The influence of rate- and displacement-dependent shear resistance on the response of rock slopes to seismic loads

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 19, pp. 1 to 8, 1982

Printed in Great Britain. All rights reserved

0148-9062/82/010001-08503.00/0 Copyright © 1982 Pergamon Press Ltd

The Influence of Rate- and Displacement-Dependent Shear Resistance on the Response of Rock Slopes to Seismic Loads A. M. CRAWFORD* J. H. CURRANI"

An evaluation of the potential practical importance of rate- and displacementdependent frictional resistance of discontinuities on the stability of rock structures has been investigated with the aid of a rigid sliding block slope stability model. Constitutive relationships for rate-dependent frictional resistance, based on experimental observations, have been utilized. It is shown that the existence of a rate effect may be an important consideration in the design of structures in jointed rock masses.

INTRODUCTION

stability of a rock slope (Fig. 1) subject to dynamic loads. In the pseudo-static approach [8], failure is assumed to have occurred once the shear resistance on the surface between the slope and the block is less than that required for limiting equilibrium. Since a seismic loading varies with time, the resistance necessary for equilibrium between the block and slope may be only momentarily exceeded (that is, experience a factor of safety of less than 1.0 for a short period of time). During this time, a finite relative displacement of the slope and block may occur, hence, a factor of safety calculated by the pseudo-static method does not give a reasonable measure of the stability of a slope. The distinct element technique proposed by Cundall [3] may be used to model the behaviour of an assemblage of rigid blocks. Each block is treated as a separate entity, with the contacts between blocks being modelled by normal and shear force-displacement relationships or frictional resistance~ Displacements of

In an experimental programme [1,23, it was determined that the frictional resistance of two sliding surfaces was, in general, rate (velocity)-dependent. The effect (an increase or decrease in shear resistance) and the magnitude of this rate dependency was found to be primarily a function of the normal stress applied to, and the mineralogy of the sliding surfaces. The significance of this phenomena to the response of rock structures has been investigated with the aid of a rigid sliding block slope stability model in which the displacements of a rigid block on a sloping surface, subjected to seismic or dynamic (inertia) forces has been investigated. In this analysis, the resistance of the sliding surfaces has been taken to vary with relative displacements between the surfaces (that is, a strain softening behaviour) or to be dependent on the relative velocities between the surfaces. Using appropriate resistance-displacement and resistance-velocity relationships derived from the testing programme, it is shown that the existence of rate effects may be an important consideration in the design of engineering structures in jointed rock masses. M E T H O D S O F DYNAMIC SLOPE STABILITY ANALYSES There exist numerous methods for estimating the * Department of Mineral Resources Engineering, Imperial College, London SW7 2BP, U.K. t Department of Civil Engineering, University of Toronto, Toronto, Canada, MSS IA4. R.MM.S. 1 9 ; 1

A

Fig. 1. Model of rock slope with sliding block.

1

2

A . M . Crawford and J. H. Curran

the blocks are calculated by integrating the equations of motion using a time-stepping procedure. Recently [9], the method has extended for solving simple dynamic slope stability problems. McKinnon [10] developed a dynamic finite difference model to study the influence of blasting on the stability of jointed rock slopes. With this model, the change in stresses on a sliding surface due to the propagation of stress waves from an explosion may be estimated. However, its usefulness is restricted since the displacements along joints due to sliding cannot be modelled with this code. In the Fifth Rankine lecture, Newmark [-11], proposed a method for estimating the expected permanent displacements, with time, of a sliding mass of rock under the influence of a dynamic loading. In this model, which is based on the principle of limiting equilibrium, the resistance along the sliding surface is assumed to follow a rigid-plastic behaviour, and that the sliding mass moves as a rigid body. Based on this mGdel, and using four earthquake records, Newmark produced a chart for estimating the upper bound of permanent deformations of slopes subjected to seismic loadings. Subsequently, Franklin & Chang [5] extended this work to a large number (179) of strong motion earthquake records. In this work, a method of analysis based on the Newmark approach is presented in which the shear resistance of the sliding surfaces is both rate- and displacement-dependent. A computer coding has been written in which a digitized earthquake accelerogram is numerically integrated to give the velocity-time and displacement-time histories of both the slope and a rigid block representing the sliding mass. The resistance to sliding may be symmetrical or asymmetrical, allowing the block to move both up and down slope in the case of symmetrical resistance, or only downslope for assymmetrical resistance. THEORY

