Fulltext-13(10) Aydan Kawamoto 1992 Flexural Toppling

Rock Mech. Rock Engng. (1992) 25 (3), 143--165 Rock Mechanics and Rock Engineering �9 Springer-Verlag 1992 Printed in Austria

The Stability of Slopes and Underground Openings Against Flexural Toppling and Their Stabilisation

By

O. Aydan I and T. Kawamoto 2

1 Department of Marine Civil Engineering, Tokai University, Shimizu, Japan 2 Department of Geotechnical Engineering, Nagoya University, Nagoya, Japan

Summary

The stability of slopes and underground openings during and after excavation is always of great concern in the field of rock engineering. Depending upon the geologic conditions and material properties of rock and discontinuities, and the geometry of excavations and topography, various kinds of instabilities are likely to be encountered. One of these is the flexural toppling failure which has become to be known recently. A stability analysis method for slopes and underground openings under various loading conditions against the flexural toppling failure is proposed. In addition, for the stabilisation of structures, a method is suggested to take into account the reinforcement effect of fully grouted rockbolts. The applicability and validity of the proposed method is checked through model tests carried out in laboratory under well controlled conditions.

1. Introduction

The stability of slopes and underground openings during and after excavation is always of great concern in the field of rock engineering. Depending upon the geologic conditions and the geometry of slopes and underground openings, the slopes and underground openings may be permanently stable, locally instable but globaly stable, or locally as well as globaly instable. Instability modes encountered in rock excavations are many in kind and the governings factors associated with their mechanism differ depending upon the loading conditions and physical nature of rock mass and the geometry of the excavations and topography. One recently recognised failure mode is flexural toppling failure which has become well- known in the case of actual (Mtiller, 1968; de Freitras and Watters, 1973; Bukovansky et al., 1974; Goodman and Bray, 1976) and model rock slopes (Erguvanh and Goodman, 1972; Erguvanh, 1973; Hoffman, 1974; Kawa- moto, 1982; Aydan et al., 1988); such failures had formerly been wrongly

144 (J. Aydan and T. Kawamoto

Fig. 1. Flexural toppling failures observed in model slopes and actual rock slopes

The Stability of Slopes and Underground Openings 145

known as gravitational flow or creepage (Sharpe, 1938; Zischinsky, 1966) (Fig. 1).

Rock slope failures near the village Elm in Switzerland and in Turtle Mountains in Canada are two well-known slope failures involving the flexural failure of rocks (Zaruba and Mencl, 1976).

Except one case which is reported by Heslop (1974), there was no recognition of the flexural toppling failure in underground openings. This may be due to the local nature of the flexural failure in underground excavations in comparison with widespread occurrence in surface slopes. Flexural failure was observed by the authors in laboratory model tests using a base friction apparatus and at several sites in Japan and Turkey (Aydan et al., 1988) (Fig. 2).

Fig. 2. Flexural toppling failures observed in model underground openings and in an actual underground opening

Because this type of failure has only recently been recognised, mechanical models to analyse the stability of slopes and underground openings against flexural toppling are very sparse in the literature. How- ever, Aydan and Kawamoto (1987) have developed a stability analysis- method for rock slopes against the flexural toppling and for their stabilisa-

146 O. Aydan and T. Kawamoto

tion. The present authors have extended this method to underground openings and taken into account more complex loading conditions. The validity and applicability of the method have been checked by laboratory model tests carried out under closely controlled conditions.

2. Theoretical Model

Flexural failure is, potentially one of the most common modes of failure in the case of excavations in layered sedimentary or schistose metamorphic rocks and occurs through bending of layers of rock in a similar manner to that of cantilever beams. Therefore, it would be instructive to investigate and discuss the behaviour of a cantilever beam explaining the proposed method of analysis.

2.1 Cantilever Beam

Let us consider a cantilever beam subjected to only its own weight under two loading conditions as shown in Fig. 3. Employing the column theory known in the theory of elasticity (i. e. Timoshenko, 1974), the distribution

UNCI:tACKED

" \ . A e,

CRACKED �9

OVERHANGING Fig. 3. Models for suspended and overhanging cantilever beams with or without a crack


o f f iber stress ~x at the base of the co lumn with a unit thickness can be given as :

N M = - + - 7 + -7- y (1)

where N: Norma l force, M: Moment , I : Iner t ia modulus .

