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NOTES
A simple stress path analog for the triaxial test
Y. P. VAID AND DAWIT NEGUSSEY Department of Civil Engineering, University of British Columbia, Vancouver, B.C. , Ca~zacicc V6T 1 W5
Received March 29, 1983 Accepted July 12, 1983
A simple pneumatic stress path triaxial testing device has been designed. The components of the apparatus and the operation of the system is described. Typical results of drained tests on a sand are presented in order to illustrate the capabilities of the device as a stress path analog and to highlight briefly the stress path dependence of soil behaviour.
Keywords: triaxial equipment, pneumatic, stress path.
Un appareil simple de contr6le pneumatique du cheminement de contraintes triaxial a CtC dCveloppC. Les composantes de I'appareil et le fonctionnement du systkme sont dCcrits. Des rCsultats typiques d'essais drainis sur un sable sont prCsentCs pour illustrer les capacitks de I'appareil et pour mettre en Cvidence brikvement l'influence du cherninement de contrainte sur le comportement du sol.
Mots-cl&: Appareil triaxial, pneumatique, chemin des contraintes. [Traduit par la revue]
Can. Geotech. J . 20, 827-832 (1983)
Introduction Stress path is one of the many factors that influence
deformation response of soils. Deformation problems of soil masses are generally solved in two different ways. The first approach, called the stress path method (Lambe 1964), consists of subjecting the soil to the same stress increments as anticipated in a given field problem and then measuring the resulting deformations. The second approach involves using a comprehensive constitutive model of soil behaviour, in which the influence of stress path variation is inherently built in. Nevertheless, in both approaches stress path testing of the soil is necessary, either to estimate deformation directly as in the stress path approach, or to check or formulate comprehensive constitutive models.
Because of the complexity associated with soil testing under a generalized stress system, the triaxial test is the most commonly used equipment for shear testing in the laboratory. The conventional triaxial apparatus permits only a limited number of stress path choices. In the past, several stress path analogs of the triaxial test have been developed and used in research studies (Law 1981; Lewin and Burland 1970; Menzies and Sutton 1980; Wong and Mitchell 1975). These devices, which vary in complexity to different degrees (some are provided with a front end computer control), permit triaxial testing along various stress paths in the triaxial stress plane (axial versus radial stress). This note describes a simple and inexpensive pneumatic system which enables stress path triaxial testing without the complexity and some- times the inconvenience associated with some of the
current techniques. Some typical results of drained tests on a saturated sand are presented in order to illustrate the performance of the device and briefly highlight the stress path dependence of soil behaviour.
Description of the apparatus A schematic layout of the stress path device is shown
in Fig. 1. The loading system consists essentially of adjustable-ratio and reversing pneumatic relays, three pressure regulators, and a double acting loading piston. The signal pressure regulator R3 directly controls the cell pressure P3 which is also the input pressure to the pneumatic relays. The relays ultimately deliver the output pressure to the air piston, which applies the deviator load to the sample. The signal pressure regula- tor R3 can be operated either manually or coupled to a variable speed motor if a constant rate of loading of the sample is desired.
The pneumatic relays transform the signal pressure P3 to an output pressure Po in the following manner:
[ l l Po = RP3 . . . adjustable ratio relay
[2] Po = K - P3 . . . reversing relay
in which R and K are positive constants. If the reversing relay is used in series with the ratio relay, the output pressure obtained will be:
[3] Po = R ( K - P3)
In [3], K > P3 since all pressures are positive.
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828 CAN. GEOTECH. J . VOL. 20. 1983
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DOUBLE A C T I N G A I R P I S T O N
L E G E N D :
A R R A D J U S T A B L E RATIO R E L A Y
L C LOAD C E L L
LVDT D I S P L A C E M E N T T R A N S D U C C R
P P R E S S U R E
B A S E C L A M P R R E G U L A T O R
R R R E V E R S I N G R E L A Y T P R E S S U R E T R A N S D U C E R
@ S H U T O F F V A L V E
?//////////////A FIG. 1. Schematic diagram of the pneumatic stress path analog.
