INVESTIGATION OF CREEP OF ROCK FORMATIONS

BY THE METHOD OF TRIAL STATIC LOADING

V. L. Kubetskii and S. B. Ukhov UDC 624.191.1: 624.131.25.001.5~

A dependable and economic planning of the linings of hydraulic pressure tunnels requires a maximum ut i l i za - tion of the mechanical properties of the surrounding formation. The widespread bel ief of purely e las t ic behavior of a rock formation does not correspond to fact because the major i ty are capable of deformation to some extent with t ime when pressure is applied to them, i .e . , they have the property of creep. When the lining of a pressure tunnel acts together with the surrounding formation, creep may have a substantial effect on the stressed state of the lining. I t is therefore necessary to investigate the theological properties of rock formations [1].

Creep is usuaUy studied on specimens and the results of the investigation are applied to the massif [2]; it is assumed that the specimens represent sufficiently accurately the behavior of the rock massif. However, the deforma - tion properties of fissured heterogeneous rock massifs which serve as foundations of hydraulic structures can be deter- mined only by methods of large-scale field investigations. This paper discusses a method of determining the regula- rities of creep of fissured rock formations,* using as an example investigations with tr ial stat ic loading of stratified sandstone which is one of the investigated types at the site of construction of a dam in the Kirgiz SSR.

The course of the experiment is shown in Fig. 1.

The pressure at each stage was kept constant until the deformation of the surface of the rock was s tabi l ized; after that, the pressure was reduced to zero and the reversal of the creep phenomena was investigated.

The dependence of condi t ional-momentary deformations on the average pressure (Fig. 2) was obtained by plotting deformations corresponding to each step. Since the loading in each case was done from zero, the abscissa of the graph corresponds to full condi t ional-momentary deformation of the surface of the rock as a function of the applied pressure (the proceeding unloading in each case was not considered). In Fig. 2 is shown also the relationship (curve 3) plotted with the results of unloading at each step. An analysis of the curves shows that the condkional - momentary deformation in loading and in unloading are s imilar .

Curve I is clearly non-linear, but it can be represented by two linear intervals, 0-10 and10-40 k g / c m 2. The deviation from a linear relationship in the fkst interval is small , in the second it does not exceed 10 percent. Curve 2 pract ical ly coincides with the relationship constructed with all loading cycles. This shows that the formation does not change its state in the course of the test.

mm o,z

Uo:E

p.q5

IC'

s j! [ IJi t 1 ! I _I_A

i , : I f I I I

"!Litl 5 I0 ~5 ZO ~ 30 ~ ~ ~5 50 ~'~ CO S5 7Oh

Fig. 1. Deformation of the surface of the formation with t ime as a function of the applied pressure (punch No. % stratified sandstone).

Condit ional-momentary flexures at different stages of loading are shown in F ig. 3. A compar i - son of the experimental curves with the theoret ical ones shows that they are close in position. The theo - rer ical curves were constructed from the relationship (1) obtained on the basis of l inear theory of elast ici ty for a rigid 1.1 x 0.9-m punch by S. B. Ukhov [3] and Z. G. Ter-Martkosyan:

* The investigations were conducted by the labora- tory of mechanics of rock formations of the chair of mechanics of soils, foundations and substructures of the V. V. Kuibyshev MISI, directed by N. A. Tsytovich, corresponding member ,Academy of Sciences of the USSR.

Translated from Gidrotekhnicheskoe Suoi te l ' s tvo, No. 8, pp. 31-34, August, 1968.

698

INVESTIGATION OF CREEP OF ROCK FORMATIONS BY TRIAL STATIC LOADING 699

kg / cm z

.r p 3 /

ia / ,

jr #.OZ #.oq Q, OS ~os # l m m

Fig. 9. Conditional-momentary deforma- tions of the formation as a function of pressure. 1) Loading at all cycles; 2) loading in the last cycle (ps = 40 kg/ cm2; 3) unloading at all cycles.

