Rock Slope Engineering Civil and mining 4th edition phần 10 docx

K Number of sets • The mechanical behavior of a rock mass and its appearance will be influenced by the num-ber of sets of discontinuities that intersect one Table II.8 Persistence dimensi

390 Appendix II

J Persistence

• Persistence implies the areal extent or size of a

discontinuity within a plane It can be crudely

quantified by observing the discontinuity trace

lengths on the surface of exposures It is one

of the most important rock mass parameters,

but one of the most difficult to quantify

• The discontinuities of one particular set will

often be more continuous than those of the

other sets The minor sets will therefore tend

to terminate against the primary features, or

they may terminate in solid rock

• In the case of rock slopes, it is of the greatest

importance to attempt to assess the degree

of persistence of those discontinuities that

are unfavorably orientated for stability The

degree to which discontinuities persist beneath

adjacent rock blocks without terminating in

solid rock or terminating against other

discon-tinuities determines the degree to which failure

of intact rock would be involved in eventual

failure Perhaps more likely, it determines the

degree to which “down-stepping” would have

to occur between adjacent discontinuities for

a slip surface to develop Persistence is also

of the greatest importance to tension crack

development behind the crest of a slope

• Frequently, rock exposures are small

com-pared to the area or length of persistent

dis-continuities, and the real persistence can only

be guessed Less frequently, it may be possible

to record the dip length and the strike length of

exposed discontinuities and thereby estimate

their persistence along a given plane through

the rock mass using probability theory

How-ever, the difficulties and uncertainties involved

in the field measurements will be considerable

for most rock exposures

Persistence can be described by the terms listed in

Table II.8

K Number of sets

• The mechanical behavior of a rock mass and

its appearance will be influenced by the

num-ber of sets of discontinuities that intersect one

Table II.8 Persistence

dimensionsVery low persistence <1 mLow persistence 1–3 mMedium persistence 3–10 mHigh persistence 10–20 mVery high persistence >20 m

another The mechanical behavior is especiallyaffected since the number of sets determinesthe extent to which the rock mass can deformwithout involving failure of the intact rock.The number of sets also affects the appearance

of the rock mass due to the loosening anddisplacement of blocks in both natural andexcavated faces (Figure II.4)

• The number of sets of discontinuities may be

an important feature of rock slope stability,

in addition to the orientation of tinuities relative to the face A rock masscontaining a number of closely spaced jointsets may change the potential mode of slopefailure from translational or toppling torotational/circular

discon-• In the case of tunnel stability, three ormore sets will generally constitute a three-dimensional block structure having a con-siderably more “degrees of freedom” fordeformation than a rock mass with less thanthree sets For example, a strongly foliatedphyllite with just one closely spaced joint setmay give equally good tunneling conditions

as a massive granite with three widely spacedjoint sets The amount of overbreak in a tun-nel will usually be strongly dependent on thenumber of sets

The number of joint sets occurring locally (e.g.along the length of a tunnel) can be describedaccording to the following scheme:

I massive, occasional random joints;

II one joint set;

III one joint set plus random;

IV two joint sets;

V two joint sets plus random;

VI three joint sets;

Figure II.4 Examples illustrating the effect of the number of joint sets on the mechanical behavior and

appearance of rock masses (ISRM, 1981a)

VII three joint sets plus random;

VIII four or more joint sets; and

IX crushed rock, earth-like

Major individual discontinuities should be

recorded on an individual basis

L Block size and shape

• Block size is an important indicator of rock

mass behavior Block dimensions are

determ-ined by discontinuity spacing, by the number

of sets, and by the persistence of the

discon-tinuities delineating potential blocks

• The number of sets and the orientation

determine the shape of the resulting blocks,

which can take the approximate form of

cubes, rhombohedra, tetrahedrons, sheets,

etc However, regular geometric shapes are

the exception rather than the rule since the

joints in any one set are seldom consistently

parallel Jointing in sedimentary rocks usually

produces the most regular block shapes

• The combined properties of block size and

interblock shear strength determine the

mech-anical behavior of the rock mass under given

stress conditions Rock masses composed

of large blocks tend to be less deformable,

and in the case of underground construction,

develop favorable arching and interlocking

In the case of slopes, a small block sizemay cause the potential mode of failure toresemble that of soil, (i.e circular/rotational)instead of the translational or toppling modes

of failure usually associated with tinuous rock masses In exceptional cases,

discon-“block” size may be so small that flowoccurs, as with a “sugar-cube” shear zones inquartzite