Inertia force

AmgL..~ I~ng/ A g

/

~

~

f

Ground acceleration

Fig. 2. Inertia force acting on block.

factor of safety during an earthquake is then (Fig. 3) F = (cos c~ - A sin (c~ - 0)) tan q5 (A cos (~ - 0) + sin cd

(2)

Similarly, for a downhill pulse F=(C°S~+ Asin(~-0))tanq5 (A cos (a - 0) - sin ~)

(3)

The ground acceleration, Ag, at the point of sliding, that is, when the driving force equals the resisting force, has been called the yield or critical acceleration Acg [6,13]. Taking F equal to unity in equations (2) and (3) gives for the uphill pulse cos ~ tan q~ - sin c~ A c,.p = cos (0 - c0 - sin (0 - ~) tan ~

(4)

and downhill pulse Ac, aow, =

cos ~ tan q~ + sin cos (0 - ~) + sin (0 - c~)tan q~

(5)

When the earthquake acceleration Ag is greater than Acg, the factor of safety is less than one and the driving force is greater than the resisting force and a relative displacement and velocity occurs between the block and the slope. The equation of motion of the block relative to the slope, by Newton's Law is mY = Driving force - Resisting force

(6)

where x is in the direction of the slope. Equation (6), for the downslope movement may be written as [12]

Consider the two dimensional problem of the dynamic stability of a rigid block resting on an inclined :~ = gcos(~ - 0 - ~) slope (Fig. 1). If the block is stationary, then the fric(A - Ac) (7) COS t~ tional and gravity forces are in statical equilibrium; and the shear force, S, available for equilibrium is limited by Sarma [12] indicates that both the factor of safety the shear resistance along the sliding surface. Hence, if and displacements are relatively insensitive to the inclithe mass of the block is m, the slope is inclined at an angle ct to the horizontal, q5 is the frictional resistance of / the surface and N is the force normal to the sliding / / st~ surface, then S cannot exceed N tan (/5 without the / / block experiencing an acceleration. / The static safety factor of the block is given by /

F = Resisting force = (mo cos c~)tan 4) _ tan 4' (1) Driving force (my sin cd tan ,=

/ / / A~m9

A e mgsin ( c z - O l / k ~ ' ~

If an earthquake ground acceleration A O at an angle 0 to the horizontal acts on the slope, this causes an inertia force rnA O on the sliding block (Fig. 2). The

l J 0

S----T--

Fig. 3. Forces acting on block for an upslope acceleration pulse.

Rock Slope Response to Seismic Loads

3

.= e

z "~.I00

3%= ~,

1.00 (Ac/A m) 0.95 (A¢/A~ 0.90 (Ae/Am) 0.08 (Ac/Am)

oggo

0.60 (Ac/Aml

~'~

~o

°o

~o

40 Shear displacement, mm

E

°-~E~ =oouE~] = ~

~ -

[ X~<5Omm X > 50mm

Y • {Ac/Am) 11" (I-PI(X/50)} ] Y • P (Ac/Am)

I 50

Fig. 4. Direct shear test on a rough Dolomite surface (normal load, 50 kN).

X,

Relative shear displacement, mm

Fig. 6. Displacement-dependent shear resistance model.

