9-bieniawski_time-dependent behaviour of fractured rock_1970

Ra& Mechanies 2, i23--137 (1970)

@ by Springer-Verlag 1970

Time-Dependent Behaviour of Fractured Rock By

Z. T. B i e n i a w s k i

With l0 Figures

(Reeeived on March 9, 1970)

Summary -- Zusammenfassung -- Résumé

Time-Dependent Behaviour of Fractured Rock. Knowledge of the time-dependent behaviour of fraetured rock is partieularly important in designing rock struetures for long- term stability. This paper is an attempt to determine the time-dependent behaviour of fraetured rock as it might prevail under in-situ eonditions. The following eascs are eonsidered:

(i) Gradually increasing eompression at different but eonstant rates of deformation;

(ii) GraduMly increasing eompression at ehanging rates of deformation;

(iii) Censtant load applieation for various time durations.

It is shewn that fraetured rock does have a long-term strength and while its resistanee •and deformation are initially affeeted by time, stability is reached at a eertain stage. It is thus possible to determine long-term stabi]ity eurves for fraetured rock. This may yield design data for stable rock struetures subjeeted to low rates o.f loading.

Das zeitabhängige Verhalten von gebroehenem Gestein. Für den Entwurf von Fels- konstruktionen, die über lange Zeitr~ume stabil sein sollen, ist es besonders wichtig, das zeitabhängige Verhalten von gebrochenem Gestein zu kennen. In der vorliegenden Arbeit wird versucht, das zeitabhängige Verhalten von gebro&enem Gestein zu ermitteln, wie es unter untertägigen Bedingungen in Erscheinung treten könnte.

Folgende Fälle werden behandelt:

(i) Allmählich steigende Druckbelastung bei verschiedenen, jedoch konstanten Ver- formungsgeschwindigkeiten ;

(ii) Allmählich steigende Druckbelastung bei veränderlichen Verformungsgeschwin- digkeiten ;

(iii) Konstante Lastaufbringung während verschieden langer Zeiträume.

Es wird na&gewiesen, daß gebrochenes Gestein eine Langzeitfestigkeit besitzt und daß Stabilität zu einem gewissen Zeitpunkt erreicht wird, obwohl Widerstand und Verformung anfänglich durch die Zeit beeinflußt werden. Es ist daher möglich, Langzeitstabilitätskurven für gebroehenes Gestein zu bestimmen. Dies kann zur E.rarbeitung von Entwurfsdaten fiir stabile Felskonstrnktionen führen, die geringen Belastungsgeschwindigkeiten ausgesetzt sind.

Le comportement de la rache fraeturée en fonetion du temps. La eonnaissanee du eomportement de la rache fraeturee en fonetion du temps ost partieulierement utile au ealeul de la stabilité prolongée des struetures rocheuses. Aussi a-t-on cherché ä déterminer le eomportement probable, en fonction du temps, d'une rache fraeturée en régime naturel. On a envisagé les éventualités suivantes:

(i) une eompression eroissante avee différentes vitesses de déformation eonstantes;

(ii) une eompression eroissante avec des vitesses de déformation variables; (iii) une eha.rge de grandeur eonstante de durée variable.

Rock Mechanics, Vol. 2/3 9

124 Z.T. B i e n i a w s k i :

On a d6montr6 que la roehe fraeturée po ss~de en effet une stabilité de longue durée. Bien qu'au début sa résistance et sa d6fol~ation soient affectées par le temps -- sa stabilité est atteinte ä une certaine époque. I1 est ainsi possible de déterminer des courbes de stabilité de longue durée des roches fracturées. Ceci pourrait fournir des données pour le calcul de stabilité de structures rocheuses soumises ä des tanx de chargement modérés.

lntroduction

Knowledge of the behaviour of f rae tured rock is of eonsider'able prae t iea l interest . Fo r instanee, at gr'eat depths, f rae tur ing of rock around exeava~ions is usua l ly unavoidable. In other eases, f r aemred rock may eren be desirable; f r acmred rock is less prone to sudd.en and violent fa i lure than strong, solid rock and henee the fo.rmer is less l ike ly to lead to rockbursts . Fur ther , if in the ease of room- and-p i l l a r s toping with panels , y ie ld ing of indiv idual p i l l a r s in a panel is intro- dueed bu t the s tab i l i ty of the panel as a who.le is not endangered, greater extrae~;ion of ore will be aehieved. Thus, provided the behaviour of f rae tured rock ean be unders tood and eontro.lled, its presenee eould reduee hazards and eren resul t in s ignif ieant eeonomie advantages.