The underlying principle For the triaxial test setup shown in Fig. 1, let A, =
upper chamber area of loading piston, A2 = lower chamber area of loading piston (this is less than A I by an amount equal to piston rod area), As = sample area, A, = sample loading rod area, PI = pressure in the upper piston chamber, P2 = pressure in lower piston chamber, and u = porewater pressure in the sample.
Vertical equilibrium of the sample then requires
[4] Asah= PIAl + P z A 2 + P3(A, -A , ) -Asu
and also
stress path, AB. Then the effective stress changes ha: and 4 4 , from [5] and [6], are
Since the porewater pressure is constant in drained tests, 4 u = 0. By defining the ratio 4a:/4uk of the principal effective stress increments to be equal to tan 0 , [7] and [8] can be combined to yield:
[6] crk = P3 - u If P2 is held constant, AP2 = 0 and [9] reduces to
In [5] and [6], a: and a;, are respectively the vertical [lo] 4P1 = C14P3
and lateral effective stresses in the triaxial sample. in which Consider, in the a:, ak stress plane (Fig. 2), an
arbitrary initial effective stress state A. It is desired to [111 C , reach an effective stress state B along a straight line AS
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NOTES
a;' in which
FIG. 2. Simulation of specific and arbitrary stress paths.
If instead PI is held constant, API = 0 and [9] reduces to
in which
Since A,/A, << 1, it may be neglected in calculating C values. Both C 1 and C2 can be positive or negative depending upon the value of 0 and, hence, the direction of the stress increment. C 1 is negative when (tan 0 - 1) < 0, which occurs for all 0 except 90" > 0 > 45" and 270" > 0 > 225", and [lo] may be written as
in which ICIJ is the absolute value of C 1 . If P,,,, PIU and P3,, P3, represent respectively the pressures PI and P3 at A and B, [I41 becomes
in which
which is numerically positive. On the other hand, C2 is negative when (tan 0 - 1) >
0, which occurs for 90 > 0 > 45 and 270 > 8 > 225, the values of 0 not covered by C l . Therefore, for negative C2 values, relations similar to [14]-[16] can be written as follows:
which is numerically positive. It may be noted that the relationships expressed by
[15] and [18] are identical to [3], which represents the combination of the adjustable ratio and reversing pneu- matic relays. In these equations, P3 corresponds to the signal and P1 or P2 to the output pressure Po. The constants ICI and D correspond respectively to constants R and K of the pneumatic relays. Thus, an arbitrary stress path AB can be followed by varying the cell
P3 and choosing appropriate constants R and K on the pneumatic relays. Clearly, a nonlinear stress path can be followed by a series of small linear segments, if new R and K values are determined at the st& of each linear segment. It may be noted that, for a given system, R is a function only of 0 and is independent of the magnitude of P I , P2, and P3 at A. K on the other hand is a function both of 0 and the pressures PI or P2 and P3 at A.
Pneumatic relays are commercially available with a variety of choice. In the apparatus described in this note, the ratio relay R factor could be varied between 1/30 and 30 and the K factors of the reversing relay could be set in the range 15-1000 kPa.
The R factor range available on the ratio relay, together with suitable choice of sample size A, and loading piston size A l , enables stress path probes for various values of 0. However, for 0 in the vicinity of 7r/2 and 3 ~ 1 2 , tan 0 tends to x and the corresponding values of R that are in excess of 30 cannot be provided by the pneumatic system. These stress path- directions, however, correspond to conventional compression and extension loadings at constant cell pressure and can be followed by loading through regulator Rl or R2. The pneumatic relay loading analog is kept disconnected for such stress paths. Stress probes very close to these limiting 0 values (to within t 5 " ) can easily be applied by this pneumatic analog.
Operation After a triaxial sample has been set up and connected
to the volume changelpore water pressure measuring device, it is under a small hydrostatic (isotropic) effective stress (point E in Fig. 2). This effective stress is a consequence of the small vacuum which is needed for confining reconstituted sand samples or it is due to residual capillary tension in clay samples. The stress state E is, therefore, the initial state from where the stress path loading (consolidation or shear) commences.
