1) theoretical basis. 1.1 stereographic projection of rock ... · 1.1 stereographic projection of...

PROGRAM GEO - Mecrocce ver.3 for Windows

1) Theoretical basis.

1.1 Stereographic projection of rock discontinuities.

The planar discontinuities in the rock mass (bedding planes, joints andfaults) can be graphically represented and analyzed through hemisphericalprojection methods, whereby three-dimensional oriented planes can bedrawn in two dimensions. Orientations of the rock planes can be recordedusing a couple of numbers, dip direction and angle of dip. They representthe orientation, expressed as trend and plunge, of the line of maximum dipof an inclined plane. Plunge is the acute angle measured in a vertical planebetween the given line and the horizontal; it varies from -90° to +90°. Trendis the azimuth, measured in clockwise sense, from the north of the verticalplane containing the line of given plunge; it varies from 0° to 360°.Plane orientations can usually be displayed through polar or equatorialprojections. Among the most used projections, there are the polar equal-areanets (Schmidt's projection) and the equatorial equal-angle nets (Wulff'sprojection).

1


Schmidt's projection

It's used to represent the rock plane through its pole, that's the normal of theplane itself. This projection preserves the ratio between the areas. Thus it'suseful to process statistically the orientation of the rock planes and identifythe main discontinuities sets and their mean orientation.

Polar equal-area projection (the pole of a plane with orientation 345°/45° is displayed)

2

Polar equal-area net

180°

0°

270°90°

IsodensitygfedcNetgfedcbBordergfedcbSlope facegfedcSlope face polegfedcbUpper slope facegfedcUpper slope face polegfedcbJoint polesgfedcbBedding plane polesgfedcbFault polesgfedcbLocal maximagfedc


Wulff's projection.

It can be used to represent the mean orientation of the main sets of rockplanes. It preserves the ratio between the angles and, consequently, it'suseful to display and analyze potential instabilities of rock wedges.

Equatorial equal-angle projection (a plane with orientation 345°/45° and its pole aredisplayed)

3

Equatorial equal-angle net

0°

180°

90°270°

NetgfedcbBordergfedcbSlope face: 0/0gfedcSlope face polegfedcUpper slope face: 0/0gfedcUpper slope face polegfedcbedding plane: 345/45gfedcbJoint polesgfedcb


In case of linear survey, a sampling bias has to be taken in account as afunction of the acute angle between the sampling line and the normal to themeasured discontinuity planes. In fact the number of measureddiscontinuities reaches the maximum value when the sampling line isperpendicularly to the planes and becomes null when the sampling line isparallel. Consequently a weighting factor must be applied to the number ofdiscontinuities of each discontinuity set. This factor (w) can be calculatedby the following formula:

w = 1 / |cos(n-s)cosn coss + sinn sins|

where:n = trend of the normal to the discontinuity;s= trend of the sampling line;bn = plunge of the normal to the discontinuity;bs= plunge of the sampling line.

The correct number of discontinuities (Nc) is given by:

Nc = w Nm

where Nm is the number of measured discontinuities.

1.2 Spacing and frequency of rock discontinuities.

Spacing is the average distance between two adjacent rock discontinuitiesbelonging to the same joint set, measured perpendicularly to thediscontinuities themselves. Frequency is the reciprocal of spacing. The mean spacing is given by:

S (m) = L / N

where L is the length of the sampling line and N is the number of measureddiscontinuity intersections. Consequently the discontinuity frequency is:

(m-1) = 1 / S4


The spacing distribution is normally associated to a negative exponentialdistribution, which has a probability density function given by followingexpression:

f(S) = e-S

where the mean value and the standard deviation are both equal to 1/.Mean values of the joint set spacings can be used to calculate the volumetricjoint count (Jv), which is a measure for the number of joints intersecting avolume of rock mass. Jv can be estimated through the following formula:

where S1, S2, S3, Sn are the mean spacings of the discontinuity sets.The volumetric joint count (Jv) is correlatable to the rock block volumethrough the following expression:

where 1, 1 and 3 are the intersecting angles of the main three joint sets.The parameter is the block shape factor, given as:

with 2=S2/S1 and 3=S3/S1( S3>S2 >S1).