One o.f the most useful approaehes to studies of f rae tured rock is the determinat ion of the so ealled "eomplete load-defo.rmation eurves" (B i e n i aw s k i et al, 1969; J a e g e r and C o o k , 1969, p. 167). Diagrammat iea l ly , this is i l lus t ra ted in Fig. 1: Up to the s t rength failure, marked A, the rock remains prae t iea l ly intaet. The rock behaviour beyond poin t A is governed by the st iffness of the loading

PB j A

\

O

4 S

~c

Fig. 1. Speeimen resistanee-deformation di&gram and testing machine load-deformation diagrams for soft (AB) and stiff (AD) machines

X = load P or resistanee R; Y = deformation

Widerstands-Verformungs-Diagramm von Prüfkörper und Last-Verformungs-Diagramm von Prüfmaschinen für ,weiche" (AB) und ,steife" (AD) Maschinen

X = Last P oder Widerstand R; Y = Verformung

Graphique résistanee-déformation d'un éehantillon et graphiques charge-dépla.eement pour des machines d'e.ssai molle (AB) et raide (AD)

X = &arge P ou résista.nee R; Y = déplaeement

maehine, represented by lines AB and AD, respeetively, for a soft and a st if t maehine. Beyond s trength failure, inerease in speeimen deformation, As, results in deereased resistanee, Re. In the ease of the soft (conventional) loading maehine (about 50 MN/m stiffness), the appl ied load PB would be higher than the speeimen resistanee Rc. The speeimen thus fails violent ly at point A -- a behaviour often observed in rock. In a stiff loading maehine (more than 500 MN/m stiffness), the appl ied load PD would be lower than the specimen re.sistanee Re. This s i tuat ion

Time-Dependent Behaviour of Practured Rock 125

is "stable" and with increasing specimen deformation, the resistance of the speeimen gradually decreases along line A C E in Fig. 1.

Sinee the ro& material is in a fracmred state after strength failure, having passed its maximum load bearing capacit.y, a valuable mea.ns of obtaining an insight into the behaviour of fraetured rock is provided by studying the eomplete "load-deformation" or rather "resistanee-defornlation" eurves obtained in stift testing maehines.

The eoncept of eomplete Ioad-deformation eurves originated in studies on concrete in compression conducted by R ü s e h (1960) in Germany. He also obtained the first complete load-deformation curves in tension, again for eoncrete (Rüseh , 1963). The desirability of obtaining information on the behaviour of rock after strength failure was first pointed out by C o o k (1965), followed up by F a i r h u r s t and C o o k (1965). The first complete load-deformation cuIa~es for rock were obtained by B i e n i a w s k i (1966) and by C o o k and H o j e m (1966). Since then rauch attention has been devoted in rock mechanies to this subject and besides extensive laboratory tests ( B i e n i a w s k i , 1967; W a w e r s i k , 1968; B i e n i a w s k i , 1969) the first underground tests have now also been undertaken ( B i e n i a w s k i , 1970).

It now only remains to establish the faetors whieh influence the behaviour of fracmred rock.

Factors of Inf luence on the Behav iour of Fractured Rock

The behaviour of fraetured rock is charaeterized by the resistance-deformation curve after strength failure, in the region where the curve has a negative slope. As diseussed above (see Fig. 1), the specimen will fail violently if the negative slope of the rock resistance-deformation cui~:e is steeper than the (negative) slope rcpresenting the stiffness of the loading machine. Knowledge of the faetors which determine the solpe of the resistance-deformation eurve after strength failure is therefore of part icular importance if violent failure of fractured rock is to be avoided.

Results of reeent studies have shown ( B i e n i a w s k i , 1969) that increasing confining pressure tends to flatten the negative slope of the rock. Thus, a fraetured rock seeimen is more stable in triaxial compression than in uniaxial compression. In addition, it was found then that a change in specimen shape, for example, a higher width to height ratio, also results in a flattening of the negative slope. A reetangular (oblong) speeimen is more stable than a eubieal specimen having the same cross-section.

It was also reeently shown ( B i e n i a w s k i , 1970) that rock materials having a higher uniaxial compressive strength er a higher modulus of elasticity, or both, have steeper negative slopes than materials with lower strength or modulus values. Consequently "hard" high-strength rock will be more prone to violent failure than ' soft", low-strength rock.

The factor whi& has not as yet been investigated is the influenee of time on the behaviour of fractured rock. This faetor is of particular interest as it is associated with the !ong-term stability of fractured rock. It is the purpose of this paper to contribute to the knowledge of this aspeet.

While no information on time-dependent behaviour of fractured rock eould be found on the literature, some data of interest to the present study have been reported for eoncrete by R ü s e h (1960). In Fig. 2, after R ü s c h , stress-strain relationships are given for concrete subjeeted to various constant strain rates. It will be noted from this figure that, with increasing rate of straining, the uniaxial compressive strength as well as the modulus of elasticity (in the unfraemred stare) increase. It is interesting to reeall that identical behaviour is also typical of rock ( S e r d e n g e c t i and B o o z e r , ]961).

9*

Amir

Highlight

Amir

Highlight


The most important observatio.n to be made from Fig. 2 is that the negative slope of the st.ress-strain curves after strength failure gradual ly becomes flatter as the rate of s t ra ining deereases. Should a similar trend be also observed for rock, this could be of considerable practical signifieance, as will be diseussed Iater.