nation, 0, of the inertia force. Thus it is reasonable to consider the acceleration as acting parallel to the slope without much error. For this case, (0 - ¢) = 0, the rela- strain-softening until the residual shear strength is tive velocity and relative displacement between wedge reached. and slope may be obtained by double integration of In the Newmark analysis, it is convenient to ignore equation (7). For earthquakes, where A is a function of the elastic deformation component since permanent time, (A(t)), the total relative displacement is the sum of deformations are, in general, several orders of magnidisplacement increments occurring during a number of tude greater than elastic deformations. A strain-softenseparate pulses of ground motion when the yield accel- ing shear stress-shear deformation yield resistance reeration is exceeded. Sliding occurs intermittently, with lationship (that is, Ac = At(x)) may be readily incorporthe block coming to rest relative to the slope some ated into a computer analysis by updating the yield time after the driving force on the sliding surface resistance, based on the relative displacement of the becomes less than the available shear resistance. During sliding surfaces, at each time step. this time, the sliding block is decelerated until the abIt has been found [1,2], that the yield resistance of solute velocities of the block and the slope are the discontinuities is dependent on the relative shear velsame. Hence it is often acceptable for the Factor of ocity, as well as the relative shear displacements of the Safety to be less than one, provided that it lasts for a sliding surfaces (that is, Ac = Ac(x, ~)). Typical experirelatively short time and the relative displacements are mental observations for smooth syenite surfaces are limited. given in Fig. 5. With an appropriate rate-dependent yield resistance relationship, and using an updating approach similar to that described for the displacement YIELD ACCELERATION dependency, it is possible to incorporate a rate-depenA knowledge of the critical acceleration of the sliding dent yield acceleration relationship into the Newmark surfaces is essential for the computation of relative dis- analysis. placements. A rigid, strain-softening, constant post peak displaceThe yield resistance of two surfaces is mobilized only ment shear-deformation model has been adopted for when there is relative displacement between the two Ac(x) (Fig. 6). The residual shear resistance is assumed surfaces. For natural rough discontinuities tested with a to develop at a displacement of 50 mm, a representative constant normal stress, the shear stress-shear deforma- value obtained from consideration of published shear tion curves for quasi-static shearing have the form of test data [7]. The rate-dependency of shear strength Fig. 4. Deformations are essentially linear elastic until (A¢(~)), was taken to occur only beyond a threshold the peak shear stress is attained, after which there is velocity of 10 mm per sec, and to vary logarithmically with velocity beyond this threshold (Fig. 7). The break point velocity of 10 mm per sec is in the mid-range of threshold velocities observed in the experiments de0.8 scribed by Crawford [1,2]. 0.7 R

E e

a

~

Mean

vo,ue Ac /Am

0.6

= o = ~, ~Oot: ~

~

.~ .¢: c u S.~

O5

.i

,~ ;~ ~',

V~< IOmm/sec V > IOmm/sec

~

~ ~ (Ae/Am)(I.O + 0.20 log (V/lO)) 10g (V/IOll

Y= (A c/A m) Y = (Ac/Am}( 1.0 + R log (V/lOll

,

128

Shear velocity, mm/sec

Fig. 5. Variation of dynamic coefficient of friction with shear velocity for flat Syenite surface.

I000 V, Relative velocity, turn/see

Fig. 7. Velocity-dependent shear resistance model.

4

A.M. Crawford and J. H. Curran

-40.00 -30.00 I -20.00

-o_ ,~

~J -,o, oo

"

~

Slope changes as resistance changes relative velocity

with

.~Ground _

o.oo

~ "

li)

0.00

,

I0.00

,J3°°V 4.()0

soo

~,e:~o ',,.,azoo

eoo

I

9',OOv

O0

20.00

30.00

V I

40.00

I Ac/A

m = O. I

Fig. 8. Ground and block velocity responses for El Centro earthquake, asymmetrical rate-dependent resistance.

COMPUTER

PROGRAMME

The programme comprises two steps. The piecewise linear (digital) ground acceleration record is numerically integrated twice using the trapezoidal rule, to determine the ground velocity and ground displacement with time. The velocity and displacement of the sliding mass is determined from the ground velocity and ground displacement-time records, and the yield acceleration, which is a function of the relative velocity and relative displacement between sliding block and ground. It has been assumed that the yield acceleration is constant throughout the integration time step At, and that it equals the value at the end of the previous time step. A flow chart of the programme is given in the Appendix. Corrected Digitized Accelerograms of the California Institute of Technology [4] have been utilized throughout. These records are digitized at 0.02sec intervals. Numerical integrations were performed using time steps of 0.001 sec. The versatility of the programme is illustrated in Fig. 8, in which the velocity time curves of both the ground and sliding block are shown. The input accelerogram is the North-South component of the 1940 El Centro earthquake and the sliding resistance is taken to be rate-dependent. The effect of the rate-dependency is best illustrated by the slope of the sliding block velocity--time curve (Fig. 8). As the relative velocity between ground and sliding mass increases, the resistance and hence the slope of the velocity time curve decreases. With a decrease in relative vclocity, the yield resistance increases and the slope of this curve approaches the original

value when the ground and sliding mass velocitie~ are once again equal. Displacements of the ground, sliding mass and relative displacements between the two surfaces are obtained by integration of the respective velocity-time curves.