Based on these two variables, it's possible to classify the shape of the blocks(Palmstrom, 1985):

5


1.3 Roughness of rock discontinuities.

The term 'roughness' indicates the non-regularity of the discontinuitysurface, that's the deviation from the perfect planarity. From a practicalpoint of view, roughness can be quantified using the Joint RoughnessCoefficient (J.R.C.) parameter (Barton and Choubey, 1977). Roughness canbe measured through the Shape Tracer and J.R.C. can be estimated using theBarton's J.R.C. Profiles.

6


J.R.C. varies from 0 (planar surface) to 20 (extremely irregular plane).J.R.C. can be theoretically calculated by the following expression :

ZLogCRJ 1047.322.32... where Z è given by:

n

iii yy

dxnZ

112

1

with:n = Number of scanning steps;dx= Width of single step;y Height of the profile from the mean line.

7


As an alternative, J.R.C. can be gotten through a laboratory test (tilt test)suggested by Barton and Choubey, 1977. J.R.C. is given by the followingformula:

n

r

SCJLog

CRJ

...

)(...

10

where:

(°)= angle of initial slidingr (°)= residual friction anglen (MPa)= normal lithostatic pressureJ.C.S.(Mpa) = Joint Compressive Strength (Miller, 1965) The residual friction angle can be approximately set equal to the angle ofsliding of a perfectly planar rock surface. As an alternative, it can beestimated using the following formula:

s

abr SCJ

SCJ

...

...2020

where:b = basic friction angle of the discontinuity;J.C.S.s=J.C.S. of unaltered discontinuity;J.C.S.a=JCS of altered discontinuity.

The basic friction angle of the discontinuity is relative to a smooth andunaltered discontinuity surface as a function of the rock mineralogy andtexture. In the following table are displayed some values of b for severallithologies.

8


Litologia b(°)Amphibolite 31Sandstone 25 - 35

Basalt 31 - 38Limestone 33 - 40

Conglomerate 35Dolomite 27 - 31Gypsum 30Granite 23 - 39Gneiss 29 - 35Marl 27

Porphyrite 31Siltstone 27 - 31

1.4 Shear strength of rock discontinuities.

The shear strength of discontinuities is estimable using the empiricalrealtion by Barton et al. (1985):

The parameter J.C.S. (Joint Compressive Strength) can be calculated bymeasures carried out through a Schmidt's Hammer or, as an alternative, aconcrete sclerometer. The instrument gives an index, correlatable to JCSthrough the following relation:

01.100088.0).(..10 rMPaSCJLog

where:(kN/mc)= Rock unit weight;r= Schmidt's Hammer index.

9


As an alternative, JCS can be estimated using the following chart, whichtakes in account also the angle between the hammer and the horizontal.

Measurements must be performed both on unaltered and altered surfaces.The ratio gives an indication about the weathering degree of the jointsurface.Using a concrete sclerometer, the hammer index must be corrected to takein account the different characteristics of the instruments. A correlation

10


between the Schmidt's hammer (ISH) and the concrete sclerometer (ICS) issuggested by Bagalà (1998):

ISH = (ICS -22.1)/0.7

1.5 Rock Quality Designation (R.Q.D.).

R.Q.D. (Deere, 1963) is defined as the percentage of the sampling lineconsisting of spacing values greater or equal than 10 cm.

R.Q.D.(%)= Spacing values>=10 cmTotal length of the sampling line

For a negative exponential distribution of the discontinuity frequency,R.Q.D. can be estimated through the following relation:

R.Q.D.(%)= 100(0.1+1)e-0.1

Deere suggests the following classification of the rock mass quality:

R.Q.D. (%) Quality0 - 25 Very poor26 - 50 Poor51 -75 Fair76 - 90 Good91 - 100 Very good

1.6 Uniaxial compressive strength of intact rock (Point Loadtest).

The uniaxial compressive strength of intact rock specimens can be estimatedthrough Point Load tests. Using irregular shaped or cylindrical specimens,the Is50 parameter can be calculated through the Gremminer's formula:

11


75.0

138.0))(50(

DL

FMPaIs

where:Is(50)(MPa)= Point load index relative to the reference diameter (50 mm);D(mm)= Distance between the instrument points;L(mm)= Specimen length along the failure surface;F(N)= Ultimate failure load.