25

20 ~ s e c I YEAR

15 ~ I MONTH see

X z 10 1670xi ~ IDAY

1 0 0 0 0 x 10-

5 ~ 1HOUR 10 MINUTES

i o ~ ~- ; ~, ~ 6

Y

Fig. 2. Stress-strain relationships fo,r eoncrete for various eonstant strain rates (a,f~er R ü s c h , 1960)

X = uniaxial eompressive stress (MN/m2); Y = axial strain (10-~); Z = loading dara.tion needed to attain 6" 10 .3 strain with the strain rates indieat.ed

Spannungs-Dehnungs-Diagramm für Beton unter verschiedenen konstanten Dehnungs- ges.ehwindigkeiten (nach R ü s e h, 1960)

X = einaehsige Druekspannung (MN/rn2); Y = axiale Dehnung (10-a); Z = erfo.rderliehe Belastungsdauer zur Erreiehung einer Dehnung vo.n 6 • 10 3 bei den angegebenen Dehnungs-

geschwindigkeiten; year = Jahr; mont.h = Monat; day = Tag; hour = Stunde

Courbes eontrainte défo.rmation d'un béton pour différentes vitesses de déformation eonstantes (d'après R ü s e h, 1960)

X = eontrainte de compressio,n monoaxiale (MN/m2); Y = déformation axiale (10-~); Z = durée de &arge nécessaire [out atteindre une déformation de 6 ' 10 -a aux vitesses

indiquées; yea.r = an; month = mois; day = jour; hcur = heure

An investigation was therefore init iated the purpose of whieh was to derer- mine the load-deformation behaviour of fraetured rock at various strain rates and at eonstant load for var'ious durations. Thre,e eases were eonsidered, believed to be applieable to in-si tu behaviour of rock struetures:

(i) gradual ly inereasing eompression at different bu t eonstant s train rates.

(ii) gradual ly inereasing eompression at ehanging strain rates.

(iii) eonstant load. applieation for various time durations.

Experimental Procedures

The equipment omd teehniques used in this study have been deseribed in detail eisewhere ( B i e n i a w s k i et al, 1969). Briefly, the equipment eonsisted of a speeially designed eompressive loading ma~hine with variable longitudinal stift- ness o.f up to l i0O MN/m. The axial load on the speeimen as weil as its axial deformation were automatieally plotted on a high sensitivity X - Y reeorder. No measurement,s o.f lateral deformation were mode. All tests were eonducted in uniaxial eompression.

The rock material used was a f ine-grained sandstone. The speeimens, after being aecurately prepared, were plaeed for over one month under eonditions of eonstant temperatur'e (20°C) and eonstant humidi ty (50 °/0). Some 30 speeimens

Time-Dependent Behaviour of Fractured Rock 127

were test ed, the specimens being cylindrical in shape and having a diameter of 21.6 mm and a height of 10.8 mm, thus yielding a d i ane te r to heigh~ ratio of 2. This geometry was ehosen sinee mine pil lars in South Afriea have generally width to height ratio of 2 and are square in plan.

firadually lncreasing Compression at Constant Strain Rates

The condition of gradual ly increasing compression at a constant s train rate is orten encountered in mining such as, for example, in mine pil lars dur ing the systematic extraction of ore in the panel in which they are sit.uated.

In order to dctermine the influence of s train rate on the eomplete load- deformation curve of ro& subjected to gradual ly inereasing eompressive deformation, a series of tests - similar to those eondueted by R ü s e h (1960) for eoncrete -- was undertaken.

The experimental procedure was as follows. Pre l iminary tests ware first eondueted to determine the average maximum speeimen deformation at rupture. This deformation was found to be 0.15 mm. Then speeimens were subjeeted to graduatly increasing eompressive deformation at c o n s t a n t strain rates such that 0.15 mm deformation would be attained after 7 minutes, 2 hours a l d 9 hours respeetively. This eorresponded to s t ra in rates of 33 • 10-6/see, 1.94" 10-6/see and 0.43 - I0 -6 per sec respeetively or to rates of deformation of 2 2 • 10-3 mm/min, 1.25 • l0 -'~ mm per min and 0.28 mm/min, respectively. In this manner, data for rock eomparable to those obtained for eonerete by R ü s e h (see Fig. 2) -- were obtained.

The results are given in Fig. 3, in which eaeh eurve, for a specifie s train rate, represents an average of a number of tests.

50

3C / ~/0-«

X {~,«

2G

005 0'10 0"15

Y

Fig. 3. hffluence of rate of strain on the complete load-deformation curves of sandstone in uniaxiM eompression

X = load (kN); Y ~ deformation (min)

Einfluß der Dehnungsgeschwindigkeit auf die vollständigen Last-Verformungskennlinien von Sandstein bei einaehsigem Druck

X = La.st (kN); Y = Verformung (mm)

Effet de la vitesse de déformation sur les eourbes effort-déformation eomp]ètes pour un grès en eompression monoaxiale

X = &arge (kN); Y = déplaeement (min)


It will be seen from Fig. 3 that a higher strain rate results in a higher modulus of elastieity (steeper positive slope before strength failure) and in a higher strength failure stress of the speeimens. The mo,st important obsmwation is, however, that the lower the strain rate (longer durat ion of loading), the flatter is the slope o.f the load-defo,rmation eurve after strength failure. This is a very signifieant f inding as it indieates that the likelihood of violent failure of fraetured rock, if the strain

I¢o

.o~iI~_ _ - Il -~ --Il; ~,ooo 80 80 X I eo so

/+20 /*0

'3 10 -10 10 -a 10-6 i0 ~~' iO -~ IO s IÕ a

Y

Xz " ~tl fi'i °'-~'"-~'-'"-~ ' " ' '~ : I ITX-UF ~ t t - ~ - q ~ . ,o iÕ «o 10 -g lO-e I0-7 I0 -s 10 -] fÕ~