• Rock quarrying and blasting efficiency are

related to the in situ block size It may be

helpful to think in terms of a block size tribution for the rock mass, in much the sameway that soils are categorized by a distribution

(volumetric joint count Jv)

Table II.9 lists descriptive terms give animpression of the corresponding block size

Values of Jv > 60 would represent crushedrock, typical of a clay-free crushed zone

Rock masses Rock masses can be described by

the following adjectives to give an impression ofblock size and shape (Figure II.5)

392 Appendix II

(i) massive—few joints or very wide spacing

(ii) blocky—approximately equidimensional

(iii) tabular—one dimension considerably smaller

than the other two

Table II.9 Block dimensions

Very small blocks >30

(iv) columnar—one dimension considerablylarger than the other two

(v) irregular—wide variations of block size and

shape

(vi) crushed—heavily jointed to “sugar cube”

II.2.5 Ground water

M Seepage

• Water seepage through rock masses resultsmainly from flow through water conduct-ing discontinuities (“secondary” hydraulicconductivity) In the case of certain sedimentary

Figure II.5 Sketches of rock masses illustrating block shape: (a) blocky; (b) irregular; (c) tabular; and

(d) columnar (ISRM, 1981a)

rocks, such as poorly indurated sandstone,

the “primary” hydraulic conductivity of the

rock material may be significant such that

a proportion of the total seepage occurs

through the pores The rate of seepage is

proportional to the local hydraulic gradient

and to the relevant directional

conductiv-ity, proportionality being dependent on

lam-inar flow High velocity flow through open

discontinuities may result in increased head

losses due to turbulence

• The prediction of ground water levels, likely

seepage paths, and approximate water

pres-sures may often give advance warning of

stability or construction difficulties The

field description of rock masses must

inev-itably precede any recommendation for field

conductivity tests, so these factors should

be carefully assessed at early stages of the

investigation

• Irregular ground water levels and perched

water tables may be encountered in rock

masses that are partitioned by persistent

impermeable features such as dykes, clay-filled

discontinuities or low conductivity beds The

prediction of these potential flow barriers and

associated irregular water tables is of

con-siderable importance, especially for projects

where such barriers might be penetrated at

depth by tunneling, resulting in high pressure

inflows

• Water seepage caused by drainage into

an excavation may have far-reaching

con-sequences in cases where a sinking ground

water level would cause settlement of nearby

structures founded on overlying clay deposits

• The approximate description of the local

hydrogeology should be supplemented with

detailed observations of seepage from

indi-vidual discontinuities or particular sets,

according to their relative importance to

sta-bility A short comment concerning recent

pre-cipitation in the area, if known, will be helpful

in the interpretation of these observations

Additional data concerning ground water

trends, and rainfall and temperature records

will be useful supplementary information

• In the case of rock slopes, the preliminarydesign estimates will be based on assumedvalues of effective normal stress If, as a result

of field observations, one has to conclude thatpessimistic assumptions of water pressure arejustified, such as a tension crack full of waterand a rock mass that does not drain readily,then this will clearly influence the slope design

So also will the field observation of rock slopeswhere high water pressures can develop due

to seasonal freezing of the face that blocksdrainage paths

Seepage from individual unfilled and filled continuities or from specific sets exposed in atunnel or in a surface exposure, can be assessedaccording to the descriptive terms in Tables II.10and II.11

dis-In the case of an excavation that acts as a drainfor the rock mass, such as a tunnel, it is helpful ifthe flow into individual sections of the structureare described This should ideally be performedimmediately after excavation since ground waterlevels, or the rock mass storage, may be depleted

Table II.10 Seepage quantities in unfilled

discontinuities

Seepage rating

Description

I The discontinuity is very tight and

dry, water flow along it does notappear possible

II The discontinuity is dry with no

evidence of water flow

III The discontinuity flow is dry but

shows evidence of water flow, that

is, rust staining

IV The discontinuity is damp but no

free water is present

V The discontinuity shows seepage,

occasional drops of water, but nocontinuous flow

VI The discontinuity shows a

continuous flow of water—estimate

l/ min and describe pressure, that is,

low, medium, high

I The filling materials are heavily consolidated and dry,

significant flow appears unlikely due to very lowpermeability

II The filling materials are damp, but no free water is

present

III The filling materials are wet, occasional drops of water

IV The filling materials show signs of outwash, continuous

flow of water—estimate l/ min.

V The filling materials are washed out locally,

considerable water flow along out-wash

channels—estimate l/ min and describe pressure that is

low, medium, high

VI The filling materials are washed out completely, very

high water pressures experienced, especially on first

exposure—estimate l/ min and describe pressure.