The uniaxial compressive strength is correlated to Is50 through thefollowing formula:

c(MPa)=24Is50

1.7 Instantaneous angle of shearing strength and cohesion ofrock mass and discontinuities.

Hoek and Brown criterion.

The Coulomb criterion

= c + tan ;

wherec = cohesion; = effective pressure; = angle of shear strength.

cannot be applied to the rock, where the correlation between shear strengthand effective pressure is not linear. However it's possible to estimateinstantaneous values of cohesion and angle of shear strength, relative to aspecific value of effective pressure, through the empirical Hoek and Browncriterion.

12


The criterion is expressed as

a

cbc sm

3'31 ;

where:s, a, mb = Constants for a specific rock type;c = Uniaxial compressive strength of the intact rock;1 3 = Major and minor principal stresses.

The s, a and mb rock constants can be correlated to GSI (Geological StrengthIndex).Three cases are distinguished based on the GSI value.

Undisturbed rock and G.S.I.>25:

28

100

GSI

iemm

9

100

GSI

es5,0a

Undisturbed rock and G.S.I.25:

28

100

GSI

iemm

0s

20065,0

GSIa

Disturbed rock any value of G.S.I.

14

100

GSI

ir emm

13


6

100

GSI

r es5,0a

where:mi= variable depending on the rock mineralogy and petrographiccharacteristics, derivable from the following table:

14


Instantaneous cohesion (ci ) and angle of shear strength (i ) of the rock mass.

The parameters ci and i can be obtained through an implicit numericaltechnique. The calculation steps are the following: Using the Hoek and Brown criterion, 1 is calculated, making 3 variable

from a value close to 0 to a maximum value approximately equal to 0.25c. The incremental step of 3 (3) is given by the ratio 3 = c/210. Ton steps 3 correspond n couple of 1, 3 values, through the Hoek andBrown formula, and n sets of values 1/3 , n’, , given by the Balmerrelations:

n

3

1 3

1

3

1;

n 31

3

;

313

1

21

cbm

(GSI>25, a=0,5).

1

3

3

1

1

amba

c

a

(GSI25, s=0).

By the linear regression formula:

i

nn

nn

arc n

n

' tan

2

2,

15


cn ni

ni' tan '

,

Inside the calculated intervals of n values (n), the interval where fallsthe searched n’ value is identified. n is associated to the intervals ofcohesion and shear strength (ci’ and i’), whereby:

''

in

nbci cc

,

''

in

nbci

,

Instantaneous cohesion (ci ) and angle of shear strength (i ) of the discontinuites.

The shear strength of the discontinuities, expressed as ci and i values, canbe estimated through the relations suggested by Barton.These the calculation steps:

'tan' 10

nbn

JCSJRCLog

;

1

'tan

10ln180'tan 10

210

nb

nb

n

JCSJRCLog

JRCJCSJRCLog

;

ni arc

tan ;

inic tan .

16


1.8 Rock mass classifications.

Rock mass classifications are simplified empirical schemes, which allow toestimate the quality of the rock mass through the calculation of a numericalindex, correlatable to the strength characteristics of the rock.

Bieniawski (C.S.I.R. Rock Mass Rating).

The Bieniawski's rock mass classification takes in account five parametersrelative to the condition of the rock mass and a corrective index as afunction of the discontinuity orientation and of the considered problem.(tunnel, slope and foundation).

RMR = (A1 + A2 + A3 + A4 + A5) - Ic;

The five parameters are the following::

A1 <Co> (Uniaxial compressive strength);A2 <RQD%> (Rock Quality Designation);A3 <s> (Discontinuity spacing);A4 Discontinuity conditionA5 Ground water conditionIc Corrective index

A partial rating is assigned to each parameter and then an overall index iscalculated, summing the ratings and applying the corrective index. Theyexist several version of this classification: 1976, 1979 and 1989.