Y

~~I ;;4~ b~l~ ':i~-~J ~ ~~~ ×3:1- - -] ]Il] ] Il i si 10"~o I tT 9 10-a i0-~ 10 -r, l f f 5 1O -~

Y

Fig. 4. Inflnenee of strain rate oll strength f~ilure, strain at strength failure and modulus of clasticity for sandstone in uniaxial eompression

X 1 = strength failure (MN/m2); X 2 = strain at strength failure (10-3); X 3 = modulus of elastieity (GN/m2); Y = strain per seeond

Einfluß der Dehnungsgesehwindigkeit auf Bruehfestigkeit, Bruehdehnung und Elastizitäts- modul von Sandstein bei einaehsigem Dru&

X~ -- Bruehfestigkeit (NM/m2); X2 = Bruehdehnung (10 a); X~ = Elastizitätsmodul (GN/m'~); Y = Dehnung pro Sekunde

Effet de la. vitesse de déformation sur la résistanee ä la rupture, la déformation ä la rupture et le module d'élasfieité pour un grès en eompression monoaxiale

X 1 = résistanee ä la rupture (MN/m2); X2 = défo.rmation ä la rupture (10-3); X~ = module d'élastieité (GN/m2); Y = déformation par seeonde

increases -- as found in yielding mine pil lars left s tanding for a long period of time -- will be less the slower the rate of strain. Fraetured ro& subjeeted to slow strain rates is, therefore, more stable than that subjeeted to higher ones. This can also now explain why rock subjeet to a sudden and violent energy release, such as in a rockburst, fails uneontrol lably -- its stabil i ty is diminished due to its steep load-deformation eharaet,eristies after strength failure, at high rates of

strain. The results given in Fig. 3 are applieable to rates of strain down to 0.43 • 10 -»

per seeond (0.15 mm defonnat ion after 9 hours). This is, of course, not eomparable to in situ eonditions where rauch smaller rates of st.rain are eommon sinee the pil lars may be subjeeted to prolonged loading with inereasing strain over

many years.


It would, therefore, be d esirable to determine such da ta for rock for very slow s t ra in rates (in this ease 0.15 mm deformat ion in one year) , as was achieved for concrete by R ü s e h (see Fig. 2). This would unfor tuna te ly be a very time eon- suming process and would also present considerable experimental diffieultes. In fact, the author ' s appara tus is not equipped to dem with such small straän rates.

Due to the importance of the matter , however, ext rapola t ion of the results was attempted. I t was thought that if definite t rends eould be es tabl ished eon- cerning the influenee of s t ra in rate on s t rength failure, modulus of elasfieity, s t ra in at fai lure and the negative slope of the load-deformat ion eurve after s trength failure, then the relevant da ta for slower s t ra in rates eould be predieted.

Ext rapola t ion of the data on the f i rs t three var iables , as obta ined from Fig. 3, i . e . s trength fai lures stress, s t rength fa i lure s t ra in and modulus of elast iei ty, is shown in Fig. 4 (experimental da ta -- solid lines, extr~polated da ta -- b roken lines). This f igure shows a definite t rend in these quanti t ies , thus enabl ing their determinat ion for very small rates of stra/n.

To reeognise a t rend that would permi t ext rapola t ion of the negative slope after s t rength fa,ilure to very small s t ra in rates, the results on sandstone are eompared, in Fig. 5, with those obtained for eonerete b y R ü s e h (1960), who tested down

2 t 1010- u 10-9 1~ s 1~ ~ ~o ~ I f f '1 t0 .6

Y

tO s

10'~ ..

X

1°3'~" ~ -

Fig. 5. Effeet of strain rate on stress-strain slope after strength failure for sandstone and eonerete in uniaxial eompression

X = slope of stress-strain eurve after strength failure (MN/m2); Y = strain per seeond

Einfluß der Deb~nungsgeschwindigkeit auf die Neigung der Spannungs-Dehnungs-Kurve nach erfolgtem Bruch für Sandstein und Beton bei einachsigem Druck

X = Neigung der Spannungs-Dehnungs-Kurve nach Überschreiten der Bruchfestigkeit (MN/m2); Y = Verformung pro Sekunde

Effet de la vitesse de déformation sur la pente de la eourbe effort-déformation après la rupture pour un grès et un béton en eompression monoaxiale

X = pente de la eourbe effort-déformation après la rupture (MN/me); Y = déformation par seeonde

to a st.rain rate of 0.19 - 1O-9/sec, i . e . loading durat ion of one year, as shown in Fig. 2. I t will be seen from Fig. 5 that it is reasonable to assume that the da ta for sandstone (experimental - - sol id line, exrapolated -- broken line) would give


a similar trend as that for eonerete, par t icular ly in view of the phenomenological simil~rity between eonerete and rock.