The normal state of the loading system represents zero deviator load. This is maintained by applying suitable nonzero base pressures P1 and P2 in the chambers of the
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830 CAN. GEOTECH. J . VOL. 20. 1983
o loading piston. These pressures are fed directly by - - 0 U In regulators R 1 and R2 with valves Vl and V2 open and V3
and V4 closed, thus shutting off any input from the relays. In the subsequent operation procedure it will be assumed that either PI or P2 is held constant by the respective regulators R I or R2 and the pneumatic relay output admitted through either valve V3 or V4, as appropriate. Consequently either of [15] or [18] will govern stress path loading, depending upon the direction 0 of the stress increment.
(a ) Consolidation Triaxial samples are consolidated under either iso-
tropic (ah = a;) or anisotropic (a: + a;) condi- .- tions. Isotropic consolidation to F, for example (Fig. 2),
is carried out as in a conventional apparatus, by suitable - .. selections of cell pressure P3 and back pressure u. M 9 Valves V3 and V4 are kept closed, thus shutting off any m 9 input pressure from the relays during this stage. A small
0 - - 0 G reduction in a, may occur on application of P3 due to an L".., a
U 'Y) .. .', '.. , Y) '. -
'. N '.. 0
0 m a uplift on the loading ram. This may be corrected, if V1
o considered important, by slightly increasing PI . 0
- V1
0)
Anisotropic consolidation is usually carried out under d
G a constant stress ratio Kc = u{/o;. Since the initial o 0 2 stress state E of the sample is hydrostatic, it would first - O n a t .a be necessary to bring the sample to the desired stress
"a ratio Kc. This is most conveniently achieved by follow- -bz C- ing a vertical stress path EE' on the compression side or
X EE" on the extension side, and involves adjustments in
3 PI only. The input from the relays is still maintained m disconnected. Anisotropic consolidation from E' to F'
under the desired stress ratio Kc now simply constitutes a stress path loading with tan 0 = Kc, and will be
'- o discussed later under shearing. However, such consoli- a 2 dation can be more simply achieved by bypassing the . d
reversing relay and using the pneumatic system with a V) adjustable ratio relay only. The R value on the ratio relay
o is adjusted until it delivers a pressure PI corresponding - 0 -
-
-
-
- o
I I
- 0 a
0 V) to the states E' or E". Transducer hookup locations TI "I - E and T3 enable precise measurements of Pl at states E' or
0 V) E" and also presetting it by the ratio relay. Valve Vl is
0 * 5 c now closed and V3 opened in order to admit pressure Pl through the pneumatic loading system. Since the sample
8 2 is now under a stress ratio Kc, with the selected constant * < R on the ratio relay, consolidation under Kc along E'F'
-bs 2 or E"F1' is automatically achieved by monotonically 0 IU '
increasing P3 with R maintained constant at all times
g until the desired stress state is reached. 0 U 0 . (b) Shear rn
Following isotropic consolidation to F or anisotropic consolidation under Kc to F' , the shear stress increment
0 0 0 " 0
0 0 0 can be applied along an arbitrary stress path, defined by 0 0 0 0 0 - the angle 0. However, following isotropic consolida- 0 w m IU tion, shear loading along conventional stress paths -
O d Y ' fo axial compression (0 = n/2) and axial extension (0 =
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NOTES 83 1
TABLE 1. Stress path dependent response of Ottawa sand
Initial Final Volumetric strain* Shear strain:* stress state stress state (V x lo3) (Y x lo3)
(a:, 4) (a:, 4,) Stress OW (kpa) path Total Incremental Total Incremental Remarks
0 0 Consolidation or shear O(50, 50) B(340, 160) OAl Bl 2.30 1.30
OA2B2 2.40 2.30 OA3B3 2.70 1.20
B(340, 160) C(460, 280) B l C , 3.20 0.90 1.30 0.00 Shear at constant shear stress B2C2 3.60 1.20 2.30 0.00 B3C3 3.80 1.10 1 .50 0.20
C(460, 280) D(340, 340) C I D l 3.60 0.40 0.70 -0.60 Shear at constant mean normal C2D2 3.60 0.00 2.30 0.00 stress C3D3 4.00 0.20 0.90 -0.60
D(340, 340) E(240, 390) D I E l 4.30 0.70 - 1 .O - 1.70 Extension loading at constant D3E3 4.90 0.90 -0.8 - 1.50 mean normal stress
*The measurement accuracy of volumetric strain was 3 X IO-' and that of linear strain 1 x 1 W5.
