mainly caused by river scour at the slope foot or unloading, tension, and loosening of the leading edge of mechanical excavation. The front and rear edges of the starting slide area combine and act together. Landslide caused by construction excavation, building, or artificial loading, reservoir storage, etc. A landslide formed by natural geological processes

Rock landslide Sliding surface and rock surface

Bedding landslide

Relationship

Cutting layer landslide

Sliding body thickness

Shallow landslide Intermediate landslide Deep landslide Ultradeep landslide Push slide

Starting position and migration pattern

Loose (traction) type landslide

Mixed landslide Inducing factors

Engineering landslide Natural high-speed railway landslides

Continued

Table 7.2: Evidence division Forming age

General classification table for high-speed railway landslides—cont’d Category

Characteristics

A new landslide

A new, well-documented or landslide-shaped, well-preserved landslide A landslide with no historical record or unclear track since Late Pleistocene High-speed railway landslides formed before the Late Pleistocene (125,000 years ago)
10  104 m3 10  104 m3–100  104 m3 100  104 m3–1000  104 m3 >1000  104 m3 Ancient landslide, old landslide whole or partial activity again First landslide

Old landslide Ancient landslide Volume of sliding body

Landslide issue

Small high-speed railway landslides Medium high-speed railway landslides Large high-speed railway landslides Extra-large high-speed railway landslides Revived landslide Nascent landslide

Geological basis 159 distance is long and the energy is fully released; there is no new source of deposit loading around the landslide, and the front edge of the landslide has formed a stable slope type of river erosion. After analysis and field investigation, the overall stability is reduced under special working conditions, but only local deformation damage may occur. (3) Potentially unstable state: the landslide appearance features are not greatly reconstructed in the later stage, and the landslide depression in the back margin is closed or semiclosed; the average slope of the sliding body is moderate (20–30 degrees). The free face of the landslide is steep (height 30–50 m, slope 20–30 degrees), and the cutting degree of the valley in the landslide body is moderate; the sliding surface is in the shape of a chair or a plane, and the average inclination of the sliding surface is 20–30 degrees, with a sliding surface impedance ratio of 0.4–0.6. The landslide debris is generally permeable, the slip distance is not too long, and the energy release is not sufficient. The loading source is load accumulation at the back edge of the landslide or a certain amount of dangerous rock mass. The erosion at the front edge of the landslide has not yet formed a stable slope type, and there is partial collapse. According to the field investigation and qualitative analysis, it is stable under general working conditions, but the safety reserve is not high, and overall instability may occur under special working conditions. (4) Unstable state: the landslide features are obvious, and the landslide depression is generally obviously closed; the average slide slope is steep (>30 degrees), and the front face of the landslide is steep (height >50 m, slope >30 degrees). The groove cut in the sliding body is shallow and the sliding surface is in the shape of a chair or a plane, with a sliding surface average inclination >30 degrees, and a sliding surface impedance ratio of 1.25, no monitoring.

7.3 Chapter summary (1) The purpose and content of geological work in landslide and rockfall monitoring and prediction are to provide necessary basic data and to create the conditions needed for comprehensive prediction. (2) The slope in a stable state need not be monitored through field judgment; surface inspection should be carried out in the relatively stable state; potentially unstable slopes and unstable slopes should be monitored professionally.

Geological basis 161 (3) By calculating the stability coefficient K of landslides and rockfalls, the necessary level of monitoring can be determined: if K < 0.95, it is necessary to focus on tracking and monitoring; if 0.95
K
1.05, monitoring should be strengthened; if 1.05 < K
1.25, surface inspection should be carried out; if K > 1.25, no monitoring is necessary.

CHAPTER 8

Landslide and rockfall monitoring technology for high-speed railway 8.1 Summary of slope information monitoring Slope stability has always been one of the main points in geotechnical engineering. A great deal of research has been done on slope stability over many years, but researchers still have not found a very accurate evaluation theory and method. At present, the most reasonable method for dealing with the slope stability problem is the comprehensive judgment method, which combines the results of theoretical analysis, expert decision-making, and effective monitoring and control. Thus slope monitoring and its feedback analysis are an important part of slope engineering problems. Slope monitoring and prediction are necessary technical measures to ensure the safety of slope engineering construction and operation. In addition, slope monitoring is an important basis for improving slope engineering design and construction technical decisions. Therefore slope monitoring technology has been rapidly developed and is widely used. Long-term and systematic monitoring of slopes is needed for diagnosis, improvement, prediction, and research. (1) The need for diagnosis. The main purpose of slope monitoring is to provide reliable data for slope stability evaluation, which is needed for the evaluation and improvement of new construction technology and the early detection of hidden safety hazards. (2) The need to improve engineering design. In addition to indicating the “health” of the slope, the monitoring data can also help to modify the slope engineering design. The rationality of the design is evaluated by comparing the monitoring data with the theoretical analysis results. (3) The requirements of forecasting and prediction. Through the long-term accumulation of monitoring data, the change rules of various parameters of the slope can be mastered and the working characteristics of the slope can be predicted promptly and effectively. Rock Mechanics and Engineering. https://doi.org/10.1016/B978-0-12-822424-3.00008-6 # 2021 Central South University Press. Published by Elsevier Ltd. All rights reserved.

163

164 Chapter 8 (4) Research needs. Based on the geotechnical material structure characteristics and the use of conservative assumptions, complex mechanical analysis is carried out in the engineering design and theoretical study of the slope. The slope monitoring data can reflect the working traits to some extent, and provide valuable quantitative information for design and analysis, which is helpful in gaining in-depth understanding of the slope failure mechanisms, improving the analysis technology, and putting forward a safer and more economical design. Based on these needs, slope safety monitoring has been garnering greater attention in recent years, making it an important part of engineering construction and management.

8.1.1 Monitoring purposes (1) Monitoring can allow a timely grasp of the characteristic information of landslide and rockfall deformation and failure, analysis of its dynamic change rules, and correct evaluation of its stability. It can provide valuable assistance in predicting and forecasting the space, time, and scale of a landslide and rockfall disaster, and provide reliable technical data and a scientific basis for disaster prevention and reduction. The purposes of monitoring in each stage of high-speed railway construction are shown in Table 8.1. (2) Provide objective standards for modifying the design and guiding construction. (3) Provide data for inversion analysis of engineering geotechnical physical parameters. (4) Provide data for grasping the characteristics of landslide and rockfall deformation and guide the emergency treatment when the landslide undergoes severe deformation.

8.1.2 Basic principles of monitoring Slope information monitoring must be based on a detailed engineering geological study of slope stability. Through a brief description of the five monitoring methods mentioned previously, it Table 8.1 The purpose of each stage of monitoring. Stage

Purposes

Survey and design stage

Combined with geological survey, to provide information for design and construction, that is, to determine the sliding surface, inversion calculation c, φ, to achieve the purpose of economical, reliable, and symptomatic management of design To provide data for dynamic design and safety guarantee for construction To ensure the safety of vehicles, lives, and property, and deal with hidden dangers in a timely manner

The construction phase Operations stage

Landslide and rockfall monitoring technology 165 can be seen that the determination of different types of landslide monitoring methods should not only consider the basic characteristics and applicable conditions of the various monitoring methods, but also should consider the organic combination of various monitoring methods, so as to obtain the best monitoring effect. Therefore, slope monitoring should generally follow several basic principles. (1) Monitoring methods should fully consider slope characteristics, geological conditions, and the external environment in choosing the appropriate monitoring methods, so as to combine soil and ocean, instrument monitoring and macro monitoring, and manual monitoring and automatic monitoring. Through the comparison and combination of various schemes, the monitoring work should not only be economical and safe, but also practical and reliable, so as to avoid the unilateral pursuit of high-precision, automatic, or multiparameter monitoring schemes divorced from the actual project. In the selection of monitoring methods, it is better to be small and precise rather than big and ambiguous. (2) The selection of monitoring instruments should be a combination of electronic and mechanical instruments and both high-precision and low-precision instruments, so the instruments can supplement and provide checks on each other, to improve the reliability of the monitoring data; in general, very high-level, extremely precise, multidimensional instruments should not be pursued. Considering the poor environmental conditions of slope engineering monitoring and the large vibration interference during slope construction, in order to improve the reliability of the monitoring results, the monitoring equipment chosen should be optical equipment, mechanical equipment, or electronic equipment with strong anti-interference capability and the ability to perform in poor environmental situations. The set of instruments chosen should be as few in quantity as possible and only as precise as needed, under the premise of ensuring any actual needs are met. In designs for monitoring the construction period, the precision requirement can be slightly lower, or simpler instruments can be used. Long-term monitoring instruments generally should be adapted to larger deformations, and the principles of precision, reliability, and durability must be overall considerations. In order to provide sufficient data for analysis, the instruments should not be distributed over larger areas but should be centrally arranged. (3) Monitoring content should be based on the geological structure, the space form and slope phase, the choice of key monitoring area, and as previously stated, on the principle of less is more; in choosing monitoring parameters, pay attention to the monitoring points, both overall and in a rational layout of the monitoring network, keeping the surface monitoring and surveillance in balance. A combination of geotechnical monitoring and a load monitoring system is advised, incorporating electronic testing, mechanical testing effect measure monitoring, and environmental monitoring; and combining geometrical quantity monitoring with relevant physical parameters, to form a three-dimensional crossmonitoring network system combining with point, line, and plane.

166 Chapter 8 (4) The monitoring accuracy is often taken as a reference for the monitoring accuracy of the same type of landslide at home and abroad, and the appropriate monitoring accuracy is determined by comprehensive analysis of such indicators as field reconnaissance, slide history, formation mechanism, importance degree, deformation development trends, and monitoring instrument accuracy. In different stages of landslide formation, different parts of deformation monitoring have different precision requirements, and the key points of the monitoring need to be adjusted accordingly. Generally speaking, an instrument with high precision is suitable for monitoring a landslide with small deformation, but for a landslide in a state of rapid change and imminent slide, the precision can be relaxed and controlled flexibly according to the specific situation. According to the error theory, the observation error should be 1/5 to 1/10 of the deformation, and the monitoring accuracy should be adjusted appropriately to improve the monitoring plan by predicting the deformation state and development trend of the sliding body through a period of monitoring practice (1–2 years) and analysis of observation data. (5) The monitoring period is mainly determined according to the deformation stage of the sliding body and the properties of different monitoring methods. Generally, if the landslide does not enter into a rapidly changing state and the deformation amount is small, the observation period is long and the observation accuracy is high. When the deformation rate of the landslide increases or there is an abnormal change, the observation period should be shortened, the number of intensive observations should be increased, and the accuracy can be relaxed appropriately. During rainy seasons with heavy rainfall, monitoring cycles should also be adjusted to provide accurate and reliable information in a timely manner. In short, slope monitoring should not only focus on the overall stability of the slope, but also attention should be paid to the monitoring of the local stability of the slope. Through the timely feedback of monitoring data, the relationship between and mutual influence of local stability and overall stability can be analyzed, and the local stability problem can be solved by rational control and timely construction under the guidance of monitoring information, which is conducive to the overall stability of the slope. Slope monitoring must run through the whole process of engineering activities. Therefore, the most important point in monitoring is timeliness, namely timely monitoring, timely analysis, timely feedback, and timely decision-making. Any negligence in these four links will reduce or lose the significance of the monitoring, and even bring irreparable losses to the project or to people’s lives and property. Therefore, good slope information monitoring is an important link in slope engineering. It is needed to make slope engineering construction safer, more economical, and more effective.

8.1.3 Requirements for slope monitoring technology (1) Pertinence of monitoring. The monitoring and design of a slope should be targeted according to the geological conditions, design, construction, and reinforcement needs of the project. In general, the deformation and failure mechanism of the slope should be predicted according to the engineering geological

Landslide and rockfall monitoring technology 167 conditions and shape of the deformation of the slope, and the size of the monitoring parameters should be predicted based on the deformation and failure mechanism of the slope, and the monitoring items and instruments should be selected accordingly. (2) Monitoring stages. Monitoring design should be divided into stages, with different monitoring items in different stages. The safety monitoring of slope engineering is carried out at the same time from the beginning of excavation. During the remediation, the safety and inspection of the remediation effect shall also be monitored; moreover, the monitoring of the construction period is as important as that of the operations period, not only because the safety problems of the construction period are more prominent and crucial, but also because the initial value of the monitoring should be established as early as possible in the construction period. (3) Timeliness of monitoring. One of the key points of monitoring implementation is the need for each link of the monitoring implementation to be performed in a timely manner. These links include monitoring feedback and monitoring information. The purpose of monitoring is to ensure construction safety; the remaining links are the means to achieve the end. (4) Guidance of monitoring design. The design should be carried out based on the inherent characteristics and requirements of the slope engineering. For example, the borehole of inclinometer on landslide is required to pass through the sliding bedrock below the predicted sliding surface, otherwise, the drilling will be far away from the slope surface, losing the monitoring function. Water is often an induced factor for edge (slip) slope instability. It is even more urgent to tighten the timely monitoring on rainy days. But during the rainy seasons or on low atmospheric visibility days, the odolite can be relatively difficult to measure, so the edge (slip) slope monitoring and the other measurement methods which are not heavily influenced by the severe conditions are needed to fill in the gaps.

8.2 Major monitoring instruments At present, the techniques and methods of landslide monitoring are developing to a higher level. In the past, simple monitoring, such as surface measurement with manual tape measure, was used and now the field is gradually moving to automatic and high-precision telemetry systems. As shown in Table 8.2, monitoring content is rich and monitoring methods and instruments are varied. They reflect the dynamic information on landslides and other information closely related to landslide deformation. With the developments in electronics and computer technology, monitoring methods and instruments have been continuously improved, and the monitoring content has become even richer.

168 Chapter 8 Table 8.2 A list of landslide monitoring methods used in China and abroad. Main monitoring methods

Characteristics of monitoring methods

Suitability assessment

Geodetic method

Theodolite, level, rangefinder

Fast input, high precision, wide monitoring surface, intuitive, safe, easy to determine the direction of landslide displacement and deformation rate

Close-up photography

Total station type tachometer, electronic theodolite, etc.

GPS method

GPS receiver

Seam measurement

Steel tape, vernier caliper, crack measuring instrument, telescopic recording instrument, joint measuring meter, displacement meter, etc.

High precision, high speed, high degree of automation, easy to operate, saves manpower, can track automatic continuous observation, monitoring large amount of information High precision, rapid return on investment, easy operation, allweather observation, not limited by topographic access conditions, the current cost is high, considerable development prospects Manual and selfrecording joint measurement method has the advantages of quick input, high precision, adjustable range, simple and intuitive method, and reliable data;

Suitable for displacement monitoring at different deformation stages and cannot be continuously observed due to the influence of topographic visibility and climatic conditions Suitable for displacement monitoring in different deformation stages and is limited by terrain visibility conditions

Content Surface deformation

Suitable for surface 3D displacement monitoring at different deformation stages of landslide

Manual selfrecording method is suitable for monitoring the opening, closing, dislocation, and rise and fall of rock and soil mass on both sides of the crack;

Landslide and rockfall monitoring technology 169 Table 8.2

A list of landslide monitoring methods used in China and abroad—cont’d Main monitoring methods

Content

Underground deformation

Survey method

Borehole inclinometer, multipoint inverted hammer, and so on

Seam measurement (shaft)

Multipoint displacement meter, shaft wall displacement meter, dislocation meter, etc.