Based upon the data presented in Figs. 4 and õ, an attempt was made to eon- struet the curves depieting the influence of s train rate on the eomplete stress- s train for rock. The results are given in Fig. 6. The data are presented in terms

125

0.127 X lO}sec

10 /sec

-9 ~5~×10 /sec

33 300 xlO /sec

e(lrs

lye¢lr

Imonth

Iday

thour

I0 min

5 10 15 20

Y

Fig. 6. St ress-strain rel~tionships for sa~dstone for various constant strain rates

X = uniaxial eompressive stress (MN/me); Y = axial strain (10-s); Z = duration of loading needed to attain 20" 10 -3 strain with the strain rates indieated

Spannungs-Dehnungs-Diagramme fiir Sandstein bei verschiedenen konstanten Dehnungs- geschwindigkeiten

X = einaehsige Druckspannung (MN/m 2); Y = axiMe Dehnung (10-s); Z = erforderliche Belastungsdauer zur Erreiehung einer Dehnung von 20 ' 10 s bei den angegebenen Deh-

nungen; year = Jahr; month = Monat; day = Tag; hour = Stunde

Courbes eontr~inte-déformation d'un grès pour diverses vitesses de déformation eonsta~ntes

X = eontrainte de eompression monoaxiale (MN/m2); Y = déformation axiale (10-~); Z = durée de ehargement nécessaire pour atteindre nne déformation de 20-10 -3 aux

vitesses de déformation indiquées

of stress versus s t ra in to facilitate comparison with Fig. 2 for conerete. In Tab. 1, both stress-strain and load-deformation data are summarized for eonvenience.

It will be seen from Fig. 6 that a five-year loading durat ion at the constant s t ra in rate of 0 . 1 2 7 ' 10-9/sec, yields a much flatter slope of the eurve after strength failure thax4 that for l0 minutes loading durat ion at the constant s train rate of 33 300 • 10-9/sec. In faet, from Tab. 1, there is a change from 509.5 MN/m for 33300" 10-9/sec to 42.1 MN/m for 0 .127 -10 -9 / sec -- this is over 12-fold improvement. It is also interesting to note from the last eolumn of Tab. 1 that there is a gradual increase in the radio of the positive to negative slope with inereasing rate of strain. It is believed that these data will be of praetieal signi- fieaaaee for determining the long-telzn stabil i ty of structures involving fraetured rock,


Table 1. l n f I u e n e e of s t r a i n r a t e on s t r e n g t h a n d d e f o r m a t i o n e h a r a e t e r i s t i e s o f s a n d s t o n e

(extrapolated from tests with strain rates down to 0.43 • 10-6/see)

© •

10 minutes 1 hour 1 day 1 month 1 year 5 years

Strain rate

10 ' / s ee

33 300.00 5 560.00

232.00 7.72 0.64 0.13

Stress-strain behavionr

IN/m ~

115.1 106.4 101.5 96.7 94.8 92.1

r D

10 -3 !GN/m 2

5.4 ]29.1 6.l 26.7 7.3 22.8 8.6 19.0 9.6 15.9

10.2 13.5

Z

GN/m

15.1 11.7 8.4 4.2 2.1 1.3

Load-deformation behaviom

~ ~ ~®

kN mm MN/m

42.1 0.058 981.8 38.9 0.066 900.9 37.1 0.079 769.3 35.4 0.093 641.1 34.7 0.104 536.5 33.6 0.111 455.1

MN/m

509.5 494.7 283.4 141.7 70.9 42.1

O ¢g

~s

1.92 2.28 2.71 4.52 7.57

10.83

Gradually Increasing Compression at Changing Strain Rates

In mining, the s toping operat ions may, in the case of pi l lars , eause redis t r ibu- t ion of the load over the p i l la rs such that these may be subjeeted to inereasing or decreasing rates of s train. I t is, therefore, of prac t ica l interest to examine the behaviour of rock under g radua l ly inereasing compressive deformation at vary- ing s t ra in rates.

To explore the reaction of f rac tured t o & to a sudden ehange in s t ra in rate, two series of tests were car r ied out:

(i) Specimens were loaded at the constant s t ra in rate of 33 • 10-6/see beyond s trength failure, up to an a rb i t r a r i l y chosen deformat ion of 0.08 mm. Then the s t ra in rate was ehanged to 0 . 4 3 ' 10-6/see and kept eonstant thereafter . Load and deformat ion were reeorded throughout the test.

(it) Speeimens were loaded at the eonstant s t ra in rate of 0.43 • 10- 6/see beyond s t rength failure, up to an a rb i t r a r i l y cho,sen deformat ion of 0.12 mm. Then the s t ra in rate was changed to 33" 10-6/see and kept constant thereafter. Load and deformat ion were reeorded throughout the test.

The eomplete load-deformat ion eurves depiet ing the results of this s tudy are given in Fig. 7, from which the follo wing observat ions are made:

Comparing the ease of ehanging the s t ra in rate from 3 3 ' 10-6/see to 0.43" 10-6/see (full eurve A) with the ease of keeping the s t ra in rate eonstant at 33" 10-6/see throughout the test (dotted eurve A), shows that a sudden decrease in the s t ra in rate results in an ini t ia l steepening of the negative slope of the load deformat ion eurve, followed by a f la t tening of this slope. The reason for the ini t ia l steepening is that for a deerease in the rate of s train, the resistmaee of the speeimen also deereases (see Fig. 6). Subsequent f lat tening in the slope is due to the faet that as the deereased s t ra in rate takes effeet, the eonfigurat ion of the eurve is typical for this rate of s train. As Fig. 7 shows, the eurvatures of the full and dotted eurves for 0.43" 10-6/see s t ra in rate are the same.

