11. Application to underground rock cavern‡：このマークが付してある著作物は、第三者が有する著作物ですので、同著作物の再使用、同著作物の二次的著作物の創作等については、著作権者より直接使用許諾を得る必要があります。

11.1 Stability in excavation of rock cavern

Stress concentration is the cause of failure around a rockcavern.
In the case where the initial stress is almost uniaxial stressstate
Tensile tangential stress can occur.
Tensile fracture which is normal to the free surface will appear.

Actually, initial stress is usually triaxial stress state andcavern geometry is designed not to induce tensile stresses.
It is not likely that tensile fracture becomes severe problems.

Failure due to concentration of compressive stress is theproblem in excavation of rock cavern.

Stress concentration on roadway wallStress concentration is small near the face since the faceacts as a support. It increases when the face goes ahead andis almost converged when the distance to the face becomes
identical to the diameter of the roadway.
Instantaneous failure due to stress concentration occurs (1)in the way of excavation or (2) in the period before thedistance to the face becomes identical to the diameter of heroadway.
Rockbursts in mountain tunnels were observed in (2)After that, mainly creep deformation occurs under aconstant stress.

Special cases
CAES cavern
Fatigue deformation due to cyclic loading of the inner pressure.

Geological waste disposal of high-level nuclear waste andunderground storage of LNG.
Thermal stress.

Roadways in mines.
Stress disturbances by excavation.

Stress concentration around a circular cavernp2

A circular cavern having
the radius R in a
homogeneous andisotropic elastic medium
subjected to stress p1 andp2 at an infinite distanceτrθ

σryσθr

θROp1xStress concentration around a circular cavernStresses σ r, σθ, τrθ at apoint ( r, θ ) (Kirsch'ssolution, ex. Jaeger andCook, 1979）p24⎞⎛ R2 ⎞ 1⎛R1σ θ = ( p1 + p2 )⎜⎜1 + 2 ⎟⎟ − ( p1 − p2 )⎜⎜1 + 3 4 ⎟⎟ cos 2θ2r ⎠ 2r ⎠⎝⎝⎛1R2
R 4 ⎞⎟⎜τ rθ = − ( p1 − p2 ) 1 + 2 2 − 3 4 sin 2θ⎜2rr ⎟⎠⎝σryσθrθR⎛⎛ R2 ⎞ 11R2R 4 ⎞⎟⎟⎜⎜
σ r = ( p1 + p2 ) 1 − 2 + ( p1 − p2 ) 1 − 4 2 + 3 4 cos 2θ
⎜⎜2r ⎟⎠ 2rr ⎟⎠⎝⎝τrθ

Op1x

In the case where only p2 is applied
p23τrθ

σr2σθ
1

12

3

45x/R

Distribution along x-axis

σθr

θ

R0

O-0.5

σr00σr0.5σ / p2

σ / p2y-10σθ
12345y/R

Distribution along y-axisp1xLarge stress cvoncentration istransfered to outer area.

3σ / p2

Significant stress concentartionoccurs on the roadway wall
Rock mass near roadway wall
may failure due the stressconcentration

2σθ
1σr

↓

This is the reason why the initialstress measurement should bedone at least roadway widthfrom the roadway wall

00

1

2

3x/R

45

Lithostatic pressure case (p1 = p2 = p)

⎛ R2 ⎞σ θ = p⎜⎜1 + 2 ⎟⎟r ⎠⎝τ rθ = 02σθσ/pStress which isconcentrated on thewall deceases with rand converged to thestress at infinite.
Average stress (σr +σθ) / 2 is independent
on r and keep a
constant value p.

⎛ R2 ⎞σ r = p⎜⎜1 − 2 ⎟⎟r ⎠⎝1σr
0012r/R

345Elasto-plastic solution
Analysis which also
considers failure
A circular roadway underlithostatic pressure
Bray's solution (Goodman,1980)
Coulomb's criterion isassumed for failure andresidual strength criterion.