PRACTICAL APPLICATION A parametric study was performed to determine the relative importance of displacement- and rate-dependent yield accelerations on the predicted permanent relative displacements in a Newmark type rigid block stability analysis. Three earthquake records, which represent a range of peak accelerations have been used for the analysis. These are, the E1 Centro Earthquake (S00E component) with peak acceleration (Am) of 0.348 g, the Parkfield Earthquake (N65E, -0.489 g) and the Taft Earthquake ($69E, 0.179 g). For each earthquake record, movement has been allowed in one direction (asymmetric resistance) corresponding to a direction opposite to that of the maximum ground acceleration. All results have been compared on the basis of the first nine seconds of the earthquake records. Investigators Ell, 13], have used various methods of presenting the computed relative displacements and relative velocities. Newmark [11] and Franklin & Chang [5] utilizing normalized earthquake records, present the standardized maximum relative displacements plotted against the non-dimensional acceleration At~A,, {the ratio of yield to peak accelerations).

Rock Slope Response to Seismic Loads I0.0

1.0

0.1

,¢

A Porkfield o E( Centr0 a Taft

O.OL

I

q~ ~

A o \

Rectangular pulse

~

o

2 Half sine pulse

\ 0.001

l

L O.2

0.0

I

I 0.4

I

I 0.6

I

A¢/Am Rigid plastic resistance,

I 0.8

I

% I

I.U

P= 1.0

(a) I0.0

I.O

\

0.I ,¢

0.01

0.00~

I

0.0

0.2

0.4

0.6

I.O

0.8

Ac/Am Displacement dependent re.stance, P - O . 6 0

(b) leo

LO

O.OI

i

I

0.001 0.0

I 0.2

I

I

I

0,4

I 0.6

I 0.8

I

I.O

A=/A~

Velocity dependent resistance, R = - 0 4 0

(c) Fig. 9. Variation of dimensionless displacements with At~Am.

Sarma 1-12] and Seed 1-14] use unscaled earthquake records to determine maximum relative displacements (xm), which are plotted in non-dimensional form (4xJI'2A,,g) vs At~A,,, where T is the predominant period of the ground motion obtained from the acceleration spectrum. Similarly peak relative velocities (V,.) are plotted non-dimensionally as (V,./AmgT). For comparison, expressions for the peak relative displacement and peak relative velocity can be found for rectangular, half-sine and triangular pulses of magnitude A,.g and duration T/2. This latter method of presentation of results has been used for this work. Results of the analyses are presented in Figs 9-12. The maximum relative deformations for the extreme cases of no shear strength reduction with displacement or velocity and 40% displacement-, and rate-dependent reductions are plotted in non-dimensional form in Fig. 9. The same data are plotted in unscaled form in Fig. 10. For small ratios of AJA,. (that is, the yield acceleration is small relative to the peak acceleration) Figs 9 and 10(a) show that for the resistance parameters used in this analysis, the change in displacements caused by a reduction of strength due to displacement and rate of displacement effects are essentially the same for small parameter changes. With larger parametric changes for the higher peak acceleration earthquake (Parkfield), relative velocities between the ground and sliding mass are greater, hence the rate effect is more significant. The occurrence of a similar effect for an equal change in rate- and displacement-dependent parameters was entirely fortuitous and can be expected to be different with a different choice of displacement- and ratedependencies of shear strength parameters. Nevertheless, the approximately same effect is hseful for comparing the two phenomena. With higher yield resistances (A~/A,. large, Fig. 9 and 10(c)), the absolute magnitude of displacements are substantially lessened. The effect is similar to that just described, except that for the higher magnitude (Parkfield) earthquake, both rate- and displacement-dependent effects are significant. Peak relative velocities have been plotted as a function of the non-dimensional yield acceleration in Fig. 11 and in non-dimensional form in Fig. 12. A reduction in shear resistance due to strain softening (displacement-dependent) generally has less effect on the peak velocity than the rate-dependency effect. From these observations, it may be deduced that if the change in relative displacements due to rate and displacement effects of the yield acceleration are approximately the same, when compared with the rigid plastic case, then the increased deformation due to the rate effect is a result of increased relative velocity each time a relative displacement occurs. On the other hand, an increased relative displacement due to the displacement effect is due to the summation of a greater number of relative displacement events (Fig. 13).