Comparing the ease of changing the s t ra in rate from 0 . 4 3 - 1 0 - 6 / s e e to 3 3 ' 10-6/see (full eurve B) with the case of keeping the s t ra in rate cons~aa~t at


0 . 4 3 - 1 0 - 6 / s e c throughout the test (dotted curve B), an ini t ia l inerease in the specimen resistanee is apparent , followed immediate ly by a steep negative slope. This slope is apparen t ly the same as that of the 33" 10-6/see s t ra in rate eurve. The curvatures of the full and dotted eurves for 33 • l0 6/see s t ra in rate are the same.

I t may be commented from the above observat ions that a ehange in the rate of s t ra in produees a two-fold effeet on f rae tured r e&:

(a) an ini t ia l deerease er inerease in the resistanee, eorresponding to a deerease er increase, respeetively, in the s t ra in rate;

(b) subsequent f la t tening er steepening of the negative slope, eorresponding to a deerease er increase, respeetively, in the s t ra in rate.

The f i rs t effeet is due to the re la t ionship between the resistanee and the s t ra in rate (see Fig. 6 -- resis tanee decreases with deereasing s t ra in rate) while the seeond effect refleets the re la t ionship given in Fig. 3, i . e . the eurvature of the curve is eharaeterist ie for a given s t ra in rate. In this ease, eonsidering Fig. 7,

3{

x

2C

lC

50

CURVE A

~;CURVE B I

,---%~ \ 'x~33 x 10-~/

0 i I 0.05 0.10 0.15 0-20

Y

Fig. 7. Effect of ehanging stra,in rate on the behaviour of fraetured rock X = load (kN); Y = defonnation (mm)

Einfluß der verä.ndeHi&en Dehnungsgesehwindigkeit auf das Verhalten von gebroehenem Gestein

X = Last (kN); Y = Verformung (mm)

Effet d'un changement de vitesse de déformation sur le eomportement de la roehe fraemrée X = charge (kN); Y = déplaeement (mm)

if the dotted eurve A of 33 • 10-6/sec s t ra in rate is shifted to the r ight and super- imposed and the solid curve B of 33 • 10-6/sec, then be,th eurves will eoineide since the curvatures of these eurves are the same. In pract ice be th effeets (a) and (b) above, y ie ld a g radua l t rans i t ion from the resistance change to the curvature change, due to the fact that the s t ra in rate cannot be changed absolute ly suddenly. Should the rate of s t ra in be changed real ly suddenly from higher to lower, the resistance should immedia te ly drop to a vMue corresponding to the lower s t ra in rate and then the eurve should immediate ly follow the eurvature typical of the lower rate of s t rain.


Another observation whieh ean be made from Fig. 7 is that as a result of a ehanged strain rate from 33 • 10-6/sec to 0.43" 10-6/see, at speeimen deformation of 0.15 mm the resistance of the specimen is inereased from A~ to A 2 while, eon- versely, an incre~se in the strain rate from 0.43 • 10-6/sec to 33 • 10-6/sec yields, for the same speeimen deformation of 0.15 mm, a deerease in the resistanee from B s to B 1. This phenomenon is, of course, due to the effect (b) above. This eurvature effeet thus yields the desirable effeet of a decreased rate of strain, that is, when the eurvature influenee takes effeet in such a ease, not only the stability of fraetured rock is inereased (flatter negative slope) but also the resistance is higher. On the o.ther haad, inereasing the strain rate results in an eventual decrease in stability (steeper negative slope) as well as in a deereased resistanee.

The above findings indieate the undesirable effeet that an increased rate of stra.in has on the behaviour of fraetured rock. In praetieal design of rock structures involving fraetured rock, care should be taken to ensure that, if a ehange in strain rate is to take place, it should be decreased rather than increased.

Constant Loading for Various Durations

The eondition of constant loading (long-term loading at co nstant stress level) is orten found in praetiee, part ieularly in mining when, after eompletion of ore extraetion, rock struetures such as mine pillars are left for an indefinitely long time to support the overburden. If the pillars are of a yielding type, i .e. in a fraetured state, their behaviour under eonstant load, over a long period of time, is of part ieular interest.

In order to determine this behaviour, tests were eondueted to establish the long-term stability of fraetured rock. The experimentaI proeedure was as follows.

Rock speeimens were loaded in the stift maehine beyond their strength failure. At an arbi t rary load level after strength failure, the applied load was kept eonstant over some period of time. Since, however, the ro& was in a f raemred state at that stage, it could be expeeted that due to fraeture propagation, the resistaaee of the speeimen would decrease and its deformation increase, even if the applied load was maintained constant. The speeimen resistanee was measured by a load eell plaeed in series with the specimen while the applied load was indieated by the stift maehine pressure gauge. Automatie plotting of the speeimen resistance and its deformation was affected by an X-Y recorder. The apparatus used is fully deseribed elsewhere ( B i e n i a w s k i et al, 1969).

In Fig. 8 typieal results of this s tudy are shown. At point A in this figure, further external Ioading was stopped and the applied load was maintained eonstant. The speeimen eontinued to deform, however, as indicated by line AB. This decrease in speeimen resistance took place during 30 minutes, the rate of strain deerease gradually dropping as shown in the insert of Fig. 8, until at point B no further ehange in speeimen resistance and deformation was observed. Observation was continued for one hour, during which time no ehange oceurred. The applied load was then inereased and the specimen resistanee and deformation followed the eurve BCD, resmning its normal eharaeteristic at point C. The faet that line AB does not eoindice with line A C is due to the specimen resistance deereasing due to the decreasing rate of strain, as indicated in the insert in Fig. 8.