17


C.S.I.R. Rock Mass Rating: 1976

The sum of the five partial ratings gives the Basic RMR (BRMR). TheBRMR in the condition of dry discontinuities corresponds to the parameterGSI (Geological Strength Index):

76BRMRGSI (when BRMR>18)

The corrective index Ic is given by the following table:

Applying the Ic parameter to the BRMR the RMR index is calculated. Thequality of the rock mass is gotten through the following table.

RMR 0-25 25-50 50-70 70-90 90-100CLASS V IV III II I

QUALITY Very poor Poor Fair Good Very good

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The sum of the five partial ratings gives the Basic RMR (BRMR). TheBRMR in the condition of dry discontinuities corresponds to the parameterGSI (Geological Strength Index):

579 BRMRGSI (when BRMR>23)





19



In the 1989 CSIR classification the parameters A1, A2 and A3 are giventhrough the following graphics:

A1:

A2:

20


A3:

The A4 parameter is estimated through the sum of five partial indexes:

Finally the A5 parameter is calculated as in the 1979 classification.The sum of the five partial ratings gives the Basic RMR (BRMR). TheBRMR in the condition of dry discontinuities corresponds to the parameterGSI (Geological Strength Index):

GSI = BRMR89 - 5 (when BRMR>23)


21





The geomechanical parameters of the rock mass are directly correlatable tothe BRMR index through the following expressions

25)(

BRMR

BRMRMPac *005,0)(

40

10

10)(

BRMR

GPaE

dove:(°)= Angle of shear strength of the rock mass;c(Mpa)= Cohesion of the rock mass;E(Gpa)= Young modulus of the rock massThe RMR index can be correlated to the Q index (N.G.I. Q-System) and toRSR (Rock Structure Rating) through the following correlations:

44ln9 QRMR

77,0

4,12

RSRRMR

22


C.S.I.R. Rock Mass Rating by the Sen's formula

Sen et al. (2003) suggested a numerical correlation between RMR and some parameters of the rock mass:

RMR = 0.2RQD + 15Log(sp) + 0.075c – 2.9Log(qw) +34 + A4 – A6

where RQD = Rock Quality Designation;sp(m) = mean discontinuity spacing;c (MPa) = uniaxial compressive strength of the intact rock;qw (l/s) = hydraulic discharge along a 10 m length of the front; A4 and A6 = parameters A4 and A6 by 1989 C.S.I.R. .

Barton (N.G.I. Q-System).

The Q index is calculated by the following formula:

SRFJJ

JRQDJQ

an

wr

The parameters at the second member have the following meaning.

RQD % (Rock Quality Designation).

Jn (Joint Set Number).

It depends on the number of the joint sets identifiable in the rock mass.

23


Jr (Joint Roughness Number).

It depends on the discontinuity roughness.

24


Ja (Joint Alteration Number) .

It depends on the alteration degree of the rock discontinuities and thecharacteristics of the filling.

25


Jw (Joint Water Number).

It depends on the ground water conditions.

S.R.F (Stress Reduction Factor) .

It depends on stress condition of the rock mass.

26


The three ratios, in the formula to determine Q, have a specific physicalmeaning :

RQD/Jn: It defines the rock mass structure and gives an approximateestimate of the block size.

Jr/Ja: It takes in account the mechanical behavior of the rock mass.

Jw/SRF: It expresses the effective stress condition acting on the rockmass.

The Q-system index (varying from 0.001 to 1000) is composed by 9intervals, to which correspond as many classes of quality of the rock mass.The intervals are expressed in logarithmic scale.

27


Q system 1000-400 400-100 100-40 40-10 10-4Description Exceptionally

goodExtremely

goodVery good Good Fair

Class I II III IV V

Q system 4-1 1-0.1 0.1-0.01 0.01-0.001Description Poor Very poor Extremely poor Exceptionally

poorClass VI VII VIII XI

Wickham (Rock Structure Rating).