A. M. Crawford and J. H. Curran

r Displacement dependent a Velocity dependent

A,/A,aO.t

o

- - -

OL 2 20

Change in displacement Equal to change in parameter

, 5

IO

< 0

Dependent

, -5

I

I

- IO

resistance

-30

-20

I- R,P,

parameter,

-40

%

500

(b)

A, /A,=

E

0.3 o Displacement dependent 0 Velocity dependent

q

400

E c’ 5 E B

300

:: 5 “z ._ 5 p

200

E

z

‘Z

s

100

o_____&_JJ__P’Q--4----n_____&_~_~-‘D.‘+

/

0 20

IO

_--_

I

I

I

5

0

-5

Dependent

(4

250

8

El Centro

R _______-----

Taft

4

-30

-40

e-e--__

_8---

1

-10

resistance

-20

parameter,

I -R,e

%

r -

4 4=0.5

q

o Disptacement dependent Velocity dependent

q

Ezoof

_ 0

150 -

Q n

5 .k i

0

_-

IOO-

El Centro o

~===_-~=~~_~~~~~~~~~~_~~r__;_~~_-_~~ 20

IO

Dependent Fig. 10. Variation

I

I

I

1

5

0

-5

-10

resistance

-20

parameter,

of displacements

I-R, P,

with resistance

o

0

I

-30

-40

%

parameters.

Rock Slope Response to Seismic Loads ¢1~ \x

1200

~

o~ --\

\

& o

Rigid plastic, Displocement

~

a

Velocity dependent,

\\

~8o0

\

\

\

\

x

x

)arkfleld \ \

2

P-O.6

R-O.4

xx,,

% \

"6

P=I.O dependent,

\

tk x

E E

Critical accelerotions

\

1000

7

5~, "

""

6O0

E 4o0 E

El Centro~'-,,,,,,..=~'ta"~..L'-',.,

"'.

"~,

:E 2OO ~ " : ' " " "-..-."'~..=_---.~.~

TOf t

~

~

o.I

0.2

0.3

0.4 At/Am

Fig. 11. Variation

"'9

~ ~ . .

" " - " " ~ ""~"k' .':"--..k.----~

~

~

0.5

.

0.6

0.7

0.8

AffA,,.

of velocities with

mic accelerations great enough to cause the shear stress to exceed the dynamic shear resistance of the joint, the slope may not fail, but merely experience irreversible but finite deformations. Thus, for the design of rock

DISCUSSION Newmark [11] demonstrated that for a rock mass resting on an inclined plane which is subjected to seis1.0 05

0.5

0.2

0.2

I

t-

>E 0.05

A Parkfleld 9

~

I Rectangular pulse 2 Half sine pulse

0.02 0.01

>E 0.05

~

o El Centro

0.0

I

I

0.2

I

\ 6

I

0.4

I

I

I

0.6

002 I

[]

I

0.8

i

ooi I

I.O

0,0

i

i

0.2

I

i

i

0.4

Ac/Am

I

0.6

I

I~i

0.8

A¢/Are

Rigid plastic resistance, P-I.O

Displacement

(a)

dependent resistance, P=0.60

(b) I.O 0.5

0.2 I.E

¢1

o

o

:

I

-.\

o.t

>E 0.05

0.02 o.oi

o.o

i

i

0.2

i

i

i

04 Ac/Am

Velocity dependent

i

0.6

resistance,

i

g

0.8

i

R=-0.40

(c) Fig. 12. Variation of dimensionless velocities with

i

i.o

AffA,~.