The above express ion takes the fol lowing explicit form:

o-~=?/h + c o s c ~ + 3 s i n c ~ (2)

where ?': unit weight of column, h: co lumn height, t: co lumn thickness, c~: base incl inat ion o f the column,

+ and - signs are for a suspended co lumn and an overhanging co lumn respectively.

Let us assume that a crack has occur red and p ropaga ted to a distance t - t* along the base and let us also in t roduce the fol lowing normalis ing parameters :

h m = t O < _ m < o o (3)

n = - - 0 _< n _< 1. (4) t

Then, for each co lumn type with cracks, the fol lowing expressions are valid:

c~ + = 7 h ncos c~+ 3 {msinc~ + cos c~(1 - n)} n2 (5)

- n c o s a~+ 3 { rn s i n a i - cos c~(1 - n)} o-~- = ~/h n2 (6)

Normal is ing the above expressions with respect to the stress at the t ime of the crack ini t iat ion yields the fol lowings:

o-; 1 - { 1 + ' ~ + ( l - n ) } n2 (7)

where

T o.;;=1 - - { 1 - . ~ - ( l - n ) } n2 (8)

2 cos cc

3 m sin c~ + cos o~

2 cos c~ . ~ - = 3 m sin c~ + cos o~


It is apparent from Eq. (7), for a suspended column, the stress at the tip of the crack will monotonically increase with the crack length, provided that the load is kept constant. In the case of an overhanging column, an additional condition is introduced such that 3 m sin cr > cos cr which is the condition of tilting of a column of the same height with a throughgoing crack at the base. With that condition, it is clear that the stress at the tip of the crack will also monotonically increase with the crack length. The above expressions are plotted in Fig. 4. The implication of the above is that once the stress at the outer fiber of the column reaches the tensile strength of

SUSPEN.ED / , ! /

9 / / / / /

~ / , " i /

6 ,/ 3_2 ,, /,/>i/_ 1 . . 5

+., ,- � 8 8 . . . . ~.99

3- 9.00

2"

1

0 I I I i I I I /

0.0 0.1 0 .2 0.3 0.4 9.5 9.6 0 .7 0.8

N O R M A L I S E D C R A C K L E N G T H t -- t* t

O V E R H A N G I N G / / I I ] lO- I,i i / : 9. / / / i , !

6- ' /

~ ...._/."./;i//'--0.90 . . + ; . ; . . j . . _ . . . . o . o 3'

. i ' , . i . .

1' - - . . . . 1.00

0.0 0'.1 o'.2 o'.3 0'.4 0.5 0'.6 9:7 oJs 0:9 N O R M A L I S E D C R A C K L E N G T H t - t*

t

Fig. 4. Stress intensity at the crack tip in relation with the crack length


rock, a crack will be initiated and if the load is not decreased, the crack will propagate, and the failure of the column will be inevitable. In other words, the maximum resistance which can be offered by a column is equal to the load which causes crack initiation at the outer fiber. Figure 5 shows a plot of the required ratio of the thickness to the height of a column at the limiting state related to the inclination of the base plane.

90. i t ~ / . - " /.</...//...

'~ . . - ' " "1 1

1 - - - / / . ; / J - . - 50

'~ 10 / "

o 0 1'0 2'0 3'0 4b 5'0 ~'0 ~'0 ~0 9'0

INCLINATION OP BASE c~ (~

Fig. 5. The required ratio of the column height to the width for stability in relation with the inclination of the base

2.2 Slopes and Underground Openings

A mechanical model for analysing the flexural stability of slopes and underground openings can be developed employing the theory of columns. To start with, a fundamental assumption has to be made, that is, the layers of rock above on a plane, which is normal to the dip of layering and ema- nating from the toe in the case of slopes and the corner in the case of underground openings which we call "the basal plane", bend only (Figs. 6 and 7). This assumption arises from laboratory and field observations of flexural failures in surface and underground excavations. Another assumption regarding the action position of forces at the interfaces among columns is also necessary to determinate the system. For an elastic distribution, the stress distribution will range from a triangular to a point distribution as illustrated in Fig. 8. The position of application of the normal forces due to gravitational and seismic sources can be given in the following general form:

Xi = / ]hi (9)

150 C). Aydan and T. Kawamoto

H

Fig. 6. Model for limiting equilibrium analysis of flexural toppling of a slope

ROOF

/ / K . ~ / " /U

SIDEWALL

Fig. 7, Model for limiting equilibrium analysis of flexural toppling of an underground opening

where hi is the height o f co lumn i. Observa t ions on mode l tests indica ted tha t the va lue of rl should be be tween 0.75 and 1; in any event assuming 7/ as 1 will err on the safe side.


t

3"

Z/:

Fig. 8. Illustration of the position of action of normal force on a column for various distribu- tions of normal stress

In the fol lowing subsections, expressions for the stability of slopes and u n d e r g r o u n d openings against the f lexural toppl ing fai lure are developed.

2.2.1 Slopes

Let us consider a slope composed o f layers o f rock dipping into the hill- side. The forces acting on a representa t ive layer with a unit thickness are i l lustrated in Fig. 6. The outer f iber stress at the base of co lumn can be shown to be:

a~=,/2= _ _ + __. Ai 2

,i ( lo )

where

w,=

A i = UiS+ 1: U[_ I:

u~: h~ : hi_ 1:

li+ l: li_l: t~: o~"

W / c o s ~z

W,. sin 7ti (hi + hi_ 1)/2 (hi --k hi_ 1)/2 ti water pressure at side i + 1

water pressure at side i - 1

un i fo rm water pressure at base

eccentr ic water pressure at base co lumn height at side i + 1 co lumn height at side i - 1 eccentr ici ty o f water pressure at base height o f water force act ion at side i + 1 height o f water force act ion at side i - 1 co lumn thickness incl inat ion of the layer normal


Assuming now that the following relations between side forces exist:

~+~ = ~g.+l , ~ -1 = ~P,-1 (11)

where # = t an ~b, and introducing the yield condition such that the outer fiber stress of the layer is equal to the tensile strength 05 of the rock with a factor of safety as:

o5 F S - cyy=ti/2 (12)

the following expression for Pi-1 can be derived:

Pi-~= (1"] t~ hi s 1 2Ii(~_.~ Ni-Ui b )

ei+l hi-lt q - S i T +(uiblt, iq- US+ill- U~-I i-1}-~7-i -~ T

( ,h,_~ + ~ ) (13)

Starting from the uppermost column numbered n which is unstable under its own weight, the above expression is solved by a "step by step method". The criteria for slope stability are as follows:

P0 < 0 stable P0 = 0 at the limiting state (14) P0 > 0 unstable

2.2.2 Underground Openings

Let us consider an underground excavation in a layered rock mass. The force system acting on a representative layer with a unit thickness is illustrated in Fig. 7. The outer fiber stresses for layers in the side wall and in the roof take the following forms:

Sidewall (T~ I=ti/2 __ l~e - Ni w

Ai

t, P~+~ rlh~- (T~+I + T~_a)~ + (S 2 + $7)- ~- - P,_~ qh,_l

+ 5 - i, (15)

Roof ~ - N~ ~ ~Yx = ti/2 __

Ai

t~ P~+~qh~-(T~+~+ T~-O-~ +(S~+S~)-2 --P~-~rlh~-I

+5- z, (16)


where ~ = IV//cos S~ ~ = IV//sin of

Nie= ~ sin (c~ + fl)

sg = K cos (c~ + fl) Wi = y t i (hi + hi_ 1)/2 A i = ti Ei= ~W,. ~:: seismic coefficient hi: column height at side i + 1 h~_l: column height at side i - 1 ti : column thickness a~: inclination of the layer normal fl: inclination of the seismic force

Assuming that the following relations between side forces exist:

~+1 =/~1`+1, Ti_l =/~1`-1 (17)

where /.t = t a n r and introducing the yield condition such that the outer fiber stress of the layer is equal to the tensile strength or, of the rock with a factor of safety as:

F S - a~ = ,i/2 (18)

the following expression for 1`-1 can be derived:

S i d e w a l l

,_1=

( ti t 1`+1 , h i - , .-~ + (s?' + s~) -~ t, ~ +

Roof

1`_1=

(19)

1`+1 ~h, -/z + (s?'+ s~) ~ ti -~- + ai

(,h,_l + ~ ) (20)

The solution procedure and the criteria for stability are as set out previ- ously for surface slopes.