A signifieant finding of this study was that specimen resistance and deformation remained eonstant at point B. Thus, the fraetured rock attained stabitity and beeame free from any fraeture propagation and time effeets.

The question now arises as to whether point B is one of a locus of points forming a long-term stability curve for fraet.ured rock. If such a eurve does exist, the further quesüon is, whether or not it depends upon the rate of strain.


In order to answer these questions, further tests were conducted the results of whieh are given in Fig. 9 and 10.

In Fig. 9 it is shown that a eurve does exist representing a locus of points at whieh fractured ro& does not ehange its resistance and deformation with time. The port ion of the curve after strength failure was found from a series of tests, as per Fig. 8, all "points B as per Fig. 8" being joined. For elarity of presentation

SO

~I &o~o ,~ ~.o ~o

0 0 05 0'10 0 ' 15 0'20

Y

Fig. 8. Dcformational hehaviour of fractured rock under constant ]oad in uniaxia,] compression

X = load or resistance (kN); Y = deformation (mm); Z = changes in rate of strain between points A and B; S = rate of strain (10-6/sec); T = time (minutes)

Verformungsverhalten von gebrochenem Gestein unter konstanter Last bei einachsigem Druck

X = Last (L) oder Widerstand (R) (kN); Y = Verformung (mm); L = R = Last = Widerstand; Z = Änderung der Dehnungsgeschwindigkeit zwischen den Punkten A und B;

S = Dehnungsgeschwindigkeit (10-8/sek); T = Zeit (Minuten)

Comportcment de la roche fracturée sous une charge constante en compression monoaxiMe

X = charge ou résistance; Y = déplacement (mm); Z = changement de la vitesse de déformation entre A et B; S = vitesse de déformation (10-8/sec); T = temps (minutes)

only four of these "points B" are shown in Fig. 9. The long term strength of intact rock, indicated in Fig. 9, was obtained by subjecting the specimens to long term loading using the techniques and apparatus described elsewhere ( B i e n i a w s k i , 1967, p. 435).

The long-term stabil i ty curve (lower curve in Fig. 9) for fractured rock was obtained for specimens subjected to s tandard laboratory rate of loading (0.7 MN/m 2 per sec = 100 lbf / in u per sec) corresponding to a rate of deformation of 0.02 mm per min or a s train rate of 31 • 10-6/sec.

It is interesting to note from Fig. 9 that the negative slope of the long-terin stabil i ty curve is somewhat steeper than that of the original curve: 625 MN/m as compared with 510 MN/m. This difference, 22.5 °/0, could perhaps be at t r ibuted to an experimental error. It is thus possible that the long-term stabil i ty curve has the same negative slope as the corresponding eurve for a given rate of defor-


mation. On the other hand, there eould be a s l ight differenee due to the effeet of g radua l ly deereasing s t ra in rate (from poin t A to B in the inser t in Fig. 8).

I t may fur ther be noted that the broken lines in Fig. 9 are para l le l and their slope of 1110 MN/m is elose to the stiffness of the test ing maehine used, namely, l I00 MN/m. Consequently, under a eonstant appl ied load, f rae tured rock eontinues

5O

~0

S L ~

30

X

20

10

F I

0'04 008 0'}2 0'I«

Y

Fig. 9. Long-term stability of fractured rock subjeeted to a rate of deformation of 0.02 mm per minute

X : load (kN); Y = deformation (mm); SL = long term strength of solid to&

Langzeitstabilität von gebroehenem Gestein bei einer Verformungsgeschwindigkeit von 0,02 mm pro. Minute

X ~ Last (kN); Y = Verformung (mm); SL = Langzeitfestigkeit von festem Gestein

Stabilité ä long terme de la roehe fraeturée soumise ä une vitesse de déplacement de 0,02 mm par minute

X = &arge (kN); Y : déplacement (mm); S 1 = résistance de longue durée de la roehe intaete

to defonn and decreases in resistance along the charaeterist ic of the test ing machine. This f inding gives a convenient means of cheeking the stiffness of the loading maehine.

The results given in Fig. 9 thus provide long-term data on which the design of rock s t ruemres ean be based. The value of these results lies in their app l ieab i l i ty to the predie t ion of the behaviour of rock s t ruetures after a very long per iod of time.

The data, in Fig. 9 app ly only to the rate of deformat ion of 0.02 mm/min. Does it also app ly to other rates of deformation, that is, is i t appl ieable to the t o & type as such?

Fig. 10 indieates that i t does not app ly to other rates of deformat ion and that eaeh rate of deformat ion has its own eharaeterist ie long-term s tab i l i ty eurve. This is elear from Fig. 10 where the long-term stabilit ,y points (broken lines) for s t ra in rates of 33" 10-6/see and 0.43" 10-6/see ean be seen not to eoineide.

I t may thus be stäted that under eonstant appl ied load f rae tured rock even- tua l ly reaehes s tab i l i ty but is affeeted by the previous s t ra in ing his tory. The long-


term behaviour of f rac tured ro& may be es tabl ished from a systematic determina- t ion of pos t - fa i lure s t ress -s t ra in curves of rock, at various s t ra in rates.