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to RockMechanics, John Wiley and Sons, p. 234,
Fig. 7.10』を省略させて頂きます。

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to Rock Mechanics, John
Wiley and Sons,
p. 236, Fig. 7.11』を省略させて頂きます。

Radius of plastic zone R1⎤Q⎡⎧⎪
φ j ⎞ ⎫⎪2⎛⎟ ⎬ S j cot φ j ⎥⎜21tan
45−+++pq⎢u ⎨⎟⎪⎜2⎪⎩⎥⎢⎠⎭⎝
R = a⎢⎥⎫
⎧
φ⎛⎞⎪
⎪
j2
⎢ 1 + tan ⎜ 45 + ⎟ ( p + S cot φ ) ⎥jj⎜⎟ ⎬⎪ i⎢ ⎨⎪⎥
2⎝⎠
⎭
⎣ ⎩⎦
tan δ−1
Q=tan(δ − φ j )δ = 45 +φj
2
R : radius of the elasto - plastic boundarya : radius of the holep : external pressurepi : internal pressure
qu : UCS
S j : cohesion in residual strength

φ j : friction angle in residual strength

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to RockMechanics, John Wiley and Sons, p. 234,Fig. 7.10』を省略させて頂きます。

Stress in elastic zone
σr = p−σθ = p +br2
br2

⎤
⎡⎧ 2⎛
φ⎞ ⎫⎢ ⎨ tan ⎜ 45 + ⎟ − 1⎬ p + q u ⎥2⎠ ⎭⎝
⎩
⎥R 2
⎢
b=⎥⎢
φ⎞
2⎛
tan ⎜ 45 + ⎟ − 1
⎥
⎢2
⎠⎝⎦⎣

φ : angle of internal friction

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction toRock Mechanics, John Wiley andSons, p. 234, Fig. 7.10』を省略させて頂きます。

Stress in the plastic zone
著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction toRock Mechanics, John Wiley andSons, p. 234, Fig. 7.10』を省略させて頂きます。
Q⎛r⎞
σ r = ( pi + S j cot φ j )⎜ ⎟ − S j cot φ j⎝a⎠Qtan δ⎛r⎞σ θ = ( pi + S j cot φ j )⎜ ⎟ − S j cot φ jtan(δ − φ j ) ⎝ a ⎠Example
Calculate the radius of the plastic zone. Uniaxial
compressive strength and the angle of internal friction are60 MPa and 30°, respectively. Cohesion and friction angle
after failure are 0and 30°, respectively. Eaternal andinternal pressure are 40 MPa and 1 MPa, respectively.
Answer: 2.24 a.

Analysis for practical cases
Actual cavern is not always circular.
Inhomogeneity, anisotropy, planes of weakness etc.
sholwd be considered.
There are some analytical soltions.
Circular roadway in an anisotropic rock mass (Hefny & Lo, 1999)
Circular roadway in a rock mas having Hoek-Brown type failure
criterion (Carranza-Torres & Fairhurst, 1999)
Roadway having a geometry which can be approximated by amathematical function (Exadaktylos & Stavropoulou, 2002), etc等

Generally, numerical analyses can be adopted.
Finite Element Method
Boundary Element Method

Finite element method
The entire region of interest isdivided into finite elements.
Simultaneous equations whichrelates displacement and forc atelements' nodes are solved
according to the boundary
conditions.

FEM is suitable to finite,
inhomogeneous and elastoplastictc. prblems.
It was very tedius to the datapreparation, displaying thecalculated results.
Excellent pre- and postprosessors have been developed.

You can find many entry leveltexts in university library.
Very easy texts are recomended
oe the first one.

Tensile stress is taken as positive

Boundary element method
Only the boundary is divided into boundary elements
Ex. Simultaneous equations which relates displacement discontinuity
to traction, displacement and displacement discontinuity are solvedfor displacement discontinuity in the Displacement Discontinuity
Method.

BEM is suitable for infinite and semi-infinite problems.
DDM is suitable for analysis of fracture propagation.
Coastline
Land

Pacific Ocean
Distribution of extensile strain due to excavation
at K Coal Mine

Boundary Element Method
54y (m)BEM is not a general
purpose one since some
professional knowledge isrequired for analysis.