3. Stabi l i sat ion o f S lopes and Underground Openings

The derived expressions for stability indicate that a reduction in the magnitude of the moment and an increase in the compressive normal load will

154 13. Aydan and T. Kawamoto

have an stabilising effect on the structure against the flexural toppling failure. There are a number of ways to provide such an effect through the use of artificial support. Prestressed cables and /or fully grouted rockbolts are effective solutions. When the prestressed cables are used, they should be anchored beyond the basal plane, otherwise prestress forces may cause much higher bending stresses in layers. The alternative is to use fully grouted rockbolts or "dowels" (Martin and Kaiser, 1984; Moore and Lewis, 1982). The authors believe that the fully grouted rockbolts would be more economical than rock anchors as they are shorter and do not need to be prestressed. In this paper, the authors will, therefore, consider only the fully grouted rockbolts and present a way to introduce the effect of reinforcement into the previous expressions.

As is very well known in the literature (BjiJrstrom, 1974; Haas, 1981; Hibino and Motojima, 1981; Ludvig, 1983; Gaziev and Lapin, 1983; Egger and Fernandes, 1987; Aydan et al., 1987; Yoshinaka et al., 1987; Aydan, 1989), fully grouted rockbolts contribute to the shear resistance of discontinuities directly through the shear resistance of steel bar itself and indi- rectly through the shear and frictional components of the axial force in the

/[_

Fig. 9. Illustration of the deformation of bolted strata and a model for rockbolt crossing a discontinuity

N

t t

N

bar. The reinforcing effect of a rockbolt on the shear resistance of a discontinuity plane can be expressed in the following form (Aydan, 1989) (Fig. 9):

Tb = Ab o'b 1 + -~- tan ~ sin 2 0 (21)

where Tb: shear resistance due to bar Ab: cross section of bar O'b: magnitude of stress in the bar in the direction of shearing q5 : friction angle of discontinuity 0: angle between the bar axis and discontinuity (installation angle)


Therefore, when n number of bolts cross the discontinuity plane, the total resistance offered by bolts can be given in the following form:

Tt = Ab as 1 + ~- tan ~ sin 2 0* . (22) i = 1

The results of laboratory and field studies and numerical modelling (Bjfirstrom, 1974; Haas, 1981; Aydan et al., 1987; Yoshinaka et al., 1987; Aydan, 1989) show that the stress in the bar quickly developes and becomes almost equal to its yielding strength at very small displacements in the order of 0.1 to 3 mm. It is reasonable, therefore, to assume that the stresses in bars are same and equal to their yield strength (a0 = %). The yielding strength of a bar subjected to tension or compression and shearing have been suggested by Chesson et al. (1965) to be:

as at (23) o-~ = ]/(as cos 0) 2 + (at sin 0) 2

where at: shear strength of bar a,: tensile strength of bar

Accordingly, provided that the installation angle of all bars is the same, the total shear resistance of n bars is:

T~=nAb~rb l + ~ - t a n ~ s i n 2 0 . (24)

Figure 10 shows a plot of normalised reinforcement effect of a rockbolt as a function of 0, ~ and ~ , where ~ = cr~/~.

Considering the contribution of bolts as an additional to the shear resistance across interfaces of layers, one obtains the following expressions for a simple loading case in which gravity alone acts:

Slopes and Sidewalls e l ' _ 1 ~

P,.+I ~ h , l / 2 T + Tf- T + s;~ L a, t, -FS + A, /

(~hi_i +/2~) (25) Roof

P i - - 1 ~

P~+ I 71hi - /2-~ - {T I+1+ TI -1} -2 + S~, 2 ti F S Ai

(7h,_1 +/2~) (26)

156 O. A y d a n a n d T. K a w a m o t o

r

1 .2

1 .0 &,?