Characteristics of monitoring methods

Suitability assessment

telemetry is highly automatic, allweather observation, safe, fast, laborsaving, automatic collection, storage, printing and display of observation values, long-distance transmission, relatively low accuracy, general instrument prone to failure, poor longterm stability, data need to be used after checking with other monitoring methods High accuracy, good effect, easy to telemeter, easy to protect, less interference by external factors, reliable data; limited range, high cost, slow input

telemetry is suitable for monitoring the acceleration deformation stage and construction safety, which is greatly affected by external factors such as climate

High precision, easy to protect, slow investment, high cost, instrument, sensor easy to be soaked by groundwater, corrosion

Mainly used to measure the deformation characteristics and the position of sliding surface at different depths in the borehole and shaft at the early stage of slope deformation Generally used to monitor the relative displacement of multilayer accumulation in the shaft; at present, due to the limited instrument performance and range, it is mainly suitable for the early deformation stage, namely small Continued

170 Chapter 8 Table 8.2

A list of landslide monitoring methods used in China and abroad—cont’d Main monitoring methods

Content

Weight method

Settlement method

Ground sound

Heavy hammer, polar plate coordinate instrument, horizontal dislocation meter, etc. Sinker, convergence, static level, water pipe tilt, etc.

Seam measurement (adit)

Unidirectional, bidirectional, three-way seam gauge, displacement gauge, extensometer, etc.

Ground volume method

Acoustic emission instrument and ground sound detector

Characteristics of monitoring methods

High precision, easy to protect; machine measurement is intuitive and reliable; electrical measurement is convenient, measuring instrument is easy to carry, but affected by moisture, strong acid, alkali, corrosion, and so on

Continuous observation, rich monitoring information, high sensitivity, saves manpower; the measured acoustic emission signal of rock microfracture was 3–7 days ahead of the displacement information

Suitability assessment deformation, low rate, relatively short observation time monitoring Suitable for monitoring the horizontal shear displacement of upper dangerous rock and lower stable rock mass Suitable for monitoring the subsidence change and vertical convergence change of soft layer or fracture of inner adit upper dangerous rock relative to the lower stable rock mass Suitable for threedimensional (X,Y, Z directions) monitoring of dangerous rock fractures in adit and displacement monitoring of dangerous rock interface fractures along its axis Suitable for monitoring the middle and late deformation stage of rock slope and monitoring the safety of rock reinforcement

Landslide and rockfall monitoring technology 171 Table 8.2

A list of landslide monitoring methods used in China and abroad—cont’d Main monitoring methods

Content

Characteristics of monitoring methods

Suitability assessment

Strain

Strain method

Tubular strain gauge

Suitable to measure the displacement and position of sliding surface at different depths

Hydrological

Underground water level Pore pressure

Water level recorder Pore hydrometer, borehole percolometer Triangle weir, measuring cup Water gauge

Results can be used as basic data for monitoring different deformation stages of landslide

Rain gauge Temperature recorder Seismic monitor

Suitable for monitoring different types of landslide and different deformation stages, and provides basic data for the analysis and evaluation of landslide

Spring flow River level Environment

Rainfall Ground moisture Earthquake

The development of monitoring technology and methods has broadened the monitoring content from surface monitoring to underground monitoring and underwater monitoring, and from displacement monitoring to stress and strain monitoring, related dynamic factors, and environmental factors monitoring. This ongoing development of monitoring technology and methods largely depends on the development of monitoring instruments. With the appearance of electronic camera laser technology, GPS technology, remote sensing and telemetry, and automation and computer technology, landslide monitoring technology now has much more powerful means at its disposal.

8.2.1 Surface displacement monitoring instruments In the observation of landslide surface cracks, not only the main cracks of the sliding body should be observed, but also the secondary cracks, clarifying the source of the crack and distinguishing the type of crack. From investigating the force of the landslide and its nature and dynamics, the reason for the slide can be deduced. The shape, length, direction, and location of landslide cracks should be recorded during the investigation and observation: the width of the crack, visible depth, the presence or absence of filler and the deep development, the nature of the crack end, the properties of the fracture wall and lip and their mutual position, the relationship between hydrogeological significance and geological conditions along fractures, and the inference of the origin of the fracture.

172 Chapter 8 (1) Right angle observation ruler. A scaled horizontal ruler and a vertical ruler (either wooden or metal) are positioned on the ground on both sides of a landslide crack; they should be perpendicular and close to each other. The right-angle observation ruler can be used to check the horizontal expansion of cracks and vertical subsidence. The real data of fracture change and ground rise and fall can be obtained by measuring the readings and their changes at the intersection of 2 ft at different periods. The embedding method consists of the following: a wooden pile should be driven into the fixed body on the upper side of the crack, with the top of the pile exposed to the ground 70 mm, and a wooden pile should be driven on the lower side of the crack, with the top of the pile exposed to the ground
100–150 mm. The line between the two piles should be perpendicular to the trend of the crack. The top of the lower side of the pile should be 30–80 mm higher than the top of the upper side of the pile. As already stated, the horizontal and vertical rulers should be close to each other and at right angles. Marking the intersection of the 2 ft with red paint and recording the scale readings on the horizontal and vertical rulers in the observation and inspection record book for the first time is regarded as the original datum. When compared with the original datum, the differences in later monitoring data can be observed. (2) Skateboard observation ruler. In steep slope angle observations, mountain cracks in the foot are difficult to observe; a skateboard observation scale is available, the principle of which is the use of a belt scale for straight, when cracks expanding, lateral pile under moving, affects the skateboard sliding upwards and record their mobile numbers. This method cannot divide the mobile numbers into horizontal and vertical, so is less precise than the angle observation method. (3) Observation pile. (a) Straight line method. Two wooden piles are driven on the fixed body of the upper part of the crack, and one is driven on the lower part of the crack, so that the three piles are in a straight line and perpendicular to the direction of the crack. Iron nails are driven into the top surface of each pile, as shown in Fig. 8.1.

Fig. 8.1 Straight line diagram.

Landslide and rockfall monitoring technology 173 The distance between the three piles is measured and recorded in the recording table, which can reflect the speed of fracture change. (b)

Triangular method.

A wooden stake is placed on the upper part of the crack, and two piles are placed in the lower part of the crack to make the three piles into triangles. The relative displacement of the three points in different periods can be used to determine the true displacement and direction of the ground. (4) Strain gauge. The original mechanical strain gauge is a simple structure, essentially a wire rope fixed at both ends. One end of the wire rope is fixed to a point in the landslide, while the other end is located outside the landslide and is attached to a weight mounted on the slide. The movement of a landslide moves a weight on a graduated track. The amount and speed of movement can be measured manually. This device is very inexpensive, but if data are not collected quickly, important information on landslide development might be missed. In addition, this instrument is easily damaged by humans or animals. A strain gauge can also use a potentiometer to measure displacement. Like the rheostat control in the electric vehicle model, the strain gauge uses a variable resistance mechanism to measure movement. The sliding arm forms an electrical contact on the fixed resistance bar, and the total resistance of the whole circuit is determined by the position of the sliding arm on the fixed resistance bar. When the strain gauge is added with a steady direct current, the ground motion arc sliding arm moves on the fixed resistance bar, which causes a corresponding change of output voltage. The wiring and sensitive elements of this type of instrument can be buried underground to prevent external damage. (5) Telescopic meter. The telescopic meter (also known as landslide recorder or landslide meter) is characterized by in situ recording; it is used for monitoring surface cracks and displacements of high-speed railway landslides. The continuous displacement-time curve can be obtained directly, and it can meet the long-term stability, reliability, and robustness requirements of field work, since its sensing, recording, and rate detection are all mechanical and its recorded data curve is intuitive. This instrument is thus widely used with little interference and high reliability. However, in the case of a dangerous sliding situation, personnel should not be close to the dangerous areas. The set-up can be described as follows: several recorders are installed along the direction of landslide main axis; the first measuring device is located in the rear on the stable portion of the

174 Chapter 8 body. Connect the instrument to the stake with indium steel wire, instrument and the distance from pile with 10–15 m advisable, machine base and piles are fixed to cement mortar. The indium wire is encased in a protective tube, to prevent the impact of human activity. (6) Total station tachometer. Total station meter, also known as "electronic total station," has been favored by people in the past 10 years because of its functions of automatic rapid 3D coordinate measurement and positioning, automatic data flow in data acquisition, electronic recording of field data and automatic process from field work to internal work integration. The electronic total station is a type of field measuring instrument using photoelectric ranging and electronic measuring of angles and distances, integrated with a microcomputer and data storage/collection system. Its main precision indexes are: ranging, standard deviation (m1), and angle measurement, standard deviation (m2). The standard deviation of the ranging, which is the accuracy of the photoelectric ranging function, is classified into three grades according to the “verification regulations of photoelectric ranging instrument for trial implementation” issued by China’s State Bureau of Technical Supervision and implemented in January 1991: Class I equipment |m1 | < 5 mm Class II equipment 5 mm < |m1 | < 10 mm Class III equipment 10 mm < |m1 | < 20 mm In the formula m1 ¼  (a + bD106), a, b are fixed error coefficient and proportional error coefficient of the nominal accuracy of the instrument, respectively. In the formula jm1 j is the absolute standard deviation when ranging D51 km. The nominal accuracy of rangefinders (including the rangefinder unit of the total station) manufactured recently in China and other countries is equal to or superior to (5 mm + 5D106), with jm1j < 10 mm classified as grade II and grade I. (7) Global positioning system. Global positioning systems (GPSs) have high precision, rapid return on investment, simple operation, and can be used for all-weather observation. The three-dimensional displacements X, Y, and Z can be measured at the same time, and the speed of the point in motion can be accurately measured; it is not restricted by conditions, so it can be continuously monitored. The high cost is the main disadvantage. GPS is suitable for horizontal and vertical displacement monitoring in different deformation stages. China has established a GPS monitoring network in the crustal activity area of Beijing, Tianjin, and Tang, the dam area of the Three Gorges project of the Yangtze River and the capital international airport, and has applied GPS technology to the monitoring of high-speed railway landslides in the Three Gorges Reservoir area, dangerous

Landslide and rockfall monitoring technology 175 rock mass deformation monitoring of the Chain Cliff, and monitoring of the treatment effects of the Chuankou landslide in Tongchuan City. To address the problem of high cost in the application of conventional GPS, a GPS multiantenna data acquisition and control system was introduced in recent years, which realizes the monitoring of multipoints by one machine. Experiments and practical applications show that this system has little effect on GPS signal attenuation and GPS measurement accuracy. This technology greatly reduces the monitoring cost of artificial and natural structural deformation (such as dams and buildings) in local areas monitored by GPS and has important practical value. There are many examples of using GPS to monitor deformation and landslides and to realize automatic monitoring and management in China. The deformation monitoring of Geheyan hydropower station is one of the most successful examples. As some scholars have pointed out, the high cost of GPS instrumentation greatly restricts its application in deformation monitoring, prevention, and reduction of geological disasters. Since 1999, Nanjing University of Aeronautics and Astronautics Navigation Research Center and Hong Kong Polytechnic University have carried out research on how to decrease the cost of GPS use and designed a nationally patented device (patent number: ZL00219891.6), a GPS multiantenna technology, that is, a GPS receiver with multiple antennas, with a different antenna connected to the receiver at different times. Thus, a receiver can monitor multiple monitoring points. Expensive GPS receiver arrays instead become GPS antenna arrays, and the price drops exponentially. A receiver can now connect to 4–16 antennas. The multiantenna technology can be used to continuously monitor geological disaster areas such as landslides, rockfalls, and debris flows. (8) TCA2003 total station. The TCA2003 total station is a high-precision range finder, which is combined with the electronic theodolite of the absolute coding dial (measuring angle accuracy 0.500 ) and large capacity computer technology. Driven by a servomotor, the instrument can automatically identify targets, measure (horizontal angle, vertical angle, and distance) targets, and record observation data under the control of onboard system software. Therefore the TCA2003 total station not only has the functions of measuring angles (horizontal angle, vertical angle), ranging, and automatic recording of a general total station, but also has the following improvements and developments in the structure and function of the instrument: ①

②

A precise servomotor is installed. When measuring the horizontal angle or vertical angle, the servomotor can be controlled by manual button or programming. The servomotor rotates the instrument sighting part as needed for observation. A coaxial automatic target recognition (ATR) device is installed in the telescope, which can automatically identify the target (prism) and automatically aim at the prism for measurement, so that it can be programmed to collect observation data using artificial intelligence. The receiving system uses a CCD element, which can automatically identify

176 Chapter 8 and lock the target in the LOCK mode and can track and measure without interference of other stray light sources. With perfect on-lcxke instructions, users can conveniently use VB, VC, Pascal, and other languages to create programs according to the actual needs. In on-lcxke mode, in a certain communication mode, it is possible to control the instrument using a computer and complete various automatic measurements. A PCMCIA card is used to record the data in the carrier, which can also be recorded in the memory of the instrument. Observation data can be transmitted to the PC through an RS-232 interface for postprocessing. The measuring principle of the TCA2003 total station instrument can be expressed simply as follows: before the measurement, coordinate parameters of the measuring station and observation point are input to the instrument, and the azimuth angle from the measuring station to each monitoring point is found by automatic inverse calculation of the instrument, which is aimed at the starting direction of the observation point on the measuring station. With the function of 0.1, based on the magnitude of azimuth difference, the instrument will automatically identify and aim the target (prism) from smallest to largest in order. After aiming at the target, the instrument emits laser to the target (prism). The laser is reflected back by the prism and captured by CCD camera, and convert to horizontal angle or vertical angle correction, according to the correct number of servomotor stepping into the center of the prism, which can aim accurately and automatically record observations. For the application of a dam deformation monitoring network, the TCA2003 total station has developed and designed the external airborne IspectorV1.0 data acquisition software. The basic function of the software can be summarized as building on the TCA2003 total station measured work basis points and the dam deformation monitoring network general coordinates database, when TCA2003 total station in test site and complete the initial direction and directional, dial instrument according to the field of airborne IspectorV1.0 software set of observation points, observation sequence set at this point in advance and the regulation of grade zero, in turn to automatic search of observation target, target, and measuring angle, measuring edge; The TCA2003 total station instrument records the observation results in real time in the files of the PCMCIA memory card, and automatically compares the collected data with the limits of various specifications. When the limit is exceeded, it will automatically give an alarm, and then manually intervene through reworking and retesting, until finally obtaining qualified field observation data files. (9) RS and close-up photography. Remote sensing (RS) and close-up photography methods are suitable for the monitoring of large-scale and regional landslides. The landslide is evaluated based on remote sensing images: the changes in the landslide can be understood based on the changes in the images during different periods. Dynamic monitoring of geological hazards using high-resolution remote sensing images is becoming more common. With the continuous developments in remote

Landslide and rockfall monitoring technology 177 sensing sensor technology, the resolution of remote sensing images is increasing. For example, the resolution of Landsat TM remote sensing in the United States is 29 m, that of the SPOT satellite in France is 10 m, and that of the IKONOS commercial satellite is 1 m. The characteristics of rich ground information displayed by satellite remote sensing images and the ability to periodically obtain images of the same site can be used to compare remote sensing images of the same geological disaster site over different periods, so as to achieve the purpose of dynamic monitoring of geological disasters. The method of close-up photography makes use of land camera theodolites and other instruments. However, the accuracy is relatively low, mainly applicable to monitoring the horizontal displacement of high-speed railway landslides and the changes in steep cracks in dangerous rocks with a large deformation rate, which are greatly affected by climatic conditions. For example, this technology was used to divide and predict large landslide-prone areas in the Three Gorges Reservoir area and to analyze and predict the huge high-speed landslide in Bomiyigong, Tibet.