Table II.12 Seepage quantities in tunnels

Rock mass (e.g tunnel wall)

Seepage rating Description

I Dry walls and roof, no detectable seepage

II Minor seepage, specify dripping discontinuities

III Medium inflow, specify discontinuities with continuous flow

(estimate l/ min /10 m length of excavation).

IV Major inflow, specify discontinuities with strong flows

(estimate l/ min /10 m length of excavation).

V Exceptionally high inflow, specify source of exceptional flows

(estimate l/ min /10 m length of excavation).

rapidly Descriptions of seepage quantities are

given in Table II.12

• A field assessment of the likely effectiveness of

surface drains, inclined drill holes, or drainage

galleries should be made in the case of major

rock slopes This assessment will depend on

the orientation, spacing and apertures of the

relevant discontinuities

• The potential influence of frost and ice on the

seepage paths through the rock mass should

be assessed Observations of seepage from

the surface trace of discontinuities may be

misleading in freezing temperatures The

pos-sibility of ice-blocked drainage paths should

be assessed from the points of view of face deterioration of a rock excavation, and

sur-of overall stability

II.3 Field mapping sheets

The two mapping sheets included with thisappendix provide a means of recording thequalitative geological data described in thisappendix

Sheet 1—Rock mass description sheet describes

the rock material in terms of its color, grainsize and strength, the rock mass in terms of theblock shape, size, weathering and the number ofdiscontinuity sets and their spacing

396 Appendix II

Sheet 2—Discontinuity survey data sheet

describes the characteristics of each discontinuity

in terms of its type, orientation, persistence,

aperture/width, filling, surface roughness and

water flow This sheet can be used for recordingboth outcrop (or tunnel) mapping data, andoriented core data (excluding persistence andsurface shape)

Appendix III

Comprehensive solution

wedge stability

III.1 Introduction

This appendix presents the equations and

proce-dure to calculate the factor of safety for a wedge

failure as discussed in Chapter 7 This

compre-hensive solution includes the wedge geometry

defined by five surfaces, including a sloped upper

surface and a tension crack, water pressures,

dif-ferent shear strengths on each slide plane, and

up to two external forces (Figure III.1) External

forces that may act on a wedge include tensioned

anchor support, foundation loads and earthquake

motion The forces are vectors defined by their

magnitude, and their plunge and trend If

neces-sary, several force vectors can be combined to

meet the two force limit It is assumed that all

forces act through the center of gravity of the

wedge so no moments are generated, and there

is no rotational slip or toppling

III.2 Analysis methods

The equations presented in this appendix are

identical to those in appendix 2 of Rock Slope

Engineering, third edition (Hoek and Bray,

1981) These equations have been found to be

versatile and capable of calculating the

stabi-lity of a wide range of geometric and

geotech-nical conditions The equations form the basis of

the wedge stability analysis programs SWEDGE

(Rocscience, 2001) and ROCKPACK III (Watts,

2001) However, two limitations to the analysis

are discussed in Section III.3

As an alternative to the comprehensive

ana-lysis presented in this appendix, there are two

2 3

4

L

H1

Line of intersection

Figure III.1 Dimensions and surfaces defining size

and shape of wedge

shorter analyses that can be used for a more ited set of input parameters In Section 7.3, acalculation procedure is presented for a wedgeformed by planes 1, 2, 3 and 4 shown in Fig-ure III.1, but with no tension crack The shearstrength is defined by different cohesions and fric-tion angles on planes 1 and 2, and the waterpressure condition assumed is that the slope issaturated However, no external forces can beincorporated in the analysis

lim-A second rapid calculation method is

presen-ted in the first part of appendix 2 in Rock Slope Engineering, third edition This analysis also

does not incorporate a tension crack or externalforces, but does include two sets of shear strengthparameters and water pressure

III.3 Analysis limitations

For the comprehensive stability analysis

presen-ted in this appendix there is one geometric

limitation related to the relative inclinations of

plane 3 and the line of intersection, and a specific

procedure for modifying water pressures The

following is a discussion of these two limitations

Wedge geometry For wedges with steep

upper slopes (plane 3), and a line of

intersec-tion that has a shallower dip than the upper slope

(i.e ψ3 > ψi), there is no intersection between

the plane and the line; the program will

ter-minate with the error message “Tension crack

invalid” (see equations (III.50) to (III.53)) The

reason for this error message is that the

calcula-tion procedure is to first calculate the dimensions

of the overall wedge from the slope face to the

apex (intersection of the line of intersection with

plane 3) Then the dimensions of a wedge between

the tension crack and the apex are calculated

Finally, the dimensions of the wedge between the

face and the tension crack are found by

subtract-ing the overall wedge from the upper wedge (see

equations (III.54) to (III.57)