It's based on the estimate of the RSR index (Rock Structure Rating), sodefined:

RSR = A + B + C.

Where A, B and C are three partial indexes given through the followingschemes.

Parameter A: General characteristics of the rock.

28


Parameter B: Block size and direction of drive.

Parameter C: Physical characteristics of the discontinuities and hydrauliccondition.

To identify a rock mass class, it's necessary to correlate RSR index to RMRor Q:

RSR = 0.77 RMR + 12.4

RSR = 13.3 LogQ + 46

29


Romana (S.M.R. - Slope Mass Rating).

The classification represents the application of the 1979 Bieniawski'sclassification to the case of rock slope stability. The SMR index (SlopeMass Rating) is given by the following relation:

SMR=A1 + A2 +A3 +A4 + A5 + (F1 x F2 x F3) + F4

The A1-A5 indexes are relative to the Bieniawski's classification.

The sum of the five partial indexes gives the Basic RMR (BRMR). TheSMR index has to be calculated by the following formula:

4)321( FxFxFFBRMRSMR

The variables F1, F2 and F3 depend on the orientation of the mostunfavorable joint in the rock mass as a function of the slope orientation.F1 is given by the following expression:

2|)]sen(|1[1 fjF

where j and f are, respectively, the dip direction of the most unfavorablejoint and the dip direction of the slope.F2 is obtained by the formula:

30


jtgF 22

where j if the angle of dip of the most unfavorable joint. When F2>1, ismust impose F2=1.F3 is a correction to apply to the BRMR value as a function of thedifference between the dip angle of the most unfavorable joint and the dipangle of the slope (j - f). It practically corresponds to the Bieniawski'scorrection:

F4 is a correction to apply as a function of the excavation method.:

The SMR index is correlated to the quality of the rock mass and to thestability condition of the rock slope:

SMR 0-20 21-40 41-60 61-80 81-100CLASS V IV III II I

Recommendation about the slope support is also given:

31


Geological Strength Index (G.S.I.).

It's a fundamental parameter to envisage the mechanical behavior of therock mass (see paragraph 1.7). It can be gotten by a correlation to the RMRindex or, as an alternative, can be evaluated through the proceduresuggested by Sonmez & Ulusay (1999).

GSI is estimated as a function of two variables, SR (Structure Rating) andSCR (Surface Condition Rating). SR is given by the following scheme:

where SR depends on the volumetric joint count (Jv) of the rock mass.

32


SCR can be obtained by the sum of three partial factors, as a function of theroughness and weathering degree of the joints and of the thickness andcharacteristics of the filling.

1.9 Tunnels.

Rock-support analysis.

The rock-support analysis is here presented using the Hoek and Brownsimplified method. This procedure is based on some basic prerequisites.

Tunnel geometry: the tunnel has a circular-section of initial radius ri anda length such that the problem could be considered two-dimensional.

In situ stress field: in situ horizontal and vertical stresses have the samemagnitude p0.

Support pressure: the installed supports exert a uniform radial pressurepi on the tunnel walls.

Characteristics of the undisturbed rock mass: the rock mass has, inundisturbed condition, a linear-elastic behavior, characterized by aYoung modulus E and a Poisson's ratio ; the failure criterion is givenby:

5,02331 cc sm

Characteristics of the disturbed rock mass: the rock mass has, indisturbed condition, a perfectly plastic behavior and satisfies thefollowing failure criterion:

33


5,02331 crcr sm

Volumetric strains: the elastic-behavior regions are governed by thevariables E and v; at failure, the rock mass is exposed to a volumeincrease and the strains are calculated using the theory of plasticity.

Time-dependent behavior: both the undisturbed and disturbed rockmasses don't exhibit a time-dependent behavior.

Extent of the plastic zone: the plastic-behavior zone reaches a radius re,which depends on the in situ pressure, the support pressure and the rockmass characteristics.