O

O 0.8

fD

0.6

O

Z

0 . 4

0 0 .2 Z

0 . 0

-/" \ t ~ = 0.5 /)..----... " \

q~ (~

0

. . . . 15

30

45

' I ' I ' I 30 60 90

A N G L E B E T W E E N BOLT AXIS AND D I S C O N T I N U I T Y 0 (~

Fig. 10. Normalised reinforcement effect of rockbolts in relation with the variation of their installation angle

4. Applications and Discussions

As is apparent from the expressions, the stability of slopes and underground openings are closely related to the geometrical configuration of slopes and underground openings in relation to layer thickness, the tensile strength of rock, frictional properties along the interfaces of layers, and the drainage and seismic conditions. The present authors will first show how the expressions developed can be applied to model slopes tested in their laboratory using the base friction apparatus (Kawamoto, 1982; Kawamoto

Table 1. Mechanical properties and dimensions - - First series of models tests

Reference or, G E v y r H t tg kPa kPa M P a k N / m 3 (~ m m m m (~

Kawamoto , 1982 4 11.4 3.02 0.35 12.4 40 300 30 75 Aydan and Kawamoto , 1987 4 11.4 3.02 0.35 12.4 40 150 30 90


et al., 1983; Aydan and Kawamoto, 1987). The mechanical properties of the material and the dimensions used in the first series of layered model slopes are given in Table 1, and a sketch of the test arrangement is shown in Fig. 11 ; calculated results are plotted in Fig. 12 together with the experimental results for q = 1. As can be seen in Fig. 12, the calculated and experimental results are in fairly good agreement with experimental results. When the inclination of the normal of layers is between 10 ~ and 50 ~ the calculated results imply that the slope angle must be reduced if the stability is required, otherwise the failure is inevitable; this is confirmed in model tests. As the inclination of the normal increases, the slope angle for the stability also increases.

B A S E F R I C T I O N M O D E L T E S T

A P P A R A T U S

Fig. 11. Sketch of the test a r r a n g e m e n t

O b - '

Z o

r/?

o �9 o o �9 �9 / / 0 - - - / . /

, / - / H / - t - 5

/ ~-= i0-----

! / / Nawamoto (1982)

Aydan &: Kawamoto (1987)

[3

i----90 ~ H-~15 cm t = 3 cm

i=75 ~ H - - 3 0 c m t----3 cm

Stable Uns tab le

[2 �9

o �9

I I I I 1 i I I 10 20 30 40 50 60 70 80

I N C L I N A T I O N OF N O R M A L o: (~

Fig. 12. C o m p a r i s o n of expe r imen ta l resul ts with ca lcu la t ions

I 90

158

g~

g

Z

O r/?

O e~

(}. Aydan and T. Kawamoto

r

4 3 ~

. . . . 40 ~

[] Stable

�9 Uns tab le

H --t-- = 14

0 10 20 30 4'0 5'0 dO 40 gO 9'0

I N C L I N A T I O N OF N O R M A L O~ (o)

Fig. 13. C o m p a r i s o n o f expe r imen ta l resul ts with ca lcu la t ions

A second series of tests were carried out. Table 2 shows the mechanical properties of the material and the dimensions and slope angle. Figure 13 shows the calculated and experimental results for 7]=1. Once again, a good agreement between calculations and experiments is apparent from Fig. 13. Conclusions similar to those drawn from the first series of tests can be drawn. The good agreement between the experiments and calculations suggest that the expressions can be used to analyse the stability of slopes against the flexural toppling failure.

T a b l e 2. Me chan i ca l proper t ies and d i m e n s i o n s - - Second series o f mode l tests

~, cr C E v 7 / r H t q/ kPa kPa M P a k N / m 3 (~ m m m m (~

12.4 31.6 8.37 0.29 17.21 40 - -43 350 25 7 5 - - 9 0

A parametric study was then performed using the material properties and dimensions given in Table 3, and the calculated results plotted together with data from case histories reported in literature and sites visited by the authors (Fig. 14). Since the material properties and dimensions of the case histories are different from those assumed in the calculations, the comparison between the calculations and the case histories is bound to be in qualitative terms rather than quantitative terms. Whatever the material properties and the geometries are, the general tendency of the


calculated results and the case histories is almost the same and they imply that when the layers or schistosity planes steeply dip into the hill-side, the slope angle must be greatly reduced if the stability is desired. In addition, the rock type in the compiled case history data indicates that the flexural toppling failure is much more likely in the sedimentary and schistose metamorphic rocks than in unweathered igneous rocks. High angle under- cutting should be avoided, otherwise catastrophic flexural toppling failure may take place as observed in Elm in Switzerland, reported by Heim (Zaruba and Mencl, 1976).