32100

123x (m)45

11.2 Support

Main members forsupport
Shotcrete
Rockbolt

社団法人地盤工学会（1986）、NATM工法の調査・設計から施工まで、p. 159図-3.24

制限資料

Example of an NATM tunnelロックボルト

Why is support needed?
The limited stand-up
time of unsupportedspan.
Ex. Unsupported roof
stands up one monthat most when RMR is60.

Rock cavern can't beused for practical
purpose without
support.

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to RockMechanics, John Wiley and Sons, p. 239, Fig.
7.13』を省略させて頂きます。

Concept of support

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to
Rock Mechanics, John Wiley and Sons,p. 237, Fig. 7.12a』を省略させて頂きます。

Extent of loosening
decreases with initialsupport pressure.
1⎤Q⎡⎧⎪
φ j ⎞ ⎫⎪⎛
⎢ 2 p − q u + ⎨1 + tan 2 ⎜⎜ 45 + ⎟⎟ ⎬ S j cot φ j ⎥2 ⎠ ⎪⎭⎪⎩⎥⎢⎝
R = a⎢⎥⎢ ⎧⎪1 + tan 2 ⎛⎜ 45 + φ j ⎞⎟ ⎫⎪( p + S cot φ ) ⎥⎬ ijj⎜⎥⎢ ⎨⎪2 ⎟⎠ ⎪⎭⎝⎩⎦⎣tan δ
Q=−1tan(δ − φ j )δ = 45 +Initial support pressure and extent ofloosening (Goodman, 1980)

φj2
R : radius of the elasto - plastic boundarya : radius of the holep : external pressurepi : internal pressure
qu : UCS
S j : cohesion in residual strength

φ j : friction angle in residual strength

Concept of support
The dotted lines denoterelationship between
support pressure anddeformation of the support.
Slope of the lines represent
the stiffness of support
system.
Deformation begins fromthe left side and stops at thecrossing point of the solidline and the dotted line.

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to
Rock Mechanics, John Wiley andSons, p. 237, Fig. 7.12c』
を省略させて頂きます。

Support pressure and wall
displacement (Goodman, 1980)

Concept of support
Rock mass behaveselastically for small
displacement.
Consequently, support
pressure decreases withwall deformation.
Yield zone extends forlarge displacement.
Support pressure
increases with wall
deformation in this case.

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to
Rock Mechanics, John Wiley andSons, p. 237, Fig. 7.12c』
を省略させて頂きます。

Support pressure and walldisplacement (Goodman, 1980)

Concept of support
The earlier the support isinstalled (1, 2 and 3 than 4and 5) or the stiffer thesupport is (1 and 2 than 3
or 4 than 5), the less thewall displacement.
1、2、3 is stable since yieldzone is small.
4 is potentially stable since
the displacement exceeds
the critical value although
the solid and doted lines
cross with each other.
5 is very unstable since thesolid and dotted line do not
cross with each other.

著作権処理の都合で、この場所に挿入されていた『Goodman (1980), Introduction to
Rock Mechanics, John Wiley andSons, p. 237, Fig. 7.12c』を省略させて頂きます。

Support pressure and wall
displacement (Goodman, 1980)

Excavation and support of tunnel
NATM (New Austrian Tunneling Method)
Mainstream in Japan at present

NATM

国土交通省北海道開発局旭川開発建設部上川道路建設事業所、中越トンネルパンフレット

制限資料

社団法人地盤工学会（1986）、NATM工法の調査・設計から施工まで、p. 159図-3.24

制限資料

Example of support pattern for a NATM
tunnel (JSGS, 1986）

Excavation and support of roadway

著作権処理の都合で、この場所に挿入されていた図表を省略させて頂きます。

Example of bolted roadway in Taiheiyo Coal Mine in Japan (JCOAL and KCOAL, 1999)

著作権処理の都合で、この場所に挿入されていた図表を省略させて頂きます。

Example of bolted roadway in Taiheiyo Coal Mine in Japana (JCOAL and KCOAL, 1999)

11.3 Evaluation of stability of rock cavernTunnel and roadway after excavation
Inner size decreases with time (convergence)
Creep deformation of the surrounding rock mass
Pressure on the concrete lining due to Creep deformation of the surroundingrock mass
Ground water pressure
Earthquake
Freezing pressure
Vibration and air pressure disturbances by trains or cars
Degeneration of concrete lining due to various causes

They can't serve as expected and detachment of concrete block and/orroof fall may occur if convergence exceeds a critical value.