8.2.2 Deep displacement monitoring instruments (1) Resistance pipe strain gauge. At a certain distance (generally about 20–30 mm; this distance is reduced near a suspicious sliding surface, generally to 5–10 mm), strain gauges are attached symmetrically to the front and back walls of a rigid PVC plastic pipe placed in the borehole along the sliding direction of the landslide, connected as a half bridge. Each strain gauge is connected to the ground with thin wires. When the landslide moves, the plastic pipe and the strain gauges attached to it deform, and then the resistance value reflected by each strain gauge changes and is measured. Because the deformation of a soil mass near its sliding surface is the most noticeable, the resistance value in this corresponding part also changes the most. Based on these facts, the displacement of a landslide at different depths can be found and the position of the sliding surface can be calculated. (2) Capacitance type slip surface meter. The principle of the capacitance slip surface meter is as follows: a 1000 μf capacitor is put into a 12 mm glass tube with 10 mm inner diameter and 1 m long, with two copper wires in parallel at a distance of 20 cm, and is then bonded with epoxy resin binder and put into the inferred sliding surface position in the drilling hole. When the slide moves, the glass tube and wire are dislocated and cut at the slide surface, which leads to a change of capacitance. (3) Short circuit type slip surface meter. In the short circuit type of slip surface meter, a wire is used as a bus bar, a number of wires are used as sublines, and the sublines are welded 20 cm apart on the bus to form a wire bundle, which is configured on a long narrow glass plate, loaded into a glass tube with an outer diameter

178 Chapter 8 of 18 mm and an inner diameter of 15 mm, fixed with epoxy resin binder, and put into the borehole. When the landslide moves, the glass tube and wire shift and are cut at the slide surface, resulting in a broken circuit with infinite resistance. This determines the position of the sliding surface. (4) Inclinometer. The inclinometer was originally used to measure the inclination of boreholes, and then it was applied in landslide research. By measuring the inclination of the borehole at different depths and using a computer to convert the value into the displacement, the position of the sliding surface can be found. The inclinometer is an effective instrument for measuring displacement and sliding surface position in soil. Several types of inclinometers are used: mechanical, resistance strain, slidewire resistance, and gyro orientation types. The mechanical method is relatively simple, but it can measure only a few meters. The resistance strain inclinometer is to fix a heavy cone on a spring piece with resistance wire, affixed with a resistance wire sheet. When drilling is inclined, the spring sheet deformation caused by the heavy cone is reflected as a change of resistance and measured. The slide-wire resistance type is connected with a pendulum and a resistance coil, and the change of inclination angle is measured by the change of resistance. ①

The principle of inclination monitoring.

The working principle of the inclinometer is to measure the angle change between the axis of the inclinometer and a plumb line, and then calculate the horizontal displacement of rock and soil at different elevations. A vertical inclined pipe with four guide channels is embedded in the rock and soil using an appropriate method. When the inclined pipe is deformed by force, the inclinometer gives the included angle θi in radians of the deviation between the axis of the inclined pipe and the vertical line section by section (generally using a measuring point of 50 cm). According to the segment length of the measuring points, the increment of horizontal displacement Δdi at different elevations is calculated, namely: X Lsin θi (8.1) Δdi ¼ The actual horizontal displacement at any elevation can be obtained by gradually accumulating from the measuring point at the bottom of the inclined pipe: bi ¼

n X

Δdi

(8.2)

i¼1

In this formula, Δdi is the increment of horizontal displacement in the measurement section; L is the segment length of the measurement point, with 0.5 m generally used (the distance between the upper and lower sets of pulleys of the probe is generally 0.5 m); θi is the included angle

Landslide and rockfall monitoring technology 179 Ground

Cable

Actual hole S = L × sina

Reader

S

The guide groove

Inclinometer

Wall filler Sensor

Hole wall

Fig. 8.2 Principle of inclinometer.

between pipe axis and plumb line in a measuring section; bi is the displacement at point i above the bottom of the pipe at the self-fixing point; n is the number of test hole sections, n ¼ H/0.5, where H is the hole depth. The working principle of the inclinometer is shown in Fig. 8.2. ②

Processing the monitoring data.

The physical measurement of displacement and inclination angle is carried out by using the data read by the inclinometer. The calculation equation for horizontal displacement is as follows: AO direction displacement:

BO direction displacement:



 A0i  A180i =25, 000 ΔA0i ¼ L 2  B0i  B180i =25, 000 ΔB0i ¼ L 2

(8.3)



Actual resultant horizontal displacement at any elevation is: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 !2 !2 3 u n n X X u di ¼ t4 ΔA0i + ΔB0i 5 i¼1

i¼1

(8.4)

(8.5)

180 Chapter 8 Actual horizontal displacement at any elevation is: 8 n X > > AO ¼ ΔA0i > < i i¼1

n X > > > ΔB0i : BOi ¼

(8.6)

i¼1

In the formula: AOi(BOi)—The actual horizontal displacement in the direction of AO (or BO) of the segment length of each measuring point, units: mm. The technical parameters of several common inclinometers used in China and other countries are shown in Table 8.3. (5) TDR A common problem with inclinometers is that the acquisition of monitoring data needs to be carried out on site manually and regularly. This means that the landslide monitoring is not in real time. In addition, in many cases, unstable high-speed railway landslides are located in remote areas, difficult to reach and possibly dangerous, and data collection cannot be completed in a timely manner. By contrast, the landslide monitoring system based on TDR technology has prominent advantages. TDR is short for time domain reflectometer. It is a remote electronic measurement technology first used in the power and communication industries to determine faults and fractures of communication cables and transmission lines. In the late 1970s and 1980s, TDR technology was widely used by the US Bureau of Mines to find collapsed layers in longwall coal mines. Since the 1990s, TDR technology has been widely used in landslide monitoring in developed countries such as Europe and the United States. ①

The principle of TDR landslide monitoring.

A complete TDR landslide monitoring system is generally composed of a TDR coaxial cable, cable tester, data recorder, remote communication equipment, and data analysis software. When using a TDR system for landslide monitoring, the first step is to drill a hole at a certain location of the landslide and place the TDR coaxial cable into the hole. Then, connect the TDR cable to the cable tester. The cable tester acts as a signal source, sending a stepping voltage pulse through the cable and reflecting the pulse signal from the cable. The data recorder is connected to the cable tester, and it controls the cable tester, recording and storing the pulses reflected from the cable for later analysis. Data recorders can also be connected to remote communications devices such as mobile phones or shortwave radios to send data to distant locations. The TDR system can also be equipped with a multiplexer to monitor multiple points simultaneously.

Table 8.3 Technical parameter list of several common inclinometers used in China and other countries.

Type of inclinometer

BC-10 strain inclinometer

Range of measurement

10°

The sensitivity

The linear error Temperature drift Time drift Measuring head weight /kg Insulation resistance Transformation method Measuring head size Roller spacing With inclined tube size

  1%FS

GC-1 servo accelerometer inclinometer 20° (Special specifications 60°) F1 ¼ 0.02 mm/0.1 mV F2 ¼ 8.4 mm/0.1 mV 1%

4.8

Vertical bidirectional digital inclinometer

SX-20 servo inclinometer

CX-01 servo accelerometer inclinometer

10°

0
 50°

The vertical direction 53°

87.16 mm/m

1 m file 0.01 mm

0.02 mm/800

0.02 mm/500 mm

0.01 mm/m

0, the slope is in the accelerated deformation stage.

9.1.4 Forecast objects and range 1) Forecasting objects include the following: (1) Lots or blocks with large deformation rate; (2) Lots or blocks that can cause serious damage; (3) Lots or blocks that play a key role in the stability of the entire landslide or rockfall; (4) Representative parts or blocks of deformation and destruction of the entire landslide or rockfall. 2) Forecasting range and scope. (1) The scope of the disaster forecast shall include: The extent of the actual landslide and/or rockfall. The range reached by landslide and rockfall movement. Secondary disasters caused by landslides and rockfalls (such as surging waves, blocking rivers, blocking canals, and rapidly transforming landslide/rockfall into a debris flow under heavy rain conditions); The scope affected by earthquake, rainstorm, and other disasters. (2) The following conditions shall be taken into account when determining the disaster scope: The scale, scope, form, and direction of landslide and rockfall movement. Landform, geomorphology, geology, and hydrology conditions in the sports ground of landslides and rockfalls. The possibility of velocity and acceleration of landslides and rockfalls producing an air cushion floating effect, refraction rebound, and/or multistroke in canyon areas. The possibility and scope of secondary disasters, as the hazards caused by surges, river obstruction, and canal obstruction should be analyzed under different water levels and flow conditions with different landslide scales (soil and rock volume) and motion speeds.

9.1.5 Forecast level The prediction grades of landslide and rockfall deformation and failure are divided into three grades based on time: prediction level, forecast level, and alarm level, as shown in Table 9.1.

256 Chapter 9 Table 9.1: Grade table of landslide and rockfall prediction. Forecast level

Time

Space

Methods

Indicators

Long-term forecast (prediction level)

More than 2 years

Region, the monomer

Survey evaluation and monitoring

Threshold of danger

Medium range forecast (prediction level)

1–2 years or more

Region, the monomer

Survey evaluation and monitoring

Threshold of danger

Short-term forecast (forecast level)

A few days to 1 year

Small amount of area, main monomer

Survey evaluation and monitoring

Threshold

Disaster prediction (alarm level)

Within a few days

The monomer

Monitor

Warning value

Specific test method 1. Risk zoning and database 2. Deformation displacement monitoring 1. Risk zoning and database 2. Deformation displacement monitoring 1. Analysis of regional nature, geomorphology, geology, and social factors 2. Deformation displacement monitoring 1. Deformation displacement monitoring and physical quantity monitoring such as geophone 2. Macroscopic deformation monitoring 3. Monitoring of related factors such as meteorology, hydrology, and geology

9.1.6 Classification and practical analysis of forecast model Landslide forecasting studies began with the empirical formula of landslide forecasting proposed by Japanese scholar Saito in the 1960s, and thus it has a history of more than 50 years. During this period, many experts and scholars in China and other countries have devoted themselves to research and exploration and have proposed various theoretical models and methods for landslide prediction. The forecasting models and methods currently proposed can be divided into the four categories described in the following paragraphs (Table 9.2).

Landslide and rockfall prediction technology 257 Table 9.2: Summary of landslide prediction models and methods. Summary of landslide prediction models and methods Deterministic forecasting model

Statistical forecasting model

Nonlinear forecasting model

Saito forecast method, Fukuoka model, HOCK method, K•KAWAWURA, Su AJ model Creep spline joint model Sliding body deformation power method Landslide deformation analysis and prediction method Limit equilibrium method Grey GM [1,1] model, traditional GM [1,1] model, GM [1,1] model of nonequal time series, metabolic GM [1,1] model, optimized GM [1,1] model, Stepwise iterative method GM [1,1] model, etc. Biological growth model (Pearl model, Verhulst model, Verhulst inverse function model) Multivariate nonlinear correlation analysis Exponential smoothing Slope creep prediction model (GMDH forecast method) Orthogonal polynomial best approximation model Grey displacement vector angle method BP neural network model Collaborative prediction model Short-term forecasting of BP-GA hybrid algorithm for landslide forecasting Collaboration-bifurcation model forecast just before sliding Mutation theory prediction (apical point mutation model and grey cusp point mutation model) Dynamic fractal tracking Long-term prediction of nonlinear dynamic models Displacement dynamics analysis

Applicable stage Accelerated creep phase Probable slip forecast Probable slip forecast Short-term forecast Long-term forecast Short-term forecast

Short-term forecast Medium- and long-term forecast

Short-term and slip forecast Short-term forecast Probable slip forecast Short-term forecast Probable slip forecast Short-term forecast Short-term forecast Long-term forecast Long-term forecast

Note: The macro forecast model varies from individual to individual and is not included in the table.

258 Chapter 9 1) Deterministic prediction model. The deterministic model quantifies the various parameters related to the landslide and its environment, and uses rigorous reasoning methods, especially mathematical and physical methods, to carry out accurate analysis and obtain clear prediction judgments. Such model predictions can reflect the physical nature of landslides and are more suitable for landslide or slope single prediction. The representative forecasting models are: Saito forecast method, Fukuoka model, Su AJ forecast model, and the limit analysis method. In addition to the limit analysis method, these models can be used for long-term forecasting of landslides, and the rest are basically short-term and forward-slip forecasting models. 2) Statistical forecasting model. The statistical forecasting model mainly uses various statistical methods and theoretical models of modern mathematical statistics, focusing on the macroscopic survey and statistics of the existing landslides and their geological environment factors and their external factors, obtaining their statistical laws and fitting them. The displacement-time curve of different landslides is extrapolated according to the model built for prediction. Such models are mostly applicable to regional land use and land development planning and have macrodecision significance. Its representative forecasting models include: grey GM [1,1] model, biological growth model, regression analysis method, exponential smoothing method, and golden section method. These methods are related to the quantity and time series of monitoring data. As long as there is sufficient and equally spaced displacement monitoring data, the accuracy of the forecast can be guaranteed. In addition to the golden section method used for medium- and long-term forecasting, most of the models are applicable to short- and medium-term and forward-slip forecasts. 3) Nonlinear forecasting model. With the development of nonlinear science and its wide application in various fields, landslide researchers began to realize that the landslide system is an open system, which is grey and white, deterministic and random, gradual and abrupt, balanced and unbalanced. Complexity is the fundamental attribute of landslides, such as order and disorder. Therefore, many scholars have cited nonlinear science theory as effective in dealing with complex problems to study the prediction of landslides and have proposed a series of landslide prediction models. Its representative forecasting models include the nonlinear dynamics model, back propagation (BP) neural network model, catastrophe prediction model, collaborative prediction model, dynamic fractal tracking forecasting model, and displacement dynamics analysis. Such models have high precision in theory and have broad application prospects. Except for the fractal dimension forecasting model, the nonlinear dynamic model and the displacement dynamics analysis method for medium- and long-term forecasting, the rest are short-term and prosliding forecasting models.