However, for the wedge geometry where (ψ3>

ψi), a wedge can still be formed if a tension crack

(plane 5) is present, and it is possible to

cal-culate a factor of safety using a different set of

equations Programs that can investigate the

sta-bility wedges with this geometry include YAWC

(Kielhorn, 1998) and (PanTechnica, 2002)

Water pressure The analysis incorporates the

average values of the water pressure on the

slid-ing planes (u1 and u2), and on the tension crack

(u5) These values are calculated assuming that

the wedge is fully saturated That is, the water

table is coincident with the upper surface of the

slope (plane 3), and that the pressure drops to

zero where planes 1 and 2 intersect the slope face

(plane 4) These pressure distributions are

simu-lated as follows Where no tension crack exists,

the water pressures on planes 1 and 2 are given

by u1 = u2 = γwHw/ 6, where Hw is the

ver-tical height of the wedge defined by the two ends

of the line of intersection The second method

allows for the presence of a tension crack and

gives u1 = u2 = u5 = γwH5w/ 3, where H5w

is the depth of the bottom vertex of the sion crack below the upper ground surface Thewater forces are then calculated as the product

ten-of these pressures and the areas ten-of the respectiveplanes

To calculate stability of a partially saturatedwedge, the reduced pressures are simulated by

reducing the unit weight of the water, γw That

is, if it is estimated that the tension crack is

one-third filled with water, then a unit weight of γw/3

is used as the input parameter It is consideredthat this approach is adequate for most purposesbecause water levels in slopes are variable anddifficult to determine precisely

III.4 Scope of solution

This solution is for computation of the factor ofsafety for translational slip of a tetrahedral wedgeformed in a rock slope by two intersecting dis-continuities (planes 1 and 2), the upper groundsurface (plane 3), the slope face (plane 4), and atension crack (plane 5 (Figure III.1)) The solu-tion allows for water pressures on the two slideplanes and in the tension crack, and for differ-ent strength parameters on the two slide planes.Plane 3 may have a different dip direction to that

of plane 4 The influence of an external load E and a cable tension T are included in the ana-

lysis, and supplementary sections are provided forthe examination of the minimum factor of safetyfor a given external load, and for minimizing theanchoring force required for a given factor ofsafety

The solution allows for the followingconditions:

(a) interchange of planes 1 and 2;

(b) the possibility of one of the planes overlyingthe other;

(c) the situation where the crest overhangs the

toe of the slope (in which case η= −1); and(d) the possibility of contact being lost on eitherplane

400 Appendix III

III.5 Notation

The wedge geometry is illustrated in Figure III.1;

the following input data are required:

ψ , α= dip and dip direction of plane, or plunge

and trend of force

H1= slope height referred to plane 1

L= distance of tension crack from crest,

measured along the trace of plane 1

u= average water pressure on planes 1

and 2

c= cohesion of each slide plane

φ= angle of friction of each slide plane

γ= unit weight of rock

γw= unit weight of water

T= anchor tension

E= external load

η= −1 if face is overhanging, and +1 if face

does not overhang

Other terms used in the solution are as follows:

FS= factor of safety against sliding along

the line of intersection, or on plane 1

S = total shear force on

planes 1 and 2 maintained on

Q = total shear

resistance onplanes 1 and 2

both planes 1and 2

j5= vector in the direction of intersectionline of 3, 5

R G

Note: The computed value of V is negative when

the tension crack dips away from the toe of theslope, but this does not indicate a tensile force

III.6 Sequence of calculations

1 Calculation of factor of safety when the forces

T and E are either zero or completely specified

in magnitude and direction.

(a) Components of unit vectors in directions

of normals to planes 1–5, and of forces

T and E.

(b) Components of vectors in the direction

of the lines of intersection of various

× tan φ1tan φ2]/R2 (III.45)

(e) Plunge and trend of line respectively of

line of intersection of planes 1 and 2:

The term−ν should not be cancelled out

in equation (III.47) since this is required

to determine the correct quadrant when

calculating values for dip direction, α

(f) Check on wedge geometry

(g) Areas of faces and weight of wedge

(i) Effective normal reactions on planes 1

and 2 assuming contact on both planes

(j) Factor of safety when N1 < 0 and

N2<0 (contact is lost on both planes)

(k) If N1 > 0 and N2 <0, contact is

main-tained on plane 1 only and the factor of

safety is calculated as follows:

(l) If N1 < 0 and N2 >0, contact is

main-tained on plane 2 only and the factor of

safety is calculated as follows:



(III.80)

2 Minimum factor of safety produced when load

E of given magnitude is applied in the worst direction.