Table 3. Mechanica l proper t ies and d imens ions used in the paramet r ic s tudy for slopes

o5 / r H t kPa k N / m 3 (~ m m

250.00 25.00 30 100 0, 1, 2, 4

The expressions developed were then used to predict the stability of model underground openings in a layered medium and tested using the base friction apparatus. The mechanical properties are the same as given in Table 2. The shape of openings was square and dimensions were varied

12 6 19 �9 j ~ 2 - ~ 0 - -

�9 ..;r 9 . ' / / " 0 �9 "" 3 �9 32 . ~ / ' I /.~z / . j

25 / " / / J 24t0 ~--28 . ' / - / 20 21 22 126 2 7 / " ~ . / / ~r~ = 0.25 MPa

I @ I l l _ _ ~

29 �9 j ~ ~-~Zi 3 ~ 30 1 0

1 / / J O Stable

H/ t

lOO . J

�9 ~ " . . . . 5 0

10 16 �9 �9

. . . . . 25

0 10 20 30 40 50 60 70 80 90

INCLINATION OF NORMAL a (o)

Fig. 14. Plot o f s lope angle versus the normal o f strata for stable and topp led slopes together with calculated results

160

z Z

0 S t a b l e

�9 U n s t a b l e

{). Aydan and T. Kawamoto

ROOF

O I I

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] 0 13

i ~ , , i i i i

0 I0 20 30 40 50 60 70 80

INCLINATION OF NORMAL o~ (o)

Fig. 15. Comparison of experimental results with calculations

9O

from 5 x 5 to 25 x 25 cm z. The calculated and experimental results are plotted in Fig. 15 for 7/= 1. It is apparent from this figure that flexural toppling failure in the roof starts to occur when the width is greater than 15 cm, and that flexural toppling failure is more likely to occur for a layer inclination angle in the range of 10 to 50 ~ . Apart from one case falling out with the calculated unstable region, the results can be said to be fairly in good agreement with those determined experimentally.

Table 4. Mechanical properties and dimensions used in the parametric study for underground openings

? ~b H L t k N / m 3 (~ m m m

25 15 10 10 Variable

A parametric study was carried out to evaluate the effect of tensile strength of rock required for the stability in the roof and sidewalls by vary- ing the width and height ratio of the opening to the layer thickness; the material properties and dimensions used are given in Table 4. The results are shown in Fig. 16. As can be seen from this figure, the tensile strength required increases as the normal becomes steeper and the ratio of the opening width to the layer thickness increases. The same conclusions are also


valid for sidewalls. However, the magnitude of the required tensile strength for stability is much lower than that in the roof and, for a certain range of layer inclinations, the required strength can be zero. The implication is that instability is more likely in the roof than that in the sidewalls and that the required forces for stabilisation should be higher in the roof than that in the sidewalls.

Fig. 16.

m

~ m m:~ 1 ~ O

r162 ~a 0

L R O O F t

. . . . 20

4 0 ~ _ .

. . . . 80 ~ \ \

100. . . . . .1-" ~ " . . . . , , ) , ~

/ . .----""" /,: _.-I"

' / /_j__ -----

~ " i i r

0 30 45 60 75 90

INCLINATION OF NORMAL (~ (~

~ v m>,

&

3-

2

1.

H

t

. . . . 20 S I D E W A L L

- - - - - 40

. . . . 80

r

15 30 45 60 75 90

INCLINATION OF NORMAL a (~

N o m o g r a m s fo r t he r e q u i r e d t e n s i l e s t r e n g t h o f r o c k fo r t he s t a b i l i t y o f o p e n i n g s a g a i n s t f l e x u r a l t o p p l i n g f a i l u r e

An example of application of the expressions for the reinforcement effect of rockbolts on the slope angle required for stability will now be considered. The material properties used in the analysis are given in Table 5 and the results are shown in Fig. 17. The reinforcing effect of rockbolts is easily apparent in Fig. 17, and implies that the rockbolts can have a considerable stabilising effect on the slopes against the flexural toppling failure.