Landslide and rockfall prediction technology 259 4) Macro forecasting model. The complexity and particularity of landslides determine that there may be large differences in the magnitude and trend of the displacement of each point on the landslide. In order to avoid the phenomenon of point-to-face and eliminate the interference of some special factors, many scholars have advocated the organic combination of the macroscopic information on the deformation and damage of the slope and the landslide monitoring data, based on various deformation and destruction forms of the landslide. Macroscopic predictions of landslides, such as signs and predisposing factors. The landslide macro forecasting model is based on various precursors and signs displayed during the landslide deformation until the final damage process. Fuzzy and critical weighting average methods are used to predict the landslide characteristics. The deformation evolution process of general landslides can be divided into four stages: slow creep, uniform creep, accelerated creep, and sharp deformation. Based on the macro forecasting model, the deformation stage of the landslide can be identified. 5) Methodological principles for establishing models. (1) Fully demonstrate the causal relationship between the conclusions of the model prediction and the deformation trace information and macropredictive information. (2) After the establishment of the forecasting model, the inversion analysis should be carried out by using similar monitoring data from landslides and rockfalls that have occurred, verifying the validity of the model, and preliminarily determining the corresponding forecasting criteria. (3) Multimodel and multicriteria forecasting should be carried out, and comprehensive analysis should be conducted to establish an appropriate and effective monitoring and forecasting model. The theoretical forecasting model should not be used solely for forecasting. (4) Actively research and improve the monitoring and forecasting model and establish models and criteria that are consistent with regional characteristics. (5) The forecasting model and forecasting criteria shall be reviewed and appraised by the competent department or unit. (6) Similar simulated tests should be carried out of the main triggering factors in the landslide to be predicted, to achieve the expected degree of deformation and damage, using the monitored data and the measured critical deformation failure parameters, to establish, select, or improve the forecasting model. (7) Use the engineering analogy analysis method to analyze successful prediction models using similar conditions.

9.1.7 Classification and practical analysis of forecasting criteria The core of landslide time forecasting is the forecasting model and forecasting criteria. At present, domestic and foreign scholars have proposed more than 10 kinds of prediction criteria

260 Chapter 9 for judging the critical instability state of slopes, such as stability coefficient, reliability probability, deformation rate, and displacement acceleration, which can be summarized as Table 9.3. The following three types of commonly used landslide prediction criteria are presented, namely safety factor and reliable probability criterion, deformation rate criterion, and macro information prediction criterion. 1) Safety factor and reliable probability criterion. The safety factor is calculated by the limit equilibrium method or the ratio of the total internal force and external force consumed when the slope has slipped, using the limit analysis method. It is generally considered that the safety factor criterion is determined to be 1 as appropriate. If the safety factor is less than 1, the slope is in an unstable state; if the safety factor is greater than 1, the slope is in a stable state; when the safety factor is equal to 1, the slope is in a critical equilibrium state. The reliable probability is the reliability index (Ps) of the slope stability calculated in recent years based on the reliability theory. It is generally accepted that it is appropriate to set a reliability probability criterion of 95%. The reliability probability criterion gives the safety Table 9.3: Various forecast criteria for landslides. Criterion name

Criterion value or range

Applicable conditions

Stability factor K

K1

Long-term forecast

Reliable probability Ps

Ps  95%

Long-term forecast

Acoustic emission parameter

K ¼ A0/A  1

Long-term forecast

Deformation rate

Due to geological conditions

Short-term prediction of rock and soil slope

Displacement acceleration a

α 0

Probable slip forecast

Creep curve tangent angle a

α 70°

Probable slip forecast

Displacement vector angle

Suddenly increase or decrease

Probable slip forecast

Critical rainfall intensity

Depending on the region

Rainstorm-induced landslide Probable slip forecast

Two parameter criterion Creep curve tangent angle and displacement vector angle Displacement rate and displacement vector angle

The displacement vector angle suddenly increases or decreases Accumulation landslide The displacement rate increases or exceeds the forecast just before sliding critical value, and the displacement vector angle changes significantly

Landslide and rockfall prediction technology 261 index of the slope. Considering the variability of the shear strength c andΦ of the rock and soil, the results are more realistic. Both the safety factor and the reliability probability criterion are applicable to the long-term forecast of landslides and are commonly used criteria for long-term landslide prediction. 2) Deformation rate prediction criterion. The occurrence of a landslide is caused by the movement of the material on the slope at a certain speed along a sliding surface. Therefore, it is more intuitive and reliable to use the magnitude of the material deformation rate on the slope as the forecast criterion for whether or not a landslide will occur. At present, the landslide deformation rate is directly or indirectly used as a criterion to predict landslide occurrence. However, due to the influence of various factors such as the composition of the landslide, deformation and failure modes, and externally induced factors, the deformation rate before the final instability is very different (Table 9.4). Obviously, it is not realistic to determine a uniform landslide deformation rate. The statistical results of some existing landslides show that the deformation rate before the occurrence of the landslide ranged from 0.1 to 1000 mm/d, so the difference is large. The critical deformation rate of a general clay slope is 0.1 mm/d; a rock slope is generally 10, 14.4, or 24 mm/d; the critical deformation rate of landslide can therefore be quite different: for example, in the Xintan landslide, the deformation rate was 116 mm/d, while the deformation rate of a Yellowstone landslide was only 2 mm/d. Therefore, when the deformation rate is used as the prediction criterion for the landslide, an in-depth engineering geological analysis of the predicted landslide is necessary. 3) Macro information forecasting criterion. Similar to other natural disasters such as earthquakes and volcanoes, landslides also exhibit a variety of macroscopic precursors before they become unstable: frequent rockfall of the leading edge, sudden changes in groundwater level, geothermal and/or geoacoustic anomalies, animal performance, etc.; due to these phenomena before a slip being intuitive and easily captured by humans, they are effective for forecasting. For the macro signs of landslides, the following points are generally recognized: (1) Macroscopic signs in the landslide area and the landslide reflect the evolution of a landslide, especially on the eve of the landslide. (2) In surface (face) deformation and ground deformation, the amplitude (strength) is different at different development stages of the landslide. (3) Creep and rupture sounds from the rock and soil and a roaring sound can often be heard before a large slide. (4) Animal abnormalities can be used as a criterion for landslides.

262 Chapter 9 Table 9.4: Deformation rate before landslide failure. Landslide name

Deformation before failure

Guizhou Qinglong landslide (rainfall triggered)

Artificial rainfall of 1.3 h, rain intensity of 60 mm/h, maximum deformation rate of 25.6/d Guizhou Qinglong landslide (excavation trigger) After 62–68 h of excavation, the sliding began. The deformation rate of the sliding slip was 13.3–21. 6 mm/d. After the crack increased to 3 mm, the sliding stopped, and then the whole body experienced rockfall under a heavy rain of 45 mm/d Guizhou Yongning landslide The atmospheric rainfall was 140 min, the rain intensity was 40 mm/d, and the deformation rate was 26.38/d Xintan landslide One month before slipping, the average deformation rate of A3 and B3 points was 85.9–399 mm/d Jiujiang River landslide One month before the sliding failure, the deformation rate was about 10 mm/d, and the crack width increased by about 10 mm every day. Lijiatun landslide 22 d before the sliding failure, the average horizontal direction deformation rate was 8.2 mm/d, and the vertical direction deformation rate was 9.2 mm/d Zipingpu Hydropower Junction 2# Diversion Tunnel The cumulative maximum horizontal displacement at Exit Landslide 10 d before the damage was 150.1 mm, and the maximum vertical displacement was 97.6 mm Yanchi River landslide 32 d before the landslide, the deformation rate of the mountain was 10 mm/d Dazhongchuan flute slope (Japan) 1 d before sliding, deformation rate was 24 mm/d Jiming Temple landslide The deformation rate was less than 1 mm/d at 15 months before sliding, the deformation rate was more than 1 mm/d at 4 months before sliding, the deformation rate was above 10 mm/d at 2 months before sliding, and the deformation rate was above 50 mm/d at 10 d before sliding. 1 d deformation rate above 100 mm/d Lichenghe landslide on Baocheng railway The average horizontal displacement rate was 8.2 mm/d for the first 22 days before the failure, and the average vertical displacement rate was 9.2 mm/d No. 377 landslide on Chengkun railway Horizontal displacement before sliding was 5–10 mm/d, vertical was 1–5 mm/d Huanglong Xicun landslide 6 d before sliding, 7 mm/d, 1 d before sliding, 300 mm/d Daye Iron Mine, Elephant Trunk Hill, North Gang The displacement rate 1d before sliding was greater landslide than 1000 mm/d

There are advantages and disadvantages to using the preceding forecasting criteria to predict the instability time of slopes. The safety factor and the reliability probability criteria can only judge whether the slope is stable. The deformation rate criterion is relatively straightforward, but it is quite difficult to determine the critical deformation rate of the slope. The information

Landslide and rockfall prediction technology 263 forecasting criteria are more accurate but have strong individual characteristics. The literature points out that some successful predictions have been made for some landslides based on the “warping tail” phenomenon of the displacement time relationship curve or the occurrence of a sudden change section and a sharp increase of the deformation rate. In China, macro information forecasting criteria were used to successfully predict the Xujiahe landslide on the Baocheng Line. However, for all landslides there are currently no reasonable or clear physical meanings, and almost all the criteria presented are imperfect. These forecasting criteria have a certain applicability but generally are not sufficient and universal, so there are some cases of false positives. The insufficiency of landslide prediction criteria is mainly due to the different deformation mechanisms of slope rock masses. There is no uniform movement behavior before the sliding of different slopes, and it is impossible to use a unified curve to characterize this behavior. It is difficult to set an objective standard for the quantitative indicators or thresholds of a forecast, and it is inevitable that there are subjective factors. Therefore, it is not a comprehensive solution to predict landslides solely by landslide forecasting criteria.

9.1.8 Principles and methods for establishing criteria (1) After the establishment of the forecasting model, the inversion analysis should be carried out using monitoring data from similar landslides and rockfalls that have occurred, to verify the validity of the model and to determine the corresponding forecasting criteria. (2) Multimodel and multicriteria forecasting should be carried out, and comprehensive analysis should be conducted to establish an appropriate and effective monitoring and forecasting model. The theoretical forecasting model should not be used solely for forecasting, as shown in Table 9.5. (3) Actively study and improve the monitoring and forecasting model and establish models and criteria that are consistent with regional characteristics. The forecasting model and forecasting criteria shall be reviewed and appraised by the competent department or unit. (4) Similar simulated testing should be carried out of the main triggering factors in the landslide to be predicted, to achieve the expected degree of deformation and damage, using the monitored data and the measured critical deformation failure parameters to study and establish the forecasting criteria. (5) Engineering analogy analysis method should be used to analyze successful prediction models and criteria using similar conditions.

Table 9.5: Theoretical forecasting model.

Table 9.5:

Theoretical forecasting model—cont’d

Notes: 1. In the table, t0 is the time of landslide instability, Vcr is the critical failure rate, determined by analogy or similar model test, and a1, a2, a3 are regression coefficients. 2. In the table, tr is the time of landslide instability; t1, t2, t3 are the times on the time displacement curve of the monitoring points in the acceleration deformation stage of the landslide. 3. Stability coefficient (K) is suitable for collapse prediction.

266 Chapter 9

9.2 Theory and method of landslide and rockfall prediction 9.2.1 GM [1,1] grey prediction A system in which some of the information is known and some is unknown is called a grey system. Grey system theory is the study of theories dealing with grey system analysis, modeling, prediction, decision-making, and control. The core of grey system theory is the grey prediction model. The salient features of this model are the generator function and the grey differential equation. Grey system theory holds that all random quantities are grey quantities and processes that vary within a certain range and over a certain period of time. The purpose of processing of grey quantities is not to find their statistical law and probability distribution, but through a certain processing method, the chaotic raw data column becomes more regular time series data, and then a prediction model can be established. The purpose of processing the original data is twofold: to provide intermediate information for establishing a model and to lessen the volatility of the original data. (1) First, the measured known data X(0) ¼ {X(0)(1), X(0)(2), …, X(0)(n)} is accumulated and generated to obtain a new data sequence: n o Xð1Þ ¼ Xð1Þ ð1Þ, Xð1Þ ð2Þ, …, Xð1Þ ðnÞ In this equation Xð1Þ ðiÞ ¼

k P i¼1

(9.23)

Xð0Þ ðiÞ newly generated data is listed as a monotonous growth

curve, which increases the regularity of the original data column and lessens its volatility. Such a data sequence generated by a certain method is called a “module” in a geometric sense; a module composed of a known data column is called a white module, and a module extrapolated by white module modeling, that is, a module composed of predicted values, is called a grey module. (2) The corresponding differential equation of the model is: dXð1Þ ðtÞ + aXð1Þ ðtÞ ¼ b dt In the equation a, b is the parameter to be estimated:

 1  T  a^ ¼ BT B B Y ^ b

(9.24)

(9.25)

Landslide and rockfall prediction technology 267 2 1 3  1  Xð1Þ ð1Þ + Xð1Þ ð2Þ 6 2 7  6 1 7 6  Xð1Þ ð2Þ + Xð1Þ ð3Þ 17 In this equation, B ¼ 6 7. 6 2 7 … 4  5 1  ð1Þ ð1Þ  X ðn  1Þ + X ðnÞ 1 2 Based on the least-squares method, the first-order differential equation can be solved as: " # ^ ^ ð1Þ b b ðkÞ ¼ Xð0Þ ð1Þ  e^aðk1Þ + b X (9.26) a^ a^ In the equation, (k ¼ n + 1,…,n + m) is a grey prediction sequence, and the predicted number m can be determined according to actual needs. bð1Þ ðkÞ reduction to data X bð0Þ ðkÞ (3) Restore X bð0Þ ðkÞ ¼ X bð1Þ ðkÞ  X bð1Þ ðk  1Þ X

(9.27)

bð0Þ ðkÞ is the grey prediction value of displacement when (k ¼ 2, …, n, n + 1, …, n + m). X (4) The accuracy test of the GM [1,1] model is usually carried out using the posterior variance bð0Þ : test and modeled by the GM [1,1] method to find X n ð0Þ o bð0Þ ¼ X b ð1Þ, X bð0Þ ð2Þ, …, X bð0Þ ðN Þ X

(9.28)

bð0Þ ðkÞ eðkÞ ¼ Xð0Þ ðkÞ  X

(9.29)

e ¼ ½eð1Þ, eð2Þ, …, eðN Þ

(9.30)

Calculate the residual:

The residual vector is:

bð0Þ and the residual sequence e are denoted as S21 and S22, The variances of the original sequence X respectively. S21 ¼

N   1X ð0Þ 2 Xð0Þ ðkÞ  X N k¼1

(9.31)

268 Chapter 9 S22 ¼ In the equation, X

ð0Þ

¼

N 1X ðeðkÞ  eÞ2 N k¼1

(9.32)

N 1X eðkÞ N k¼1

(9.33)

S2 S1

(9.34)

N 1X Xð0Þ ðkÞ N k¼1

e¼ The posterior variance ratio is:

c¼

The model accuracy level index C value can be divided into indication levels as in Table 9.6.