Distance from the edge of the punch, cm 0 Z0 q0 60 88

- -

0.08 "/

m m Smom

Fig. 3. Conditional-momentary flexures. 1)Experimental curves; 2) theoretical curves.

"~'~r (l --p.~)-2b,p (1) Eo = So '

where E 0 is the conditional-momentary modulus of deformation; q is the coefficient which depends on the movement

of the point on the surface of the rock from the edge of the punch with the measured deformation S0; t~const is the shape and rigidity coefficient of the punch [4], p is the average pressure on the bottom of the punch, and b is the

half-width of the punch.

Therefore, in describing momentary deformation of the considered formations, it is permissible to use the theory of a linearly deforming half-spaoe, with the assumptions made above considered.

Let us consider the creep phenomenon. Observations of deformations of the surface at some distance from the punch made it possible to conclude from the nature of developments of the flexture bowl in t ime when the pressure is constant, that the stressed state in the formation remains unchanged in the course of protracted loading. It is not difficult to shows that in this case the increase in deformation with time at different distances from the punch will decrease in proportion to the coefficient n -

It is seen from Fig. 4 that the creep curves at different distances from the punch are sufficiently close to the theoretical curves. Therefore, the stressed-state can be considered with a sufficient degree of accuracy to be un- changed with t ime.

Yet according the theorem proved by N. Kh. Arutyunyan [5],the condition for non-dependence of the stressed- state under the action of forces on the creep of material is the constancy of the Poisson coefficient go- It follows that in the experiment the coefficient/~0 does not change with t ime.

The nature of deformations of creep for the given formations is attenuating in the investigated range of stresses. Investigation of the restoration of the creep deformations when the pressure was removed showed their practically irreversible nature. Thus, creep deformations were produced not by the action of the absolute magni - tude of pressure at each step, but by an increase from pressure between two consecutive loadings. Thus, at steps I and II, it was produced by Ap= 5 kg /cm z, and at steps III-V by the action of Ap=10 kg / c m 9-. Creep deformations vary linearly with the acting pressure and the principle of superpositiou iS], which consists in the principle that total creep deformation with alternating stresses can be found from the sum of creep deformations produced by cor- responding inc~emenu of stresses applicable to them. The magnitude of creep deformations caused by increments of stress depends on the magnitude and duration of the action of this increment and does not depend on the magnitude and duration of action of the remaining increments. At the same instant of time counted from the instant of the ap- plication of the load at each step the ordinates of creep curves of steps I and II must in this case be equal and those of steps [II, IV and V must be equal to each other and must be twice as large as the ordinates of the creep curves of steps I and II (Fig. 4).

700 V. L. KUBETSKII AND S. B. UKHOV

cm S..I# -2 [ / 1

,, / . ~ - - r"~,';'-r~-~ - -

a

cm!& "r t I

1 , ,4 2

22 7 i ' < ' ~

I /i f &,If" '. I b | I

cm 4," 10-3 ' 4. ~ . . . .

q

/ 2

l

�9 I 0 �9

fi t 2 3 ~ sec C

Fig. 4. Creep curves of stratified sandstone. a) StepI Ap = 5 kg/cmZ; b)steplII Ap=10 kg/ cruZ; c) step V Ap= 10 kg/cm z. 1) From re- ference points on the punch; 2 and 3) from re- ference points at distances of 12 and 40 cm respectively from the punch; 4) creep curves constructed with the obtained parameters.

- l l

I ! ~ / | # l 2 3 sec

Fig. 5. Relative rate of settling as a fun- tion of t ime.

The established regularities made it possible to describe mathematically the phenomenon of creep of a given type of rock formation from results of investigations by the trial stat- ical loading method on the basis of the linear sequential theory of creep of Boltzmartn-Volterra.

All the conclusions of the theory of linear sequential creep can hold only for the regime of loading of the punch because the investigated rock formation behaves as a linear sequential body only in the case of increasing or constant stresses (but not in the case of decreasing stresses because of the clearly expressed irreversible nature of creep deformation).