162 ~. Aydan and T. Kawamoto

Table 5. M e c h a n i c a l p r o p e r t i e s a n d d i m e n s i o n s u s e d f o r r e i n f o r c i n g e f f e c t o f r o c k b o l t s in s u r f a c e e x c a v a t i o n s

M a t e r i a l s crt y r H t d s 0 n M P a k N / m 3 (~ m m m m (~

R o c k 0.5 25 - - 20 0.5 - - - -

D i s c o n t i n u i t y - - - - 30 . . . .

R o c k b o l t s 300 . . . . 25 0 .56 30 2

Fig. 17.

U"

O

Z .<

N

O

Q N

O' N

60

L T E D

5 0 - . . . . . . B O L T E D

4 0

30

20

I 0

0 �9 o 1'o ~o 30 4"o ;o ;o 70 ;o ~o

INCLINATION O F NORMAL c~ (~

C o m p a r i s o n o f slope angles at the l i m i t i n g s t a t e in b o l t e d a n d u n b o l t e d c a s e s in r e l a t i o n w i t h t h e v a r i a t i o n o f t h e n o r m a l o f l a y e r s w i t h r e s p e c t t o h o r i z o n t a l

A parametric study on the reinforcement effect of rockbolts on roof and sidewall stabilisation of underground openings was also carried out; the results are shown in Fig. 18. The material properties used in the analysis are given in Table 6. The required reinforcement effect of rockbolts in the roof is clearly illustrated to be much larger than that in the sidewalls. The figure also shows that the mobilised shear resistance of the bars will be

Table 6. M e c h a n i c a l p r o p e r t i e s a n d d i m e n s i o n s u s e d f o r t h e r e i n f o r c i n g e f f e c t o f r o c k b o l t s f o r u n d e r g r o u n d e x c a v a t i o n s

M a t e r i a l s crt 7 ~b H L t d K2 0 M P a k N / m 3 (~ m m m m m (~

R o c k 0 .25 25 - - 10 10 v a r i a b l e - - - - - -

D i s c o n t i n u i t y - - - - 15 - - - - - -

R o c k b o l t s - - - - 25 0 .56 30

Fig. 18.

T h e S t a b i l i t y o f S l o p e s a n d U n d e r g r o u n d O p e n i n g s 163

L

10 10T a o o = - - - 20 ,

N 8. J - - 30 / - ,, 0 - - - - - 40 / / / 1 ~ . . . . ~ . x

O ~ / / . j . - -,,.

, ' . . - - 'Z-- - - '_ ; - - - - - . . . . . . . . . . - - - - ) \ Om 4 ~ <-/" j - ' 7 i - \ o .~..- f- .j/

0 0 15 30 45 60 75 90

INCLINATION OF NORMAL ~ {~ H

t

10

. . . . 20

3O ~ S I D E W A L L

o ~ 6 - - - - - - 5 0

g 0 [

15 30 4 5 60 75 90 I N C L I N A T I O N O F N O R M A L (x (~

N o m o g r a m s for the requi red rockbol t force for the stabil i ty o f open ings aga ins t f lexural t opp l ing fai lure

larger in the case of thinly layered media as compared with the thickly layered media, provided that rock in the vicinity of the bars remains stable.

5. Conclusions

A method for analysing the stability of slopes and underground openings against flexural toppling failure has been proposed. Its applicability and validity have been checked through the model tests on surface slopes and underground openings carried out under closely controlled conditions in laboratory. It is found that the proposed method is valid for analysing the stability of model slopes and underground openings and it can be used to predict the stability of the actual slopes and underground openings in layered rock masses.


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Authors' address: Prof. Dr. Toshikazu Kawamoto, Department of Geotechni- cal Engineering, Nagoya University, Chikusa-ku, Nagoya 464, Japan. Asst. Prof. Dr. 0mer Aydan, Department of Marine Civil Engineering, Tokai University, 3-20-1, Orido, Shimizu 424, Japan.

Fulltext-13(10) Aydan Kawamoto 1992 Flexural Toppling

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