9.2.2 Exponential smoothing prediction The exponential smoothing method was developed by the American mathematicians Brown and Holt in the late 1950s and has been applied in many fields. This method has the advantages of a simple calculation process and easy mastering. From a fundamental point of view, exponential smoothing is a nonstatistical method. There are some basic mathematical models in each time series of the method, and the actual monitoring value is the reflection of the interaction between this basic model and random changes. The goal of the exponential smoothing method is to “smooth” the historical data to distinguish the basic data model and the random variation, i.e., by eliminating the maximum and minimum values in the historical data, the smoothed value of the time series is obtained, as the predicted value in a future period. This method continuously uses the “error-fed” principle to correct the predicted values throughout the prediction process. The exponential smoothing method currently has a variety of types, such as the mobile arithmetic average method, the single exponential smoothing method, and the quadratic curve exponential smoothing method. When the time series increases or decreases according to the shape of the quadratic curve, prediction using the quadratic curve exponential smoothing method is quite effective. It continuously adjusts the predicted value as the time series grows, and the linear growth factor of the time series and its parabolic linear growth factor are considered in the calculation. However, when the exponential smoothing method is used for prediction, the initial accuracy is slightly poor, which is mainly Table 9.6: Error check level indicator. Prediction accuracy level

Excellent

Good

Medium

Qualified

Failed

C

0.70

Landslide and rockfall prediction technology 269 affected by the initial value. As the prediction process progresses, the prediction accuracy will gradually increase. Slope sliding is a gradual process, and its deformation tends to increase with time. Therefore, the quadratic curve smoothing method can be used to predict the deformation trend of the landslide. This method can adjust the predicted value continuously with the increase of a time series, especially suitable for midterm forecasting of landslides, which generally obtains higher forecasting accuracy. The second exponential smoothing method is divided into seven steps in the calculation process: (1) Calculate the single exponential smoothing value S1t of the t period S1t ¼ α  Xt + ð1  αÞ  Sð1t1Þ

(9.35)

(2) Calculate the double exponential smoothing value S2t of the t period S2t ¼ α  S1t + ð1  αÞ  Sð2t1Þ

(9.36)

(3) Calculate the triple exponential smoothing value S3t of the t period S3t ¼ α  S2t + ð1  αÞ  Sð3t1Þ

(9.37)

(4) Calculate the horizontal value At of the tperiod At ¼ 3S1t  3S2t + S3t

(9.38)

(5) Calculate the linear increment Bt of the t period Bt ¼

α 2ð1  αÞ2

½ð6  5αÞ S1t  ð10  8αÞ S2t + ð4  3αÞ S3t 

(9.39)

(6) Calculate the parabolic increment Ct of the t period Ct ¼

α2 ð1  αÞ2

½S1t  2S2t + S3t 

(9.40)

270 Chapter 9 (7) Predicting displacement observations Ft+m (m 1, positive integer) after (t + m) periods:

1 Ft + m ¼ At + Bt m + Ct m2 2

(9.41)

The initial value of the second exponential smoothing method depends on the monitored values of the first two periods. Usually, the initial value can be taken as: S11 ¼ S21 ¼ S31 ¼ X1; S12 ¼ α  X1 + (1  α)X1; S22 ¼ α  S12 + (1  α)X1 1 1 S32 ¼ α  S22 + ð1  αÞX1 ; F2 + m ¼ A2 + B2 m + C2 m2 ; F2 + m ¼ A2 + B2 m + C2 m2 2 2

(9.42)

It can be seen that when the exponential smoothing method is applied, the first predicted value does not start from the initial time, but starts from the 2 + m time, where α is called a smoothing constant, and the value ranges from 0 to 1. Generally, when the value is large, it means that the more information contained in the recent data, the larger the correction range, and the predicted value can reflect the actual change state of the time series faster, i.e., the time series comparison of Ft+m to the greater degree of change. This method is sensitive, but the curve is not smooth enough; when the value of α is small, the predicted value Ft+m is relatively slow to reflect the time series, and the correction range is also small. The longer the data sequence, the smoother the curve is. Therefore, choosing the smoothing constant α has a great influence on the prediction result. A computer-preferred method can be used to compare the mean square error of prediction errors of different α values and select a smoothing constant that minimizes the mean square error as a calculated value. The exponential smoothing method is generally applicable to medium- and short-term forecasts. Therefore when it is used for landslide prediction, the forecast step size should not be too large; otherwise it will cause a large error. The sampling time used in the forecast is related to the forecast time. When using a medium-term forecast, the input monitoring data should adopt the monthly deformation value. When using a short-term forecast, the input monitoring data should adopt the daily deformation value. Accordingly, the accuracy of the medium-term forecast is measured in months, while the accuracy of short-term forecasts is measured in days.

9.2.3 Nonlinear regression analysis Regression analysis is an effective method to study and determine the connection and form between variables through statistical analysis and processing of observation data. According to the different functional forms between independent and dependent variables, regression analysis can be divided into linear regression and nonlinear regression. Using the nonlinear

Landslide and rockfall prediction technology 271 regression method, a mathematical model of the displacement-time relationship of a landslide can be established to further predict its deformation trend. The general form of a nonlinear regression model is: y ¼ f1 ðx1 Þ + f2 ðx2 Þ + f3 ðx3 Þ + ⋯ + fn ðxn Þ

(9.43)

where fi(xi), i ¼ 1, 2, …, n can be a polynomial, exponential, trigonometric, or other functional form. The use of exponential regression is explained in the following paragraphs. Suppose there is a set of time-corresponding landslide displacement observation data xi, yi (xi is the time of each stage, yi is the landslide displacement corresponding to each time), and the physical equation it expresses is represented by the function Y ¼ f(x). This set of data xi, yi is fitted using a polynomial. Let the regression equation of this polynomial be: y ¼ a0 + a1 x + a2 x2 + ⋯ + an xn

(9.44)

where n is a polynomial power exponent. The larger n is, the higher the fitting accuracy. Generally, n > 3 can meet the engineering requirements. In this study, take n ¼ 3; a0, a1, a2, a3 are the undetermined coefficients. Assume that there are m (mn + 1) landslide displacement observations y1y2y3…, ym corresponding to time (x1, x2, x3, …, xm), 2 3 2 32 3 y1 a0 1 x1 x21 x31 6 y2 7 6 1 x2 x2 x3 76 a1 7 2 2 76 6 7¼6 7 (9.45) 4 … 5 4 … … … … 54 a2 5 ym a3 1 xm x2m x3m 2

3 2 y1 1 x1 6 y2 7 6 1 x2 7 6 In the formula, Y represents 6 4 … 5, B means 4 … … ym 1 xm Eq. (9.45) can be written as

x21 x22 … x2m

3 2 3 a0 x31 6 a1 7 x32 7 7, A means6 7, 4 a2 5 …5 3 a3 xm

Y ¼ BA

(9.46)

Multiply both BT and BT by the B transpose matrix at both ends of Eq. (9.46): BT Y ¼ BT BA

(9.47)

Let C denoteBTB. Then the left end of Eq. (9.47) becomes a matrix of four rows and one column，For a matrix of 4  4, find the inverse matrix C1 of C, and multiply C1 by the two ends Eq. (9.47); then:

272 Chapter 9 C1 BT Y ¼ C1 CA

(9.48)

Therefore, all the undetermined a0, a1, a2, a3 of A in Eq. (9.48) can be obtained, and the landslide displacement at any time can be predicted by substituting into Eq. (9.44).

9.2.4 Grey Verhulst prediction model The Verhulst model is a biological growth model proposed by the German biologist Verhulst in 1987. He believes that the process of reproduction, growth, maturity, and death can be described and predicted by this model. Yan (1987) introduced this model into the study of landslide time forecasting, considering that the evolution of a landslide also has a process of deformation, development, maturity, and destruction, and can have development and evolution similar to a biological entity. Set the original equal-pitch monitoring data sequence X(0)(t): X(0)(t) : X(0)(t) ¼ {X(0)(1),X(0)(2), …,X(0)(n)}: Create an AGO transform to obtain a new data sequence: X(1)(t) ¼ {X(1)(1),X(1)(2), …,X(1)(n)}, and fit the Verhulst first-order whitening nonlinear differential equation to X(1)(t): h i2 dXð1Þ ðtÞ ð1Þ ð1Þ ¼ aX ðtÞ  b X ðtÞ dt

(9.49)

In the equation, a, b are undetermined coefficients, obtained by the least-squares method:  1 (9.50) a^ ¼ ½a, bT ¼ BT B BT Y Substituting the obtained undetermined coefficient into the preceding equation to solve the nonlinear differential equation gives: bð1Þ ðtÞ ¼ X

a=b  a 1  1 eaðtt0 Þ 1+  b Xð0Þ ð1Þ 

(9.51)

Eq. (9.51) is the Verhulst nonlinear differential dynamic prediction model for landslide time, and t0 is the initial time. Since the evolution of landslides is similar to the process of reproduction from organism to extinction, the critical value (inflection point value) a/2b of the transformation of the organism from mature (rapid growth) to extinction (slow growth) can be used as the critical displacement ð1Þ

b ðtÞ in Eq. (9.51) to solve for the time t of the value of the landslide. In this way, a/2b replaces X landslide damage: ! 1 a (9.52)  1 + t0 t ¼ ln a bXð0Þ ð1Þ

Landslide and rockfall prediction technology 273 The initial time t0 is generally 0, and Eq. (9.52) becomes: ! 1 a 1 t ¼ ln a bXð0Þ ð1Þ

(9.53)

The t in Eq. (9.53) is actually the number of time series of a landslide. The true landslide damage time t0 should be: t ¼ t  Δt

(9.54)

In this equation, Δt is the average time interval for monitoring data.

9.2.5 Verhulst inverse function residual correction model 1) Modeling thought. The Verhulst model prediction method described earlier is based on the similarity between the deformation and destruction of the slope and the evolution of biological growth. The prediction criterion is only derived from the language model of “similarity.” Therefore, the modeling process lacks a theoretical and quantitative basis, and the quantitative analysis of the error is not carried out in the modeling, resulting in large errors in the forecasting results. The quantitative characteristic curve of the Verhulst model is an “S” shape, and the displacement-time curve of the slope deformation failure is often an inverse “S” shape, which is exactly opposite to the Verhulst model curve (see Fig. 9.7). From the perspective of quantitative information, when the inverse function of the Verhulst model is used to fit the deformation characteristics of the slope, the modeling basis is more acceptable. At the same time, the Verhulst inverse function model can be used to fit the original data and predict the future trend. The necessary conditions for achieving higher precision are: nonnegative, equal time interval, monotonous, when the Displacement (X) Inverse function curve

X=t

Verhulst curve

Time (T)

Fig. 9.7 Verhulst forecast model and its inverse function characteristic curve.

274 Chapter 9 original data fluctuates greatly, or when the modeling conditions are not very good. When the model accuracy is low, the BP neural network prediction model can be applied to nonlinear dynamic data systems with large volatility. At this point, it simply makes up for the defects of the Verhulst inverse function prediction model. To this end, the Verhulst inverse function residual correction model is established. The residual sequence {ε(0)(i)} of the Verhulst inverse function prediction model is remodeled using the BP neural network prediction model, and the Verhulst inverse function prediction value is added to the BP neural network residual prediction value, that is, the predicted value of the Verhulst inverse function residual correction model of the slope displacement. Therefore, this chapter uses the grey Verhulst inverse function residual correction model to analyze the deformation and failure characteristics of slope instability and to establish a prediction model. 2) Modeling steps. The modeling steps of the Verhulst inverse function residual correction model are: (1) Using the five-point cubic polynomial motion smoothing filtering method, the original monitoring data is separated by signal and noise, and a filtered value sequence {X(0)(t), t ¼ 1, 2, …, n} of the original monitoring data is obtained. (2) Perform a cumulative generation transformation (1  AGO) on the filtered value sequence {X(0)(t), t ¼ 1, 2, …, n}: n o Xð1Þ ðtÞ ¼ Xð1Þ ð1Þ, Xð1Þ ð2Þ, …, Xð1Þ ðnÞ

(9.55)

(3) Fitting X(1)(t) to a Verhulst first-order albino nonlinear differential equation gives:  2 dXð1Þ ðtÞ ¼ aXð1Þ ðtÞ  b Xð1Þ ðtÞ dt

(9.56)

The solution of this nonlinear differential equation is: bð1Þ ðtÞ ¼ X

a=b  a 1  1 eaðtt0 Þ 1+  b Xð0Þ ð1Þ 

(9.57)

bð1Þ ðtÞ and exchange the variables to get the Verhulst inverse (4) Find the inverse function of X function prediction model: bð1Þ ðtÞ ¼ 1 ln ða  bt0 Þt + Xð0Þ ð1Þ X a ða  btÞt0 In the formula, t is the time sequence; a and b are the undetermined coefficients.