50

4 0

30

2C

\',, \ X ' , ~ ~'0-~ ....

I - V ",1

0 '06 0'10 0'1 "t 0'18

Y

Fig. 10. Inf]uence of strain rate o]1 long-term stability of fractured rock

X = load (kN); Y = deformation (mm); Z = long-term stability

Einfluß der Dehnungsgesehwindigkeit auf die Langzeitstabilität von gebro,ehenem Gestein

X = Last (kN); Y = Verformung (mm); Z = Langzeitstabilitä~

Effet de la vitesse de déformation sur la stabilité de longue durée de la rache fraeturée

X ~ &arge (kN); Y = déplacement (mm); Z = stabilité de longue durée

Conclusions

(1) Under g radua l ly increasing compres:sive deformation at constant s t ra in rates, decreases in s t ra in rate resul t in a f la t tening of the slope of Ioad-deformation curve of f rac tured rock after s t rength failure, th~t is, in an increased s tab i l i ty after s t rength failure.

(2) Conversely, increas ing stra.in rate results in decreased s tab i l i ty of fractured rock manifes ted by a steepening of the negative slope of the load-deformat ion curve after s t rength failure.

(3) Al though resistance and deformat ion of f rac tured rock are t ime-dependent , af ter the appl ica t ion of a constant load for a re la t ively shor t per iod of time, s tabi- l i ty is achieved and no fur ther reduct ion in the resist~nce o r increase in deformation, is apparent . F rom this phenomenon it is poss ible to construct a resul tant curve of long-term s tab i l i ty of f rac tured rock.

(4) The long-term s tab i l i ty curve for f rac tured rock is dependent on the previous loading h i s tory of the rock, i . e . the rate of deformat ion previously applied.

Acknowledgements The ~uthor is grateful to Messrs. U. W. V o g l e r and S. J. C o e t z e r for their

assis tance in the experimental tests and to Dr. H. G. D e n k h a u s for his con- structive cr i t ic ism of the manuscr ipt .

Time-Dependent Behaviour of Fraetured I[ock i37

B i e n i a w s k i , Z. ind. Res. S. Afr., MEG

B i e n i a w s k i , Z. Sei. 4, 407--430, 1967.

B i e n i a w s k i , Z. rock. Int. J. Rock Meeh.

B i e n i a w s k i , Z.

References

T.: Meehanism of rock fracture in compression. Rep. Counc. scient. 459, 1966.

T.: Mechanism of brittle fracture of rock. Int. J. Rock Mech. Min.

T., II. G. D e n k h a u s, and U. W. V o g l e r : Failure of fraetured Min. Sei. 6, 323--341, 1969.

T.: Deformational behaviour of fraetured rock under multiaxial eompression. Proe. Int. Conf. Strueture, Solid Meehanies and Engineering Design, John Wiley & Son, London, 55/1--55/10, 1969.

B i e n i a w s k i, Z. T.: Load-deformation behaviour of eoal after f~ilure. Proe. 2nd Congress Int. Soe. Rock Meeh., Belgrade, 1, 467--473, 1970.

C o o k , N. G. W.: Failure of rock. Int. J. Rock Meeh. Min. Sei. 2, 289--403, 1965.

C o o k , N. G. W., and J. P. M. H o j e m : A rigid 50-ton compression and tension testing maehine. S. Afr. Me&. Engr. 16, 89--92, 1966.

F a i r h u r s t, C., and N. G. W. C o o k : The phenomenon of ro& splitting parallel to a surfaee under eompressive stress. Chamber of Mines of South Afriea Researeh Report No. 65/65, 1965. Also in: Proe. 1st CongressInt. Soe. Ro,ck Me&., Lisbon, 1, 687--691, I966.

J a e g e r , J. C., and N. G. W. C o o k : Fundamentals of rock meehanies, Methuen & Co., London 1969.

R ü s e h , H.: Researehes towards a general flexural theory for struetural eonerete. Ameriean Conerete Institute Proeeedings 57, 1--28, 1960.

R ü s eh, H.: Verformungseigensehaften von Beton unter zentrischen Zugspannungen. Voruntersuehungen, München, Bericht Nr. 44, 1963.

S e r d en g e e t l , S., and G. D. B o o z e r : The effeets of strain rate and temperature on the behaviour of rock subjeeted to triaxial eompression. Proe. Fourth Symposium on Rock Meehanies, Soeiety of Mining Engineers, New York, 83--97, 1961.

W a w e r s i k, W.: Detailed studies of rock fraeture in eompression. Thesis, University of Minnesota, Minneapolis 1968.

Key Words: effeet, strain rate, inereasing Ioad, eonstant load, fraetured rock, sandstone, conerete, uniaxial eompression, stift testing maehine, time dependeney, long-term stability, eomplete resistanee-deformation eurve, eomplete stress-strain eurve, experimentM data.

Address of the author: Dr. Z. T. B i e n i a w s k i , Head, Rock Meehanies Division, Couneil for Seientifie and Industrial Resea.rch, P. O. Box 395, Pretoria, South Afriea.

9-bieniawski_time-dependent behaviour of fractured rock_1970

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