The scheme of calculation is listed below (Hoek e Brown, 1982).Input data:

c= Uniaxial compressive strength of the intact rock;m, s= Constants of the undisturbed rock mass;E= Young modulus of the undisturbed rock mass;= Poisson's ratio of the rock;mr, sr= Constants of the disturbed rock mass;r= Unit weight of the rock;p0= In situ pressure;ri= Tunnel radius. Calculation sequence.

The calculation sequence has to be repeated, using a pi (support pressure)value varying from 0 to p0.

842

15,0

0

2m

sp

mm

Mc

34


5,0

04

sMpm

m

mD

cc

5,0

202

r

r

cr

c

m

s

m

MpN

Per pi>p0-Mc strain around the tunnel is elastic.

ii

i ppEr

u

0

1

Per pip0-Mc plastic failure around the tunnel.

c

e

e MEr

u

1

5,0

22

r

r

cr

i

m

s

m

pN

i

e er

r

Per 3i

e

r

r:

i

e

r

rDR ln2

Per 3i

e

r

r: DR 1,1

Rr

r

r

r

r

u

e

i

e

i

e

e

e

av

111

2

2

2

2

2

i

eav

e

e

r

re

r

uA

35


5,0

1

11

A

e

r

u av

i

i

For the roof of the tunnel, plot i

i

r

uas a function of

0p

rrp ieri

For the sidewalls of the tunnel, plot i

i

r

uas a function as

0p

pi

For the floor of the tunnel, plot i

i

r

uas a function of

0p

rrp ieri

The variable cMp 0 represents the critical pressure, that's the pressurethe tunnel support has to contrast to prevent the making of a plastic zonearound the tunnel.

Rock-support design.

The design of the tunnel support (concrete or shotcrete lining, steel set androck bolt) has to be executed, proceeding by trial, first calculating thesupport stiffness and the maximum support pressure and then plotting thesupport curve on the graphic pressure vs strain. The method makes possibleto combine two supports and to process a compound support curve.The support will be considered well designed, when the support curve willintersect, in the chart, the pressure-vs-strain curves relative to the roof, thesidewalls and the floor of the tunnel.

36


Concrete or shotcrete lining : calculation of the support stiffness and the maximum support pressure.

Input data:Ec(MPa)= Modulus of elasticity of concrete or shotcrete;c= Poison's ratio of concrete or shotcrete;tc(m)= Thickness of concrete or shotcrete;ri(m)= Tunnel radius;cc(MPa)= Uniaxial compressive strength of concrete or shotcrete.

Stiffness:

22

22

211 ciicc

ciicc

trr

trrEk

Maximum pressure:

2

2

max 12

1

i

ciccsc

r

trP

Steel set: calculation of the support stiffness and the maximum support pressure.

Input data:37


W(m)= Flange width of the steel set;X(m)= Depth of the section of the steel set;As(m2)= Cross-sectional area of the steel set;Is(m)= Moment of inerzia of the steel section;Es(Mpa)= Modulus of elasticity of the steel section;ys(MPa)= Yield strength of steel;ri(m)= Tunnel radius;S(m)= Steel set spacing along the tunnel; (rad)= Angle between the blocking points;tb(m)= Thickness of the blocks;Eb(MPa)= Modulus of elasticity of the blocks.

Stiffness:

22

3 21

2

cos1

WE

tS

sin

sin

IE

Sr

AE

Sr

k b

b

ss

i

ss

i

s

Maximum pressure:

cos12

132

3max

XtrXAISr

IAp

bissi

ysssss

Bolt: calculation of the support stiffness and the maximum support pressure.

Input data:l(m)= Free bolt length;db(m)= Bolt diameter;Eb(MPa)= Modulus of elasticity of the bolts;Q(Mpa)= Load-deformation constant for anchor and head;Tbf(MN)= Ultimate failure load from pull-out test;ri(m)= Tunnel radius;sc(m)= Circumferential bolt spacing;sl(m)= Longitudinal bolt spacing.

Stiffness:

Q

Ed

l

r

ss

k bbi

lc

b 41

38


Maximum pressure: lc

bfsb ss

Tp max

Calculation of the support curve in a single support system.

Input data:k= Stiffness support;psmax= Maximum support pressure;ui0= Initial tunnel deformation before support installation.