(9.58)

Landslide and rockfall prediction technology 275 According to the principle of least squares, a and b are obtained by the following equation: 8 nX + t0 1 nX + t0 1 > > > d ðkÞ + k > > > k¼t0 k¼t0 > > > a¼ > > n " # > > nX + t0 1 nX + t0 1 nX + t0 1 > < ½ d ðk Þ  k   dðkÞ  k =n (9.59) > k¼t k¼t k¼t 0 0 0 > >b ¼ !2 > > nX + t0 1 nX + t0 1 > > > > k =n  k2 > > > > k¼t0 >  k¼t0 : dðkÞ ¼ 1= Xð0Þ ðk  t0 + 1Þk In Eq. (9.59), k has the same meaning as t, and is a time series. (5) The value of the initial time sequence t0. t0 is related to modeling and cannot be simply taken as 0 or 1. An optimal solution for its value should be searched for, using computer optimization, so that the average relative error E of the system output meets certain accuracy requirements, namely: 0

1 ð0Þ

ð0Þ

b n 1 X B X ðiÞ  X ðt0 + i  1Þ C (9.60) E¼ @ Am n i¼1 Xð0Þ ðiÞ In the equation, m is the given accuracy requirement. n ð0Þ o b ðtÞ of the Verhulst inverse function model (6) Calculate the restored value sequence X using: X

ð0Þ

ð1Þ

ð1Þ

ðt + 1Þ ¼ X ðt + 1Þ  X ðtÞ

(9.61)

(7) Find the residual sequence {ε(0)(t)}: εð0Þ ðtÞ ¼ xð1Þ ðtÞ  x^ð1Þ ðtÞ, t ¼ 1,2,…, n

(9.62)

(8) Before the residual sequence {ε(0)(t)} is trained in the BP network, in order to cause the sample parameter variation range of the parameter to fall within a given interval and enhance the display of the data information, a normalization process is performed in advance as follows: Xi0 ¼

Xi  Xmin ði ¼ 1, 2, …, nÞ Xmax  Xmin

In the equation,Xmax ¼ max {X1, X2, …, Xn}

Xmin ¼ min {X1, X2, …, Xn}

(9.63)

276 Chapter 9 0

The BP neural network modeling method was used to obtain the predicted value {ε(0) (t)} of the n 0 o n 0 o residual normalized sequence ^εð0Þ ðtÞ , and then ^εð0Þ ðtÞ was substituted into Eq. (9.63) to n 0 o obtain the predicted value ^εð0Þ ðtÞ of the residual sequence using the BP neural network. (9) The Verhulst inverse function residual correction model is: bð1Þ ðt + 1Þ ¼ 1 ln ða  bt0 Þt + Xð0Þ ð1Þ + ^εð0Þ ðt + 1Þ ðt ¼ 1, 2, …, nÞ X a ða  btÞt0

(9.64)

bð1Þ ðt + 1Þ  X bð1Þ ðtÞ ðt ¼ 1, 2, …, nÞ bð0Þ ðt + 1Þ ¼ X X

(9.65)

3) Forecast criteria and time forecast. bð1Þ ðtÞ ! + ∞， ln ðabt0 Þt ! + ∞，(a  bt)t0 ! 0. By Eq. (9.64)，when X ðabtÞt0 If in the initial time sequence t0 6¼ 0, generally t0 0, then t ! a/b. It can be seen that when the amount of slope deformation tends to infinity, the time t tends to a certain value a/b. Therefore, T ¼ a/b can be used as the forecast time when the landslide is unstable. Thus, the forecast time T from the starting point of the modeling data to the instability of the slope is: T ¼ a=b  t0

(9.66)

The T in Eq. (9.66) is actually the number of timings of a slope instability time. The true slope failure time T0 should be: T 0 ¼ TΔt

(9.67)

In the equation, Δt is the average interval between monitoring data.

9.2.6 BP neural network 1) Overview of BP network algorithm. There are many types of artificial neural network models, including the perceptron model, BP network model, radial basis (RBF) network model, self-organizing network model, and feedback network model. At present, most of the practical applications use the BP network or its variants. It is the core of the forward network, embodying the most essential part of the artificial neural network, and is the most widely used and most mature neural network model. The standard BP network model is a one-way propagation multilayer feedforward network consisting of three layers of neurons: input layer, intermediate layer (hidden layer), and output layer. The neurons between the levels form a full interconnection through the weights, and there is no coupling between the neurons in each level. The input signal passes through the input layer

Landslide and rockfall prediction technology 277 nodes, passes through the hidden layer nodes in turn, and finally passes to the output layer nodes. The output of each layer node only affects the output of the next layer node. The activation function of a node must be differentiable and nondecreasing. Usually, the excitation function of its neuron is taken as a sigmoid function. The output is therefore a continuous amount between 0 and 1, which enables an arbitrary nonlinear mapping from input to output. Because its weight adjustment is the calculation method that uses the difference between the actual output and the expected output, and the connection weight of each layer of the network is corrected from the back to the front layer, it is called the error back propagation learning algorithm, referred to as the BP algorithm. The BP network uses the gradient maximum descent method to change the weight along the negative gradient direction of the error function. The BP network algorithm consists of two major steps: ① the forward propagation process of the input, in which the input data is propagated from the input layer to the output layer by layer, and the output response is obtained; ② the back propagation process of the output error, i.e., the error of the output as the output layer begins to propagate back to the input layer, and the amount of weight change for each layer of the network is determined by the amount of error propagated to that layer. 2) BP network structure and algorithm. The difference between the BP network and other network models is mainly reflected in the excitation function. As shown in Fig. 9.8, the graphs of the two S-type excitation functions show that f(net) is a continuously differentiable monotonically increasing function. The output characteristics of this excitation function are relatively soft, and the value range of the output state is for [0,1] or [1, +1]; the hardness can be adjusted by the parameter λ. The input and output relational expression of the function is described in the following text.

f (net)

f (net)

1

1

net l=7 –1

Bipolar

l=7

5

1

0 Unipolar

Fig. 9.8 Sigmoid type function graph.

5

1 net

278 Chapter 9 Bipolar S-type excitation function: f ðnetÞ ¼

2 , f ðnetÞð1, 1Þ 1 + exp ðλnetÞ

(9.68)

Unipolar S-type excitation function: f ðnetÞ ¼

1 , f ðnetÞð0, 1Þ 1 + exp ðλnetÞ

(9.69)

For a multilayer network, the region partitioned by this excitation function is no longer a linear partition, but an area composed of a nonlinear hyperplane. Because the sigmoid function has a nonlinear large coefficient function, it can transform the input from negative infinity to positive infinity into an output between 1 and +1, so the sigmoid function can be used to achieve a nonlinear mapping from input to output. For different situations, BP network training has corresponding learning rules (i.e., different optimization algorithms), and follows the principle of reducing the error between expected output and actual output, and realizes function approximation, vector classification, and pattern recognition of the BP network. Fig. 9.9 shows the main process of BP network training. (1) Initialize the network according to the current internal expression of the network, and construct a reasonable network structure. Take the adjustable parameters (weights and threshold) as random numbers obeying uniform distribution on [1, 1], and determine the initial values of the expected error, the maximum number of cycles and the learning rate of corrected weights. (2) Use the corresponding BP network learning rules to perform forward calculation on the sample input mode, perform network training, obtain the squared error of the corrected Input layer

Hidden layer

Output layer

R P(1) P(2)

. . .

P(R)

P R×1

n1

W1

S1×R

S1×1

n1

a2 W2

S2×S1

S2×1

S1×1 1

1

b1

S1×1 R Input number of neurons

S2×1

n2

S1 S1 Number of neurons in the hidden layer

b2

S2×1

S2 S2 Output layer neurons

Fig. 9.9 BP network structure with one hidden layer.

Landslide and rockfall prediction technology 279 weight, and compare the error between the output of the network and the expected output. If the error is less than the specified value, then the training ends; otherwise, the error signal is propagated back in the original path, and the weight and threshold are adjusted layer by layer. This forward propagation and back propagation cycles until the error reaches the accuracy requirement. (3) At the end of the network training, the connection weight and node threshold will not change. At this time, if a new input is given to the network, the network can obtain the corresponding output result only by forward calculation. Fig. 9.10 shows the BP neural network learning algorithm flow.

9.2.7 Saito model The Saito model is a short-term forecasting model. Based on a large number of indoor and outdoor experimental studies, the Japanese scholar Saito proposed an empirical relationship between landslide time and creep variability of homogeneous slopes in 1968. Network initialization Given input vector and target vector

Find hidden layer, output layer unit output Find the deviation between the target value and the actual output. Does E meet the requirements?

Yes

End

No Calculate hidden layer unit error

Error transmission

Weight learning

Fig. 9.10 BP neural network learning algorithm VB program implementation flow chart.

280 Chapter 9 lgtr ¼ 2:33  0:516  lgε  0:59

(9.70)

In the equation, 0.59 includes a range of 95% of the measured values; tr is landslide occurrence time (min); and ε is distances between observation points. When the slope enters the accelerated creep phase, three points can be taken on the displacement-time curve, and the damage time is predicted using the following equation: tr ¼

ðt2  t1 Þ2 =2 + t1 ðt2  t1 Þ  ðt3  t1 Þ=2

(9.71)

When selecting three points, make the displacement between t1  t2 and t2  t3 equal. It can be seen that the Saito model is a deterministic model based on the monitoring curve and creep theory. When the landslide is in the accelerated creep stage, a more accurate forecast of the slipping time can be obtained.

9.2.8 Fukuoka model In 1985, the Japanese scholar Fukuoka used a sand and soil material to create a model and carried out a large-scale model test (model height, width, and thickness of 5  4  1 m3, respectively). Under artificial rainfall conditions, it was found that the displacement acceleration of the model soil surface was proportional to the square of the corresponding displacement velocity. Its expression is:  2 d2 x dx ¼A (9.72) 2 dt dt After transformation: i dt 1 h ¼ ¼ Aðα  1Þ1=ða1Þ ðtr  tÞ1=ða1Þ dx V

(9.73)

In the equation, A, α are constants, α ¼ 1.5–2.2; υ is the landslide sliding speed; t is the initial time; tr is the slope damage time; and a, b are constants. Compared with the time prediction formula of Saito and Fukuoka, the Ha Y S formula can predict the leading time. The higher the initial velocity v0 at each stage, the shorter the damage time, shown in the following: v ¼ v0  eat

(9.74)

ln v ¼ lnv0 + at lne

(9.75)

ln

v ¼ at v0

(9.76)

Landslide and rockfall prediction technology 281

9.2.9 Accuracy comparison and applicability analysis of model Based on the monitoring data of actual landslides, the forecasting accuracy of various models was tested and analyzed, and the applicable conditions were obtained to serve the practice. (1) (2) (3) (4)

Inspection basis 1: Yongning landslide, which occurred on May 27, 2013. Inspection basis 2: Qinglong landslide CXK13, a landslide that occurred on June 24, 2012. Inspection basis 3: Qinglong landslide CXK14, a landslide that occurred on June 24, 2012. Inspection basis 4: Qinglong landslide excavation test ZK1, a landslide occurring at 10:00 on April 20, 2011.

The actual monitoring data for these landslides are shown in Table 9.7. Table 9.7: Landslide monitoring data sheet. Qinglong landslide CXK13 (rainfall type)

Qinglong landslide CXK14 (rainfall type) (mm)

2011-11-23 2011-12-8

5.47 5.15

2011-12-5 2011-12-20

7.76 11.23

2011-12-23

6.89

2012-1-4

11.56

2012-1-7

7.15

2012-1-19

13.53

2012-1-22

4.31

2012-2-3

11.26

2012-2-6

3.99

2012-2-18

8.60

2012-2-21

3.68

2012-3-5

5.92

2012-3-8

1.77

2012-3-20

9.72

2012-3-23

4.73

2012-4-4

10.71

2012-4-7

1.62

2012-4-19

17.11

2012-4-22

5.34

2012-5-4

10.46

2012-5-7

6.22

2012-5-19

12.67

2012-5-22

10.42

2012-6-3

23.80

2012-6-6

28.36

2012-6-18

66.27

2012-6-21

150.9

Excavation simulation landslide test (excavation type) 2011-4-17 10:00 2011–4-17 16:00 2011–4-17 22:00 2011–4-18 04:00 2011–4-18 10:00 2011–4-18 16:00 2011–4-18 22:00 2011–4-19 04:00 2011–4-19 10:00 2011–4-19 16:00 2011–4-19 22:00 2011–4-20 04:00 2011–4-20 10:00

Yongning landslide ZK2 (rainfall type) (mm)

11.71 15.25

2012-12-14 2012–12-29

43.99 48.56

14.87

2013-1-13

56.47

18.23

2013-1-28

68.07

17.18

2013-2-12

82.91

16.93

2013-2-27

106.67

19.61

2013-3-14

123.21

18.83

2013-3-29

135.8

23.96

2013-4-13

151.46

26.64

2013-4-28

178.39

29.62

2013-5-13

255.01

32.84

2013-5-28

428.96

35.39

282 Chapter 9 Table 9.8: Various model forecast comparison tables. Numbering

Forecast value Saito model

CXK13 CXK14 ZK1 (Excavation) ZK2

Verhulst inverse function model Fukuoka model

2012-6-10 2012-6-7 2011-4-19 11:00 2013-5-7

2012-6-28 2012-6-21 2011-4-20 15:15 2013-5-22

2012-6-21 2012-6-25 2011-4-20 11:00 2013-5-24

Ha Y S model

Actual landslide time

2012-6-13 2012-6-14 2011-4-20 23:00 2013-5-21

2012-6-24 2012-6-24 2011-4-20 10:00 2013-5-27

The time prediction of the landslide was carried out by using the Saito model, the Verhulst inverse function model, the welfare model, and the Ha Y S model. The results are shown in Table 9.8. Analysis summary: (1) The accuracy of various model predictions is related to the frequency of actual monitoring. The shorter the monitoring time interval, the higher the prediction accuracy. Therefore, when the time is equidistant, it cannot be less than the minimum actual monitoring time interval. (2) Models that can be used for time prediction can be used for displacement prediction. (3) The Saito model is based on the empirical formulas obtained from a large number of indoor and outdoor experimental studies, and the applicability is poor, but the effect of the forecast is good. (4) The Ha Y S model is better than the Saito model, but significantly worse than the Verhulst inverse function model and the welfare model. (5) The Verhulst inverse function model and the Fukuoka model have better prediction results and are suitable for landslides in the accelerated deformation stage. (6) Only landslides in the accelerated deformation stage can use the preceding models for forecasting. In actual work, several models should be used, combined with macroscopic characterization and triggering factors for comprehensive forecasting. (7) Verhulst model is used to predict the development of uniform deformation and creep deformation.

9.3 Rockfall grey-mutation theory prediction 9.3.1 Grey theory 1) Basic concept of grey theory. Some problems in geotechnical mechanics have the characteristics of a grey system and are therefore suitable for the application conditions of grey theory. For example, if a particular

Landslide and rockfall prediction technology 283 slope rockfall problem is a grey problem, then there are many situations in that slope problem that are not understood, but some information about it can be grasped through engineering geological surveys, rock continuous testing, and acoustic emission tests. The prediction of rockfall geological disasters using grey theory is discussed in the following paragraphs. According to grey system theory, if there are known, unknown, or uncertain factors among the determining factors in a system, the system they are in is called a grey system. All the pieces of information (including random information) in the system are regarded as grey quantities, and a mathematical model describing a grey quantity is established using a special method. It has three basic links: (1) build different prediction models according to the number of variables based on a set of time series data from the system; (2) estimate model parameters, such as with the least-squares method; and (3) use the model for prediction and evaluation. The grey theory requires less original data, making the grey system suitable for cases in which the data sequence is increasing exponentially. The grey correlation analysis principle can be used to find the correlation of incomplete information through certain data processing, to determine the influence degree of each influencing factor of slope stability, and then to evaluate the slope stability by multifactor superposition analysis. Let the number of events be a data column in a time sequence. The GM (1,l) model of grey theory was established to predict possible future situations, so as to provide dynamic information for the prediction and prevention of rockfalls. First, the GM [1,1] model was discussed. In order to improve the accuracy of the model, the quadratic fitting parameter method was adopted. 2) Steps of grey theory modeling. Grey theory modeling generally is similar to Eqs. (9.23)–(9.34), so the establishment process is omitted.