(a) Evaluate N1

, N2

, S

, Q

, FS3

by use ofequations (III.61), (III.62), (III.78),

(III.79) and (III.80) with E= 0

(b) If N1

< 0 and N2

< 0, even before

E is applied Then FS = 0, terminatecomputation

404 Appendix III

If E > D, and E is applied in the direction

ψ e , α e, or within a certain range

encom-passing this direction, then contact is lost

on both planes and FS = 0 Terminate

calculation

(d) If N1

> 0 and N2

< 0, assume contact

on plane 1 only after application of E.

(e) If N1

< 0 and N2

> 0, assume contact

on plane 2 only after application of E.

(III.88)

(f) If N1

> 0 and N2

> 0, assume contact

on both planes after application of E

Check that N1 0 and N2 0

3 Minimum cable or bolt tension Tmin required

to raise the factor of safety to some cified value FS.

spe-(a) Evaluate N1
, N2
, S
, Q
by means of tions (III.61), (III.62), (III.78), (III.79)

equa-with T = 0

(b) If N2
< 0, contact is lost on plane 2

when T = 0 Assume contact on plane 1

only, after application on T ate S x
, S y
, S
z , S a
and Q
ausing equations

Evalu-(III.65) to (III.69) with T = 0

ψt1= arctan



tan φ1( FS)

(a) If N1
< 0, contact is lost on plane 1

when T = 0 Assume contact on plane 2

only, after application of T

Evalu-ate S x
, S
y , S z
, S b
and Q
b using equations

(III.72) to (III.76) with T = 0

(a) All cases No restrictions on values of N1

and N2
Assume contact on both planes

Tensioned anchor

Note that the optimum plunge and trend of the

anchor are approximately:

at the average friction angle to the line of intersection (Figure III.2).

Square mile square kilometer km2 1 mile2= 2.590 km2 1 km2= 0.3861 mile2

square meter m2 1 acre= 4047 m2 1 m2= 0.000 247 1 acreSquare foot square meter m2 1 ft2= 0.092 90 m2 1 m2= 10.7643 ft2

Square inch square millimeter mm2 1 in2= 645.2 mm2 1 mm2= 0.001 550 in2

Volume

Cubic yard cubic meter m3 1 yd3= 0.7646 m3 1 m3= 1.3080 yd3

Cubic foot cubic meter m3 1 ft3= 0.028 32 m3 1 m3= 35.3150 ft3

Cubic inch cubic millimeter mm3 1 in3= 16 387 mm3 1 mm3= 61.024 × 10−6in3

cubic centimeter cm3 1 in3= 16.387 cm3 1 cm3= 0.061 02 in3

Imperial gallon cubic meter m3 1 gal= 0.004 56 m3 1 m3= 220.0 gal

US gallon cubic meter m3 1 US gal= 0.0038 m3 1 m3= 263.2 US gal

Mass

ton (2000 lb) (US) kilogram kg 1 ton= 907.19 kg 1 kg= 0.001 102 ton

(continued)

Imperial unit SI unit SI unit symbol Conversion factor

(imperial to SI)

Conversion factor (SI to imperial) Mass density

ton per cubic

cubic foot

1 lb/ft3= 16.02 kg/m3 1 kg/cm3= 0.062 42 lb/ft3tonne per cubic

meter

t/m3 1 lb/ft3= 0.01602 t/m3 1 t/m3= 62.42 lb/ft3pound per

cubic inch

gram per cubic

centimeter

g/cm3 1 lb/in3= 27.68 g/cm3 1 g/cm3= 0.036 13 lb/in3tonne per cubic

1 tonf= 9.964 KN 1 kN= 0.1004 tonf (UK)

(continued)

cubic foot per

second

cubic meter persecond

m3/s 1 ft3/s= 0.028 32 m3/s 1 m3/s= 35.315 ft3/sliter per second l/s 1 ft3/s= 28.32 l/s 1 l/s= 0.035 31 ft3/sgallon per

kilopascal kPa 1 tonf/ft2= 95.76 kPa 1 kPa= 0.01044 ton f/ft2(US)

ton force per

pascal Pa 1 lbf/in2= 6895 Pa 1 Pa= 0.000 1450 lbf/in2

kilopascal kPa 1 lbf/in2= 6.895 kPa 1 kPa= 0.1450 lbf/in2