The support curve is obtained, making the pressures (pi) varying from 0 topsmax in the following relation:

ii

i

ii r

k

p

r

uu

0

Calculation of the support curve in a compound support system (the supports are installed at the same time).

Input data:k1= Stiffness of the support 1;psmax1= Maximum pressure of support 1;k2= Stiffness of the support 2;psmax2= Maximum pressure of support 2;ui0= Initial tunnel deformation before support installation.The support curve is obtained, making the pressures (pi) varying from 0 topsmax in the following relations:

1

1max1max k

pru asi

2

2max2max k

pru asi

2112 kk

pru ii

39


For u12<umax1<umax2: ii

i

ii r

kk

p

r

uu

21

0

For u12>umax1<umax2:

ir

kkup 211max

12max

For u12<umax2<umax1:

ir

kkup 212max

12max

40


1.10 Rock slope stability.

Markland testThis test allows to get an indication of the stability of the wedges inside ofthe rock mass as a function of their spatial orientation and of the averageshear strength along their discontinuities. The discontinuity shear strength isquantified through an average angle of shear strength.The considers five possible conditions.1. Potentially unstable wedge.

This condition happens when the wedge is oriented in the slope directionand the angle of shear strength is lesser than the angle of dip of the slidingline.

41

Friction cone

Equatorial equal-angle net. BRMR correction:-25

0°

180°

90°270°

NetgfedcbBordergfedcbSlope face: 145/80gfedcbSlope face polegfedcUpper slope face: 0/0gfedcUpper slope face polegfedcbedding plane: 205/55gfedcbjoint: 76/76gfedcbJoint polesgfedcFriction circlegfedcbTunnel axisgfedcTopplinggfedcbStablegfedcbUnstablegfedcbUncertain instabilitygfedcb


2. Stable wedge.

This condition happens when the wedge is oriented in the slope directionand the angle of shear strength is greater than the angle of dip of the slidingline or when the wedge is anti-dip slope oriented.

42

Equatorial equal-angle net. BRMR correction:0

0°

180°

90°270°



3. Uncertain instability.

This condition happens when the wedge is oriented in the slope directionand the angle of shear strength is approximately equal to the angle of dip ofthe sliding line (±2°).

43

Friction cone

Equatorial equal-angle net. BRMR correction:-25

0°

180°

90°270°



4. Block toppling.

This condition happens when the slope and one of the discontinuity areapproximately vertical and have a similar dip direction.

44


0°

180°

90°270°

NetgfedcbBordergfedcbSlope face: 145/80gfedcbSlope face polegfedcUpper slope face: 0/0gfedcUpper slope face polegfedcbedding plane: 140/78gfedcbJoint polesgfedcFriction circlegfedcbTunnel axisgfedcTopplinggfedcbStablegfedcbUnstablegfedcbUncertain instabilitygfedcb


5. Roof stability.

The presence of unstable blocks in the roof of a tunnel is put in evidence bythe intersection of three or more discontinuities which form a closed shape.

Once identified the potential slope instabilities, the Markland test makespossible to quantify the correction to apply in the Bieniawski's rock massclassification, on base of the following scheme:

45


0°

180°

90°270°

NetgfedcbBordergfedcbSlope face: 145/80gfedcbSlope face polegfedcUpper slope face: 0/0gfedcUpper slope face polegfedcbedding plane: 205/58gfedcbjoint: 76/76gfedcbjoint: 305/78gfedcbJoint polesgfedcFriction circlegfedcbTunnel axisgfedcTopplinggfedcbStablegfedcbUnstablegfedcbUncertain instabilitygfedcb


Very favourable = no unstable wedgesFavourable = uncertain instabilitiesFair = one unstable wedgeUnfavourable = two unstable wedgesVery unfavourable = three or more unstable wedges

Planar stability.

In case of potential sliding along a single discontinuity, the slope stabilitycan be performed through a two-dimensional scheme.