9.3.2 Catastrophe theory Rockfall on a slope is a discontinuous abrupt change phenomenon. With the continuous change of the control variable parameters, the stability of the slope is thus an abrupt, bifurcated, or critical phenomenon. Mutation theory is a powerful tool to study this critical behavior, because mutation theory proves through partial lemma that the instability of system structure does not depend on the total number of thousands of possible state variables, but only on a few substantial state variables. Therefore, it is more reasonable to use the catastrophe theory to describe such a discontinuous catastrophic phenomenon as a slope rockfall than any mathematical tool better used to describe continuous phenomena. Using mathematical models to describe natural phenomena is a basic method of scientific research. Ever since Newton and Leibniz invented calculus, people have become increasingly

284 Chapter 9 accustomed to using differential equations to describe natural phenomena. Over the past 300 years, various models have been successfully established and many practical problems have been solved by using calculus and differential equations. But these analytical mathematical equations can only be used to describe the phenomena involving continuous and smooth changes. However, in reality, discontinuities and abrupt changes can be found everywhere in the real world, such as earthquakes, landslides and rockfalls in nature; political revolutions in the social sciences; or sudden extinctions of species in the biological world. Mutation theory has been proposed to solve these types of problems. The mutation theory was first proposed by Thom (1972), a French mathematician. It is mainly used to explain how some variables in a system change gradually from continuous to sudden changes in the system state. In the fourdimensional space-time we live in, Thom’s categorization theorem states that there are at most seven basic forms, and only one of them—cusp mutations—is commonly used. Mutation theory uses topology and singularity theory as mathematical tools to study various mutations. The main method is to generalize all kinds of phenomena into different topological structures and discuss the discontinuous properties near various critical points. Mutation theory is a branch of nonlinear science, which studies the characteristics of a system changing with the change of control parameters, especially when the parameters change the system performance under certain conditions. At present, mutation theory has been widely used in many fields of study and has contributed many application achievements. A significant advantage of mutation theory is that it is possible to predict many qualitative or quantitative states of a system with only a few “important parameters,” based on a few assumptions, even without knowing what differential equations exist, let alone how to solve them. The theory of abrupt change has been widely used in biology, physics, medicine, chemistry, sociology, economics, military science, and other sciences since it appeared. Kang Z Y used it to analyze the Euler instability of a platy rock mass. As for the application of catastrophe theory in slope engineering, it began only recently. For example, Qin S Q used the catastrophe theory to analyze the mechanism of slope instability, reservoir-induced earthquakes, and ground pressure of a narrow coal pillar, and proposed a catastrophe prediction model of slope instability time. Kong G G deduced the energy criterion of instability failure of an excavation system in rock mass engineering using the catastrophe theory. Gao P analyzed the deformation and failure mechanism of various soil slopes using the mutation theory. Tang C N studied the instability mechanism of rock specimens by using a sharp point mutation model. Liu D W discussed the application of catastrophe theory in slope stability research. Xu Q put forward the cusp mutation model of flexural tension slope. Liu J introduced the function of water weakening to establish the sharp point mutation model of the rock mass instability of a slope near a deep cutting of a railway in Sichuan province. Qin S Q, aiming at the problem of slope

Landslide and rockfall prediction technology 285 plane sliding instability, used the catastrophe theory method to provide the criterion for the occurrence of rapid landslide and slow speed slide slope, and proposed a new theory of stiffness effect instability. Guang Z Y used the catastrophe theory method to study coal and rock stability problems under the action of horizontal force and vertical force, deriving the total potential energy of the system function and establishing the system angle type of mutation model; the horizontal force and vertical force control space was obtained in the system of a buckling bifurcation set and analyzed due to the change of their state in the coal and rock mutation process. The basic principle of analyzing dam and rock foundation stability using catastrophe theory was studied, and the criterion and calculation model of analyzing dam and rock foundation stability using a cusp catastrophe model was put forward. Fu H L used the mutation theory to predict the possibility of rock burst in an underground stope. Sun S L discussed the phenomenon of horizontal unloading cracking through a mechanical model, analyzed the abrupt change condition of the unloading crack theoretically, and put forward the displacement value of critical instability point. Based on the catastrophe theory, Li R Q established a sharp angle catastrophe model of bedding slope and deduced the critical condition of slope instability. Cao J introduced the mutation theory into the evaluation of slope stability and evaluated the slope stability of the Lancang River basin. 1) The basic idea of mutation theory. There are two fundamentally different modes of change in the objective world: one is a smooth, continuous change, such as the continuous growth of organisms, the continuous rotation of the earth around the sun, the continuous rise or fall of a reservoir water level, the slow release of heat within a dam, and so on. This smooth continuous gradient phenomenon has been successfully solved mathematically using calculus. We can say that classical calculus is a mathematical model of continuous change. However, another type of change is the discontinuous leap, such as volcanic eruptions, sudden rock ruptures, a suddenly collapsing bridge, or a sudden dam failure during an earthquake. These phenomena are examples of suddenly jumping to a radically different form of discontinuous change; there is a transient process of mutation. The discontinuity caused by this mutation cannot be described by traditional calculus methods. We know that the state of a system can be described by a set of parameters, while a dynamic system can be described by dynamic parameters or time-varying structural models. When the system is in a stable state, a certain function indicating the state of the system takes a unique extreme value (such as minimum energy, maximum entropy, etc.). When the parameters change in a certain range, the function has more than one extreme value, and then the system must be in an unstable state. Therefore, from a mathematical point of view, to investigate whether a system is stable or not, it is often required to find the extreme value of a function, and the extreme point, that is, the point where the derivative value is zero, is the simplest

286 Chapter 9 singularity, or the critical point. If function Fu, v (x) is set, where u and v are parameters, then the critical point of function Fu, v (x) is the solution of the differential equation, when the values of u and v are given: d Fu, vðxÞ ¼ 0 dx After one or more critical points z have been obtained, critical point z can be regarded as a single or multivalued function of parameters u and v. When we study the stability or instability of a system, the minimum change of the function Fu, v (x) is found, and the function Fu, v (x) is often referred to as the potential function. This method of using a potential function to study discontinuous functions leads to catastrophe theory. In mutation theory, we refer to quantities that can be mutated as state variables or internal variables, and to the factors that cause the mutation, the continuous change, as control variables or external variables. A so-called abrupt change refers to the jump from one stable state to another stable state, or in the system evolution, certain variables change gradually from a continuous to a sudden change in the system state. The occurrence and development of slope instability is a gradual process, which is then destroyed to a certain extent. Current research on this aspect is ongoing. A significant advantage of mutation theory is that it is possible to predict many qualitative or quantitative properties of a system with a few control variables, even without knowing what differential equations exist, let alone how to solve them. At present, we apply the elementary mutation theory. Table 9.9 shows seven basic mutation forms, and only one is used often in daily life, namely the cusp mutation model.

Table 9.9: Seven basic mutations. Mutation name Fold Sharp point Swallowtail Butterfly Hyperbolic umbilical point Elliptical umbilical point Parabolic umbilical point

Number of state variables

Number of control variables

Potential function

1 1 1 1 2

1 2 3 4 3

V€¼ x 3 + ux V€ ¼ x4 + ux 2 + ux V€ ¼ x5 + ux3 + vx2 + wx V€ ¼ x 6 + tx4 + ux 3 + vx2 + wx V€ðx, y Þ ¼ x 3 + y 3 + wxy  ux  vy

2

3

3 V€ðx, y Þ ¼ x3 + xy2 + w ðx2 + y 2 Þ  ux + vy

2

4

V€ðx, y Þ ¼ y 4 + x2 y + wx2 + ty 2  ux  vy

Landslide and rockfall prediction technology 287 2) Cusp mutation model. The cusp mutation model is commonly used, most widely in nonlinear dynamics. The cusp mutation model was proposed by Zeeman and its potential function is a two-parameter function: V ¼ x4 + ux2 + vx

(9.77)

The so-called two-parameter function refers to two control variables u and v (the state variable is x), and the corresponding equilibrium position satisfies:

V ¼ 4x3 + 2ux + v

(9.78)

Its graph in the (x, u, v) space is called a mutant manifold. This is a wrinkled surface so that the equilibrium position is one, two, or three in different areas. The potential function corresponding to the middle lobe takes a maximum value, so that the equilibrium position is unstable, and the equilibrium position corresponding to the upper and lower leaves is stable: V€ ¼ 12x2 + 2u

(9.79)

Obviously, there is a vertical tangent to the surface, that is, the number of equilibrium positions is different in the vicinity of the point satisfying the preceding equation. These points are called abrupt points or singular points, actually the inflection points of the curve, which is a segmentation by the u parabolic line, as shown in Fig. 9.11. They form a bifurcation set in the parameter space, as shown in Fig. 9.12. Uniting Eqs. (9.78), (9.79) gives: 8u3 + 27v2 ¼ 0

(9.80)

A set of points satisfying the control points (u, v) of the preceding equation is called a set of divergent points. The criterion is: F ¼ 8u3 + 27v2

(9.81)

x 12x2+2u

o

u

Fig. 9.11 Parabola with second-order derivative of potential function.

288 Chapter 9 B¢ A¢ B XA A O1 x

u v

Fig. 9.12 Point mutation model and fork set.

As the control variables u and v change, the corresponding points change smoothly on the surface M. However, when the control point trajectory crosses the divergent point set, the corresponding point must make a sudden jump through the middle lobe, and the rock mass is unstable: F > 0, the slope is in steady state. F ¼ 0, the slope is in a critical state. F < 0, slope rockfall. 3) Basic characteristics of the catastrophe model. (1) Sudden jump Sudden jump refers to a small change in the control parameter causing a huge change in the state variable, causing the system to jump from a critical point of a local minimum to a critical point of another local minimum. When the energy value change of the potential function occurs from a gradually disappearing local minimum value to another critical point of the global and local minimum values, the mutation energy is performed in a sudden manner, and the potential energy is a discontinuous change. (2) Hysteresis In any physical system, if the state variable jumps from the local minimum value critical point 1 to the local minimum value critical point 2, the inverse process jumps from the critical point 2 to the critical point 1 is different, which is called hysteresis.

Landslide and rockfall prediction technology 289 (3) Divergence Normally, small changes in the control parameters cause only small changes in the state variables on the equilibrium surface; and weak disturbances to the control parameters cause only small increments of the state variables, which is a generally continuous smooth change. However, in the neighborhood of the critical point of degradation, small changes in the control parameters will result in large changes in the state variables, which is divergence. (4) Dual mode behavior The dual mode is also called multimodal. It shows that a system can have two or more different physical states, such as the phase transition of a substance. The system described by potential function can have more than one local minimum under the action of external control parameters. In the cusp catastrophe, the upper and lower leaf surfaces represent two different states, and for one position of the control point, there are two (or more) possible steady-state positions. (5) Unreachability Unreachability is the system’s ability to achieve a stable balance on certain state variables. For example, the midplane surface of the cusp catastrophe model is an unstable local maximum critical point, which divides the cusp catastrophe into the upper and lower leaf surfaces, while the state variable of potential function is the unreachable state surface under the external control parameters of the middle surface. (6) Multipath State variables are in a state in the equilibrium surface that can be achieved by different traces or paths of parameter changes, which is referred to as a multipath.

9.3.3 Least-squares polynomial fitting (1) Least-squares fitting principle. The data monitored by experiments may not be completely reliable. The error of individual data may even be large, but there are many data values given. Curve fitting is used to find out the relationships between variables from a large amount of data given, to try to construct the “best fit” curve that reflects the general trend of the data points to eliminate their local fluctuations. There exists a set of observational (experimental) data for a known physical process (function process) y ¼ f(x): ðxi , f ðxi ÞÞ, i ¼ 0,1, …,N

(9.82)

290 Chapter 9 It is required to find a function F(x) as an approximation function of y ¼ f(x) in a particular function class {ϕ(x)} (for example, a polynomial), so that the error (or residual) on xi is: εi ¼ Fðxi Þ  f ðxi Þ, i ¼ 0,1,…, N

(9.83)

The minimum is a certain metric, which is the fitting problem and also the curve fitting. (2) Polynomial fitting method. For a given set of data (xi, yi), i ¼ 1, 2, …, N, seek an m-degree polynomial y¼

m X

aj xj

(9.84)

j¼0

Minimize the total error Q: Q¼

N X i¼1

yi 

m X j¼0

!2 aj xji

(9.85)

Since Q can be regarded as a multivariate function about aj (j ¼ 0, 1, …, m), the construction problem of the preceding fitting polynomial can be attributed to the extremum problem of the multivariate function: ∂Q ¼ 0, k ¼ 0,1, …,m ∂ak ! N m X X j yi  aj xi xki ¼ 0, k ¼ 0,1,…, m i¼1

That is,

(9.86)

(9.87)

j¼0

8 X X X > ¼ yi xi + ⋯ + am xm a0 N + a1 > i > X X X < X 2 m+1 a0 xi yi xi + a1 xi + ⋯ + am xi ¼ > ⋯ > X X X X > :a +1 + ⋯ + am xm xm xm x2m 0 i + a1 i i ¼ i yi

(9.88)

This is a system of linear equations for coefficient aj, often referred to as a system of canonical equations. It is known here xi ¼ yi, the coefficient aj of the equations can be obtained by solving m P the preceding equation, and the least-squares fitting polynomial y ¼ aj xj is obtained. j¼0

Landslide and rockfall prediction technology 291

9.3.4 Grey mutation prediction model The application of catastrophe theory can be roughly divided into two categories: one is quantitative description, which is mainly applied to “hard” sciences such as mathematics, physics, and chemistry. The method is to find a potential function, a function similar to the potential function, or a system in which a mutation is popular or bifurcated with the same mathematical description, and appropriate mathematical means or techniques are applied to classify it as a type in the Thom classification table. Such applications not only deepen existing understanding, but also deal with problems from a higher perspective and often lead to new results. The other type is qualitative analysis, which is mainly applied to “soft” sciences such as psychology and sociology. That is, to imagine an elementary mutation model from the observed characteristic phenomena, such as jumping and lag, and then to fit the data to see if this mathematical model can be used. With a good explanation of the observed phenomena, finally inspired by the mechanism of inferring the phenomenon, the derivation of a physical model of such applications can serve as an effective mathematical means for many problems that were difficult or impossible to process mathematically. This chapter uses the first type of quantitative description. Combining the grey theory with the catastrophe theory, first using the grey theory to process the original data and obtain the prediction data, the least-squares curve is used to fit the expansion, and then the differential homeomorphic transformation is used to convert to the basic form of the cusp catastrophe model, then used to establish a criterion for forecasting geological disasters. When making predictions, when the original data amount is small, we often use grey theory. In practice, however, it is more difficult to use grey theory to predict, and there are still major defects. Therefore, we should adapt to local conditions, according to practice. In different situations, choose and create different, simpler, and more practical methods. First, the original data is predicted according to the grey theory quadratic fitting parameter method, and the predicted data is obtained. Then the least-squares curve fitting is performed on the predicted data: y ¼ a0 + a1 t + a2 t2 + a3 t3 + a4 t4

(9.89)

t ¼ z0  q

(9.90)

q ¼ a3 =4a4

(9.91)

Make

Substituting the preceding equation: y ¼ b0 + b1 z0 + b2 z0 + b4 z0 2

4

(9.92)

292 Chapter 9 In the equation: 3 b0 ¼ a0  a1 q + a2 q2  a3 q3 4

(9.93)

b1 ¼ a1  2a2 q + 2a3 q2

(9.94)

3 b2 ¼ a2  a3 q 2

(9.95)

b4 ¼ a4

(9.96)

pffiffiffiffiffi z0 ¼ z= 4 b4 ðb4 > 0Þ

(9.97)

pffiffiffiffiffiffiffiffi z0 ¼ z= 4 b4 ðb4 < 0Þ

(9.98a)

Let

or

Substitute into the previous equation: y ¼ z4 + uz2 + vz + c

(9.98b)

In the equation,

u¼

v¼

8 b2 > ffi > < pffiffiffiffi b

b4 > 0

4

> b2 > :  pffiffiffiffiffiffiffiffi b4 < 0 b4 8 b1 > ffiffiffiffiffi > b4 > 0

b1 > :p ffiffiffiffiffiffiffiffi b4 < 0 4 b4

(9.99)

(9.100)

Eq. (9.100) is the standard form of the cusp catastrophe model with z as the state variable and u and v as the control variables. For a specific acoustic emission process, y is a certain parameter of the process (acoustic emission parameters), z is the mechanical parameter representing the mechanical process, and u and v are parameters reflecting the relationship between the mechanical process and the acoustic emission process; c is a constant. The balanced surface equation is: 4z3 + 2uz + v ¼ 0

(9.101)

F ¼ 8u3 + 27v2

(9.102)

The criterion is

Landslide and rockfall prediction technology 293 As shown in Fig. 9.12, the bifurcation set of the cusp catastrophe model is a half cube parabola with a sharp point at the (0,0) point. The bifurcation set divides the control plane into two regions. In the region E, F > 0, then the change of v only causes a continuous change of z, so the system is stable. In the region J, F < 0, then the change of v causes a sudden jump of z. Therefore, the discriminating rule for making the system abrupt is: 8 > 0 No mutation ðsteady stateÞ > > < F ¼ 0 Critical state (9.103) > > : < 0 Mutation ðunstable stateÞ Substituting the AE parameters at each time into Eq. (9.89), using the least-squares method, obtaining ai, substituting the Eqs. (9.99)–(9.103), and obtaining u, v, and F, and the case of F, and according the case of F, the prediction of rockfall is made.