The longitudinal relative deformations of a body ~(t) when a constant stress o0" ) is applied to it can be expressed by:

(0 = ~ + * ( 'O-c (t, "0, (2)

where E 0 is the modulus of elasticity (conditional-momentary modulus of deformation), C(t, r ) is the measure of creep [5] or the expression for the creep curve for unit stress o = 1, r is the instant of application of the stress, and t is any considered time instant (greater than T ).

The first member represents the elastic-momentary part of the deformation while the second is the creep deformation at the time instant t. Expression (2) can be represented in a different way:

"('0 (2') = (t) - - Es (t, "0'

where E s, the modulus of total deformation, is

E o

Es (t, ~) = -1 + C (t, ~) Eo CS)

On the basis of Volterra's principle [7] any static problems of the theory of linear sequential creep can be solved as a corresponding problem of the common theory of elasticity. Only in the final results the elastic constants are replaced by elastic operators. Therefore, we obtain from Eqs. (1) and (3) for /% =const the equation

INVESTIGATION OF CREEP OF ROCK FORMATIONS BY TRIAL STATIC LOADING 701

TABLE I

Deformat ion indices of

stat if ied sandstones

Stress interval , k g / c m z

0-I0

E0 , k g / c m z 920,000

E S, kg/cmZas t - * ~o 147,000 O, cm z / k g 5.7 �9 10 "s

k, I/sec 0.9" I0 "4

Remark. Poisson coefficient go = 0.25.

10-40

s ( t ) =

designating

~" %o.st (I - - ~02). 2b- p Eo/[1 + C (t , , ) Eo] '

167, O0 0 "~. COeons t (1 - - p-g)- 2b. p 75,500 So = E0 '

5 . 7 . 1 0 "6 0.9 - 10 .4 we obtain

(4)

S (t) = S, [1 + C (t, ~) E,]. (4')

The expression C (t, r ) is sufficiently wel l approximated for the given formations by the exponent ia l functions

C (t, ~) = 0 [1 - - e - X l t - ' ) l , (5)

where 0 and k are the creep parameters .

In the tests we have at each step a case of s imple creep, p = const; r can therefore be taken to be equal zero

and we obtained from Eqs. (4') and (5) the expression

S (t) ----- S O [1 + Eo0 (1 --e--U)], (6)

and differentiat ing

a s (7) dt = S , . Eo. O. ~. e - x t .

Applying Eq. (7) to two arbitrary points on the expe r imen ta l creep curve, we find

S O �9 E o �9 O. k . e - x t i - - Si + tit - - Si--At 2at (i = 1; 2),

where t i and Si are the abscissa and ordinate of point i, At is a sufficiently small t i m e interval .

Dividing the first equat ion of this system by the second we obtain the fol lowing working equations

2.303 Sl+tit --SJ-tit " (8) k = t, -- t, Ig S2+tit __ S2_tit ,

SI + At -- S I--At $2+ tit -- S2-M 0 = (9)

So.Eo.k.2M.e --~, =So.Eo.k.2&t.e -xt,

In this case, if as a result of the test a s tab i l iza t ion of creep deformation was reached the de te rmina t ion of the

parameters can be s impl i f ied. Thus, from Eq. (6) when t -~ ,o we obtain the equation

S, --So (I0) 8 = So.Eo '

where Sk is the final relatively stabilized deformation of the rock surface.

The parameter k can be determined graphical ly [8] provided the relationship of the relat ive speed of settl ing

to t i m e is expressed by a straight l ine on a semi- logor i thrnic scale In S ~ - - So - - t �9 In this case, the parameter

702 V. L. KUBETSKII AND S. B. UKHOV

AS k is equal to the tangent of the inclination angle of this straight line (Fig. 5). Here S (t) = ~ is the rate of creep

deform ation in c m / s e c.

The indices of deformation of stratified sandstones determined from results of experiments are given in

Table 1.