Two-dimensional scheme (after Giani, 1988)

The safety factor can be described by the following expression:

where:W = weight of the sliding wedge;V = volume of the sliding wedge;A = area of the sliding plane; = angle of dip of the discontinuity; = angle of shear strength along the discontinuity;c = cohesion along the discontinuity;U = groundwater pressure along the discontinuity;H = height of the slope;z = depth of the tension crack;

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V = groundwater pressure along the tension crack;w = unit weight of water;R = magnitude of the external force, if present; = angle from the horizontal plane of R;kv = vertical seismic coefficient;kh = horizontal seismic coefficient.

The variables c and are calculated, using the procedure seen in theparagraph 1.7.The groundwater flow inside the slope can be taken in account, in the slopestability calculation, by supposing one of the four following scenarios:

(after Lembo, Fazio and Ribacchi, 1988)1. The rock mass is heavily fractured and a real piezometric line is

formed: U = wVcos .2. The rock mass is heavily fractured and there is a saturated tension

crack: U = wVcos , V =0.5wz2; 3. The rock mass is usually drained and only a temporary flow along

the discontinuity and the tension crack occurs during rainy events: U= 0.5wz(H-z)cosec , V =0.5wz2;

4. The rock mass is usually drained and only a temporary flow alongthe discontinuity occurs during rainy events, but occasionally the

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flow can be stuck at the slope toe: U = 0.5w(H-z)2cosec+wzHcosec .

Three-dimensional stability.

The simplest scheme of three-dimensional instability is referred to the caseof a tetrahedral wedge having one or two free surfaces. For the calculationof the safety factor, the weight of the wedge, having two free surfaces, canbe decomposed in two components: T12 acting along the intersection line of the two joints; N12 normal to this line.The last one must be balanced by a tangential reaction TN and by a normalreaction N, acting on both the faces.

The reaction N determines the maximum mobilised resistance to the slidingand the safety factor can be defined as follows:

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22

12

2 N

r

TT

A

NAT

Fs

where:A = area of each joint;Tr = selected shear strength criterion.

To reach equilibrium along the normal direction to the intersection line mustbe:

1212 cos2

cos22

sen2 bWNi

Ti

N N

where:

i = angle between the joints A and B;b12 = angle of dip, in respect to the horizontal plane, of the

intersection line.From a static point of view, the problem is undetermined, because differentcombinations of N and TN can give different safety factors. By assumingTN = 0, the maximum safety factor, among the possible ones, is calculable(method of the rigid wedge).To calculate the safety factor, it's necessary, first of all, that the orientationof the joints, in respect to the slope, be able to allow the sliding. Moreover itneeds that the wedge be in contact to the underlying rock mass, that is thenormals to the wedge faces have to be directed downward.The safety factor can be calculated as follows:

1)

12

2

222

1

111

T

A

NTA

A

NTA

FsRR

where:

A1 = area of the joint 1

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A2 = area of the joint 2TR1 (N1/A1) = available shear strength along the joint 1 as a function

of the normal stress N1/A1;TR2 (N2/A2) = available shear strength along the joint 2 as a function of the normal stressT12 = component of the wedge weight acting along the intersection

of the planes 1 and 2.

The expression 1) is valid only in case both N1 and N2 are greater than zeroand the wedge moves itself along the intersection between the planes 1 and2. In case of N1>0 and N2<0 or N1<0 and N2>0 the sliding happens alongthe dip angle respectively of the plane 1 and of the plane 2 and not along theintersection line. Thus the safety factor has to be expressed in the followingways:

1

1

111

T

A

NTA

FsR

(N1>0 and N2<0)

whereT1 = component of the wedge weight acting along the dip angle of

the plane 1

2

2

222

T

A

NTA

FsR

(N2>0 and N1<0)

whereT2 = component of the wedge weight acting along the dip angle of

the plane 2

Finally, in case of N1<0 and N2<0 (wedge which is uplifted in respect to theunderlying rock mass, due to a toppling or to a very high water pressure) it'simpossible to define a safety factor and thus it assumes a instability withoutnumerically quantify it.

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1) theoretical basis. 1.1 stereographic projection of rock ... · 1.1 stereographic projection of...

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