9.4 Monitoring data processing program research 9.4.1 Programming strategy 1) Programming principle. From the perspective of actual work needs, a monitoring data information processing program can provide disaster data information and technical support for scientific research and production, and it can also provide auxiliary decision-making services for government departments. To this end, the basic principles that should be followed in the development of such a program are: based on the basic principles of software engineering methods, to avoid any adverse consequences of incorrect methods, the object-oriented programming method should be adopted in the design and implementation, and the structural procedure should be followed. This design principle adopts the programming style of top-down and step-by-step refinement and references mature algorithms and corresponding source code to complete system development with a good user interface. 2) Programming method. The development and preparation of the entire program system should be based on software engineering specifications, using the idea of centralized design and development. For modules with independent functions, relatively independent development should be carried out; and the development of the corresponding interfaces should be given close attention to ensure the integrity and relative independence of the system. For mature modules, integration and migration methods should be adopted to speed up the running of the program. With the user’s requirements as the goal, the system should strive to be simple in operation and perfect in function, and at the same time to improve data sharing.

294 Chapter 9 The specific program development method can be summarized as follows: (1) Based on the principle of system development methodology, the program is decomposed into subfunction modules, and the basic attributes and their relationships are analyzed, so as to propose the tasks, requirements, and content of the program system. (2) When designing the program system, first the specification should be formulated, that is, the standards that should be complied with in the system development, such as the unified data mode of the industry or department, the standard code system, the specification of the schema, the agreed-upon processing, and the mode and general program interface; then divide the system into module hierarchy by function, determine the function of each module, the calling relationship between modules, and its interface; finally determine the algorithms necessary to realize the system functional requirements, and the algorithms and modules needed for control. (3) Propose the framework structure of the program system, perform functional analysis, data flow, module division, and interface design, fill in the details in the framework, and express the details of the program with the source program. (4) According to the design and development situation, debug the program system to ensure the coordinated operation of each functional module and realize the corresponding logic function. (5) In the program design, maintainability should be a top-level consideration from beginning to end, and the robustness of the program emphasized. 3) Programming strategy. According to user requirements, the program development includes three functions: monitoring data preprocessing, simple statistical analysis, and predicting the establishment of mathematical models. Based on the system’s functional goals, including data continuity, cost-effective development environment software, and other comprehensive comparisons, the program uses the VB6.0 visual programminglanguageforprogramming.VisualBasicisavisuallanguageintroducedbyMicrosoftin 1991. It is basically compatible with the earlier BASIC language, but it is more powerful and simpler for beginners and nonprofessionals. Visual Basic proposes a visual, object-oriented, and event-driven structured high-level programming language. The main features are visual programming, objectoriented programming, structured programming, event-driven programming mechanism, access to databases, and dynamic data, including exchange, linking, and embedding of objects.

9.4.2 Programming 1) Overall design of the program. The purpose of the program design is to preprocess and analyze the historical monitoring data of various monitoring instruments, use some traditional or intelligent methods to establish

Landslide and rockfall prediction technology 295

Fig. 9.13 Overall design of the program.

mathematical models, and use mathematical models for nonphysical scale constraints on the monitored objects, including quantitative prediction. The overall design of the program is shown in Fig. 9.13. 2) Program structure and function design. According to the characteristics of the monitoring data, combined with the aforementioned system design ideas and technical routes, the overall framework of the design program system is shown in Fig. 9.14. It can be seen from the figure that the logical relationship between modules can be expressed as follows: the forecasting model proposes the data requirements and storage format requirements for the database, and the database serves as the data source to provide the model library with the data needed for the model operation through the interface program. The running result of the forecasting model is stored in the data file in the agreed storage format, and the database manages the running result data of the model in a unified manner.

9.4.3 Main function realization and principle of the program 1) Data preprocessing. (1) Singular value test. In any monitoring system, there are some singular values in the observed data, and it is necessary to eliminate any singular value at the beginning of the deformation analysis. Taking into account the continuity, real-time nature, and automation of the system, the easiest way is to use the “3σ criterion” to eliminate singular values. Among them, the medium error σ of the observation data can be estimated by the sequence of observations itself, or by statistical results

296 Chapter 9 Data singular value test Data filtering smoothing Data prediction processing

Data equal spacing Data accumulation

Read text data

Monitoring data information program

Graphical output

Statistical analysis Data regression modeling

Result file output

Exponential smoothing modeling

Predictive model

Gray CM (1.1) model Verhulst curve model Verhulat inverse function model BP neural network model

Fig. 9.14 System structure framework.

from long-term observations, or empirical values. There are two main methods for testing singular values. ①

Method one.

For the observed data sequence {x1,x2,…,xN}, the variation characteristic of the sequence data is described as   (9.104) dj ¼ 2xj  xj + 1 + xj1 ðj ¼ 2, 3, …, N  1Þ Thus, (N  2) dj can be obtained from N observation data. At this time, the statistical mean d and the mean square error σ^d of the sequence data change can be calculated from the dj value: d¼

N 1 X dj N 2 j¼2

(9.105)

Landslide and rockfall prediction technology 297 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  u N1  u X dj  d 2 (9.106) σ^ ¼ t N 3 j¼2 The ratio of the absolute value to the mean square error of the difference between dj and the mean d is:

dj  d qj ¼ (9.107) σ^d When qj > 3, then xj is considered to be a singular value and it should be discarded. ②

Method two.

For the observation data sequence {x1,x2,…,xN}, the first-order difference equation can be used for prediction, and its expression is:   (9.108) x^j ¼ xj1 + xj1  xj2 ðj ¼ 3, 4, 5, …, N Þ The difference between the actual value and the predicted value is: dj ¼ xj  x^j

(9.109)

Let the medium error of the observation data be m (the value can be calculated from the longterm forecast data or the value according to experience); then the actual value and prediction can be calculated by Eqs. (9.108), (9.109), and the mean square error of the difference is σ^d ¼ 2m. For the absolute value jdjj of the difference between the actual value and the predicted

value, when dj > 3^ σ d , xj is considered to be a singular value and is discarded. In addition, for the singular value discarded, it can be complemented by a data value equal to the value of the previous point, or replaced by a predicted value to maintain the continuity of the data sequence. (2) Filter processing. The main filtering method of the program uses a three-point moving average based on the moving average method, a five-point moving average, and a three-point moving average method. If yt is the t-stage monitoring value, t ¼ 1, 2, …, T, Mt is the t-phase moving average, and N is the number of monitoring values included in each moving average, N < T. The basic calculation formula for simple moving average is: Mt ¼

yt + yt1 + ⋯ + ytN + 1 N

(9.110)

298 Chapter 9 where t is the length of the time series. Because it is often the serial number of the most recent data, t is also called the current period. If the predicted value of the future L period is recorded as y^T + L , then y^T + L ¼ Mt L ¼ 1,2

(9.111)

In the moving average method, the more monitoring data contained in the moving average, the larger N, the higher the degree of smoothing, so it is relatively stable; if N is small, the more the features of the original sequence are retained, there may be more random interference. Therefore, the choice of N value is very important. Generally speaking, considering that the historical monitoring sequence contains a large number of random components, if the basic trend of the data sequence does not change much, then N should be taken as larger, so that the smoothing effect is more significant. If the basic trend of data series changes with the impact of external environment, then N should be smaller to make the moving average more adaptable to the current trend. N generally takes a value between 5 and 200, depending on the length of the sequence and the specific circumstances. If we change Eq. (9.110), the following can be found: yt + yt1 + ⋯ + ytN + 1 yt + yt1 + ⋯ + ytN + 1 + ytN  ytN ¼ N N yt1 + yt2 + ⋯ + ytN + 1 + ytN yt + ytN yt + ytN + ¼ Mt1 + ¼ N N N

Mt ¼

(9.112)

It can be seen that the moving average calculation Eq. (9.110) is actually a recursive formula. With Eq. (9.112), a new moving average can be easily obtained by simply calculating yt + ytN/N. At the same time, it can be seen from Eq. (9.112) that the new moving average predicted value is an adjustment to the previous predicted value. In addition to the arithmetic average method, the method of averaging in the simple moving average method can also be determined by a geometric mean, a weighted average, etc., depending on the specific situation. For each moving average, the resulting new sequence is lower than the original sequence by (N  1)k/2, and the moving average result is later than the original sequence (N  1)k/2 (k is the slope) ð1Þ

ð1Þ

ð2Þ

y t  Mt ¼ M t  Mt ¼

N1 k 2

(9.113)

(2) In the formula, M(1) t and Mt are a moving average and a secondary moving average, respectively. The secondary moving average is a moving average of one moving average result. The calculation formula for both is:

Mt ð1Þ ¼

ð1Þ

ð1Þ

ð1Þ

yt + yt1 + ⋯ + ytN + 1 M + Mt1 + ⋯ + MtN + 1 Mt ð2Þ ¼ t N N

(9.114)

Landslide and rockfall prediction technology 299 The trend moving average method is based on the moving average estimated value aT ¼ E[yT] of the most recent observation value, and the second moving average estimated trend is the change slope kT to establish a prediction model. Available from Eq. (9.113):  2  ð1Þ ð1Þ ð2Þ ð2Þ aT ¼ E½yT  ¼ 2Mt  Mt ; bT ¼ (9.115) Mt  Mt N1 In summary, the trend moving average filtering model is: yT + L ¼ aT + bT  L, L ¼ 1,2,3,…  2  ð1Þ ð1Þ ð2Þ ð2Þ aT ¼ E½yT  ¼ 2Mt  Mt ; bT ¼ Mt  Mt N1 y + y + ⋯ + y t t1 tN + 1 ðt ¼ N, N + 1, …, T Þ Mt ð1Þ ¼ N Mt ð2Þ ¼

ð1Þ

ð1Þ

ð1Þ

Mt + Mt1 + ⋯ + MtN + 1 ðt ¼ 2N  1, 2N, …, T Þ N

T is the current period and is also the sequence length; N is the number of monitoring sequences included in the moving average; L is the predicted length. The moving average can also be written in a recursive form: ð1Þ

Mt ð1Þ ¼ Mt1 +

ð1Þ

ð1Þ

yt + ytN M + MtN ð2Þ ; Mt ð2Þ ¼ Mt1 + t N N

(9.116)

At present, in the linear trend prediction, the second exponential smoothing method is often used. However, for a sequence with both linear and periodic fluctuations, the trend moving average is still a method that can both respond to trend changes and effectively separate periodic fluctuations. (3) Equal spacing. Some predictive modeling algorithms, such as the grey model, need to study the timing of observations to be equally spaced. In actual engineering, the observation timing is often a sequence with nonequal spacing and randomness. In this case, the nonequal spacing must be first. The sequence is transformed into an equally spaced sequence. There are nonequal spacing raw data columns: X(t) ¼ {x(1), x(2), …, x(n)}, where the time interval between each observation point is: Δti ¼ tt + 1  t, i ¼ 1,2,…, n  1 The steps of its equal spacing processing are as follows:

300 Chapter 9 ①

Average time interval Δt0:

Δt0 ¼ ②

(9.117)

Find the unit time difference coefficient μ(ti) of each time period and the average time period: μðti Þ ¼

③

n1 1 X 1 Δti ¼ ðtn  t1 Þ n  1 i¼1 n1

ti  ði  1ÞΔt0 , i ¼ 1,2,…, n Δt0

(9.118)

Find the total difference Δx(ti) for each time period

Δxðti Þ ¼ μðti Þ½xðtt + 1  ti Þ

(9.119)

④ Calculate equidistant point values

i ¼ xðti Þ  Δxðti Þ

(9.120)

(4) Accumulative processing. The purpose of the data accumulation process is to eliminate the large volatility of the observed data sequence and obtain the characteristic data of the exponential form. Although this data processing method is relatively simple, it can often highlight the regularity of the original observation data after accumulation. Set the observation data sequence: {x1, x2, …, xN} The accumulated data sequence is: fx1 , x1 + x2 , x1 + x2 + x3 , …, x1 + x2 + x3 + ⋯ + xN g

(5) Normalized processing. The normalization of data is used to scale the data down to a small specific interval. In order to prevent the attribute with large initial value range from being too heavy, effectively reducing the prediction error and analyzing the data of different dimensions, the research data needs to be

Landslide and rockfall prediction technology 301 normalized and transformed by function. Its values are mapped to a range of values. Several methods are commonly used. ①

Minimum-maximum normalization linearly transforms the raw data. It is assumed that maxA and minA represent the maximum and minimum values of the attribute A, respectively. Minimum and maximum normalization through calculation: v¼

A  minA ðb  aÞ + a maxA  minA

(9.121)

Map the value of attribute A to v on interval [a, b]. In general, minimum-maximum normalization is used in the credit indicator data. The following two functional forms are commonly used: a. The membership function of the benefit indicator (the larger the better): 8 1 > >

>