The method proposed by P. I . Vasil 'ev [9], can be used as a basis for obtaining relationships which describe the deformation of a given formation with a variable stress. If the stress increases in jumps, then we shall, on the basis of the principle of superposition, obtain.the following equality:

�9 (t,,5 = Z- . + =.c (t,, --'~.5 + (~, - -" .5 c (t,, --'=,5 + . . .

~n Jr" (:,, - - % - , 5 C (tn - - % ) : ~ -{-- %C (t. - - %1 (121"

+ ~ , (~. - - =._,) C (t . - - .~). i : l

With a continuous change in the stress, the magnitude of the deformation is expressed by the equation

=max

�9 (t,,5 = * e,(t"5 S + c ( r ) do, (13)

0

where Omax is the maXimum stress during the t ime interval O-in and T is the duration of applicat ion of t h e e l e - mentary increments of stress do .

In obtaining Eq. (13), it is assumed that the stress reaches its maximum value once during the t ime interval O-tn. Equation ( la ) corresponds to the full irreversible creep deformation.

In the case where the creep deformation is partly reversible, it is possible to obtain similarly the equation

=max. %

"" '-~ t S - - T , + c, (r) d= + C, (r) a~, (14)

0 : m a x .

where C,( t , r ) and C~(t, r ) are the measures of creep respectively in loading and unloading regimes.

In the special case when Cl( t , r ) equals Cz(t, r ) Eq. (14) can be considered a Volterra integral equation of the form

t

* (t) ~ aC (t, ~5 , ( t ) = E, - - = (*) dr

0

- - d ~ ( 1 5 )

with a nucleus

aC (t, ~)

CONCLUSIONS

1. The investigated rock formatiom show creep properties in the stress interval 0-40 k g / c m 2, Magnitudes of creep deformation reach 100-200% of momentary deformation. The creep is close to l inear and is prac t ica l ly non-reversible.

*Eq. (11) is omitted in the Russian original due to an apparent error in numerat ion-Publisher .

INVESTIGATION OF CREEP OF ROCK FORMATIONS BY TRIAL STATIC LOADING 703

2. Considering the substantial non-uniformity of rock formations in their natural condition, the non-linearity of instantaneous deformation should, for engineering purposes, be neglected; this will substantially simplify the

corn put ations.

3. The obtained regularities make it possible to use experimental data in the solution of certain problems connected with the construction of hydraulic structures and especially in the design of linings of pressure tunnels for

internal pressure.

LITERATURE CITED

I. B.S. ~ristov and V. L. Kubetskii,Accounting for Rheology of Rock Formations in Designing Linings of Hydraulic Pressure Tunnels, Report at the XXVII Scientific Technical Conference of MISI, Theses (1968).

2. Zh. S. Erzhanov, The Theory of Creep of Rock Formations and Its Application [in Russian], Izd-vo "Nauka n

Alma-Ata (1964). 3. S.B. Ukhov and N. A. Tsytovich, Some principles of mechanical properties of chloritic schists. Proceedings

of the first congress of the International Society of Rock Mechanics, Lisbon (1966). 4. N.A. Tsytovch, Soil Mechanics [in Russian], Stroiizdat (1963). 5. N. Kh. Arutyunyan, Some Problems in the Theory of Creep [in Russian], Tekhizdat (1952). 8. I.I. U1itskii, Chang Chung-Yao, and A. B. Golyshev, Design of Reinforced Concrete Structures Taking into

Account Longtime Processes [in Russian], Stroiizdat, Kiev (1960). 7. Yu. N. Rabomov, Creep of Construction Elements [in Russian], Izd-vo nNauka" (1966). 8. N.A. Tsytovh, et al., Forecasting the Speed of Settling of Foundations of Structure [in Russian], Stroiizdat

0067). 9. P.I. Vasil'ev, "Relationship between stresses and deformations in concrete in compression, taking into account

the effect of time, " Izv. VNIIG, 45 (1951).