Stability evaluation of underground cavern
Deformation of concrete lining
Axial load and bending moment of rockbolt
Stress disturbances and displacement in rock mass
AE

Strength of rock mass is given in stress
It is expected that stress measurement is required for precise
stability analysis...
Displacement is measured much more than stress analysis since theformer is much easier than the latter.

制限資料

Relationship between strain and tunnel failure （Sakurai, 1998, 1：difficulties in maintainface, 2：failure and cracking of shotcrete, 3：buckling of steel arch, 4：breakage of rockbolts,
5：roof fall, 6：floor heave of invert, 7：others)"Reprinted from Tunneling and Underground Space Technology, Vol. 12, Sakurai, S.,
Lessons Learned from Field Measurements in Tunneling, p. 456 Fig. 4, Copyright(1998), with permission from Elsevier".

1%

Estimated yield zone
StrainSchematic figure showing strain measurement in bolt
roadway in a coal mine

Example of stability monitoring

著作権処理の都合で、この場所に挿入されていた『CSIRO EXPLORATION AND MINING REPORT P2005/138, p. 2 』を省略させて頂きます。

CSIRO, 2005

Roof monitoring of ULAN Mine from Japan

11.4 Prediction of failure around rock cavern

Inward displacement of rock cavern increases graduallyeven under a constant condition.
This means gradual extension of failure zone if wall is alreadyfractured.
Creep deformation is occurring if wall is not already fractured.

Failure may occur as a result of creep deformation just as
rock specimen.Strain in the tertiary creep

Tertiary creep

ε = − B log(t F − t ) + Cεlog (tF - t)

Fukui & Okubo （1997）

Example of in-situ deformation monitoring
TargetFace

LaserSensor headε = − B log(t F − t ) + C

Differentiating the both sides
with respect to t

t = tF − B

dtdε

Failure time can be predictedmeasuring strain rate in thetertiary creep region (Fukada,1999).
Measurement of rock mass
surrounding a rock cavern can't
be started just after theexcavation.

dt / dεTotal strain is not known.
Using strain rate can be anadvantage

tOtF

Example
Strain rate was 1 × 10-4 s-1 at 1 o'clock and 1 × 10-3 s-1 at 2o'clock. Predict failure time.
Answer：2:06:40'

11.5 SubsidenceDeformation may occur on the ground surface due toexcavation of underground cavern.
Longwall coal excavation
Surface subsidence

Shallow excavation (ex. Oya-ishi, room and pillar)
Sinkhole

制限資料

Subsidence due to longwall coal mining （Whittaker & Reddish, 1989）"Reprinted from, Subsidence, Whittaker & Reddish, p. 21 Fig. 10, Copyright (1989),
with permission from Elsevier".

制限資料

Surface deformation above oldworkings by room and pillar
method (Bruhn et al., 1978）
Bruhn R. W., Mangnuson M. O. & Gray R. E. (1978), Survey of ground surface conditions
affecting structural responce to subsidence, ASCE Spring Convention, Pittsburgh, ASCE,Preprint, 3293, 26-55

制限資料

Formation of a sinkhole (Whittaker & Reddish, 1989)"Reprinted from, Subsidence, Whittaker & Reddish, p. 197 Fig. 116, Copyright (1989),
with permission from Elsevier".

11.6 Rockburst

Violent failure of rock mass sometimes occur due to deep
excavation.
The failure phenomena is called rockburst.
Rockbursts may give severe damage to machines andworkers.

著作権処理の都合で、この場所に挿入されていた写真を省略させて頂きます。

South Africa
Gold mines
breakouts

著作権処理の都合で、この場所に挿入されていた写真を省略させて頂きます。

Kolar Gold Fields