i

i

1.0

8

A . M . C r a w f o r d a n d J, H. C u r r a n (o] Displacement dependent

Time

(b) Velocity dependent

Time

Fig. 13. Effect of displacement- and velocity-dependent resistances on displacements. slopes a n d u n d e r g r o u n d o p e n i n g s which m a y experience seismic loadings, it is necessary to e s t i m a t e the p o t e n t i a l d i s p l a c e m e n t s d u r i n g the seismic l o a d i n g , then d e t e r m i n e w h e t h e r these d e f o r m a t i o n s m a y h a v e an a d v e r s e effect on the p e r f o r m a n c e o r s e r v i c e a b i l i t y of the structure, This s t u d y has s h o w n t h a t the existence o f r a t e effects on the s h e a r resistance of d i s c o n t i n u i t i e s is an i m p o r t a n t c o n s i d e r a t i o n in t h e design of s t r u c t u r e s o n j o i n t e d r o c k masses. T h e effect of s t r e n g t h r e d u c t i o n d u e to rate effects has been f o u n d to be as e q u a l l y i m p o r t a n t as t h a t of a d i s p l a c e m e n t - d e p e n d e n t resistance. T h e overall effect of r a t e - d e p e n d e n c y is t h a t the d i s p l a c e m e n t s are i n c r e a s e d a p p r o x i m a t e l y in the s a m e p r o p o r tion to the relative c h a n g e in s t r e n g t h p a r a m e t e r , except for large a m p l i t u d e e a r t h q u a k e s , w h e r e the relative velocities b e t w e e n the sliding wedge slope are high a n d as a c o n s e q u e n c e the effect of r a t e - d e p e n d e n c y is also greater. N e v e r t h e l e s s , the r e l a t i v e d i s p l a c e m e n t s a r e still b o u n d e d a n d finite a n d a r e a s o n a b l e e s t i m a t e of the effect of rate- and d i s p l a c e m e n t - d e p e n d e n c y can be o b t a i n e d with the aid of design charts, s i m i l a r to those in Fig. 9. A m o r e d e t a i l e d u n d e r s t a n d i n g of the effect o f rated e p e n d e n t resistance c a n be h a d only when m o r e c o m plete s h e a r strength d a t a for n a t u r a l d i s c o n t i n u i t i e s b e c o m e s available. Acknowledgements-. The assistance of Peng K. Leong and the financial support of NSERC, Canada and the University of Toronto, are acknowledged.

Thesis, Department of Civil Engineering, University ot T~)ronto. Canada (1980). 2. Crawford A. M. & Curran J. H. The influence of shear velocity on the frictional resistance of rock discontinuities, lnt. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 505 515 (1981). 3. Cundall P. A. A computer model for simulating progressive largescale movements in blocky rock systems. Proc. Int. Syrup. on Rock Fracture, ISRM, Nancy (1971). 4. Earthquake Engineering Research Laboratory Reports, 1971-50 and 1972-50. California Institute of Technology, California. 5. Franklin A. G. & Chang F. K. Permanent displacements of earth embankments by Newmark sliding block analysis. Miscellaneous paper S-77-17. U.S. Waterways Experimental Station, Mississippi (1977). 6. Goodman R. E. & Seed H. B. Earthquake-induced displacements in sand embankments. Proc. Am. Soc. cir. Engrs SM2, 125-146 (1966). 7. Goodman R. E. Methods of Geological Engineering in Discontinuous Rocks. West, San Francisco (1976). 8. Hendron A. J, Cording E. T. & Aiyer A. K. Analytical and graphical methods for the analysis of slopes in rock masses. NCG Technical Report No. 36. U.S. Waterways Experimental Station, Mississippi (1971). 9. Maini T., Cundall P., Marti J., Beresford P., Last N. & Asgian M. Computer modelling of jointed rock masses. Technical Report N-78-4, U.S. Army Engineers Waterways Experiment Station, Vicksburg (1978). 10. McKinnon S. D. The influence of blasting on the stability of jointed rock slopes. M.Sc. Thesis, Department of Civil Engineering, University of Toronto, Canada 11979). 11. Newmark N. M. Effects of earthquakes on dams and embankments. Geotechnique 15, 139-160 (1965). 12. Sarma S. K. Seismic stability of earth dams and embankments. Geotechnique 25, 43 76 (1975). 13. Sarma S. K. Response and stability of earth dams during strong earthquakes. Miscellaneous Paper GL-79-13, U.S. Waterways Experimental Station, Mississippi (1979). 14. Seed H. B. Considerations in the earthquake-resistant design of earth and rockfill dams. Geotechnique 29, 215 263 (!979).

APPENDIX Flow Chart of Computer Programme

I

l

SETINITIALVALUESOFGROUNDAND 1 SLIDINGMASSACCELERATIONS,VELOCITIESAND DISPLACEMENTS,T =0.0.

'l I CALCULATEACCELERATION,VELOCITYAND DISPLACEMENTOFGROUNDINTIMET,TO,T +AT, FROMACCELEROGRAM

I

t I

CALCULATE RELATIVEVELOCITYAND DISPLACEMENTOFGROUNDANDSLIDING MASSATTIME,T.

I

CALCULATE RESISTANCEONSLIDING SURFACEASA FUNCTIONOFRELATIVE VELOCITYANDDISPLACEMENT

I

CALCULATE VELOCITYANDDISPLACEMENT OFSLIDINGMASSINTIME,~,T.

t

t

Received 6 February 1981 ; revised 23 July 1981.

REFERENCES 1. Crawford A. M. Rate-dependent behaviour of rock joints. Ph.D.

I

I

I

I;