3n/2), which do not involve changes in cell pressure, is more conveniently carried out by adjustments in either PI or P2 only using the regulator R I or R2 and keeping the input from the relays shut off. For any other desired stress path, say FG, the constants ICJ and D of the pneumatic relay system are computed by means of the relationships discussed previously. These constants are then set on the respective relays, which will enable them to deliver the necessary pressure PI or P2 at F to the appropriate piston chamber. Once the settings have been established, the pneumatic system pressure is made to correspond to the current regulator pressure being fed to the piston. At this time, control of the piston chamber pressure is switched by shutting off the regulator supply and connecting piston to the system output. The desired stress path can then be followed by monotonically varying P3 until the stress state G is reached. In order to follow the next leg of the stress path, say GH, valve V3 or V4 is closed, thus shutting off input PI or P2 from the relays. P1 or P2 is, however, maintained at the current level by connecting the loading piston to regulator R1 or R2, which has been preset to the value P1 or P2 at G. Knowing 8 and Aa: for the stress path GH and the current values of P, or P2 and P3 at G, new constants (C( and D for the relays are now calculated and set on the respective relays. PI or P2 are once again transferred to the relays by opening valve V3 or V4 and shutting off VI or V2. The desired stress path GH is followed in the same manner as FG by monotonically varying P3. This procedure can be repeated for the next leg of the stress path in a similar manner.
Consolidation under anisotropic stress ratio Kc along E'F' , if desired, is in no way different from following any other stress path. For this stress path E ' F ' , the
constants J C and D of the relays will be computed for tan 8 = Kc and the values of PI or P2 and P3 at E' and F ' . Once the loading piston pressure supply PI or P2 at E' has been transferred to the relay system, stress path E'F' is followed as usual by monotonically changing P3. Application of any arbitrary stress path increment, after consolidation to F ' , is identical to any other stress path increment described in the preceding paragraph.
Performance and typical results Results from three drained triaxial tests on Ottawa
sand C-109 at a relative density of 50% (void ratio = 0.66) are now presented. These tests illustrate the capability of the apparatus as a stress path testing device. The stress paths chosen are also aimed at briefly highlighting some aspects of path dependent deforma- tion response of the sands.
Considerable improvements in the deformation mea- surement techniques have been incorporated (Vaid and Negussey 1982) in order to obtain experimental data with confidence, particularly when the deformation increments are small. This includes more rational methods of correction of volume changes due to membrane penetration. Consequently, the accuracy of the deformation measurements in the results reported here is of the highest order.
Figure 3 shows comparisons between the stress paths desired (solid lines) and those actually applied (data points) in three triaxial tests on identical samples. The stress paths chosen represent a wide spectrum from isotropic and anisotropic consolidation, conventional compression and extension, constant shear under in- creasing and decreasing mean normal stress to loading and unloading under constant mean normal stress. It
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832 CAN. GEOTECH
may be noted that excellent reproduction of the stress paths desired is possible by the new stress path device.
Following back pressure application and saturation, the initial state 0 of all samples corresponded to a hydrostatic effective stress of 50 kPa. Each sample was then loaded using a different stress path until all samples reach an identical stress state at B. Stress paths BC and CD are common to each sample. Further sections of stress paths are not common to all samples. The volumetric strains v and shear strains y = (E, - E ~ ) at each stress state referred to the initial state 0 are given in Table 1, together with strain increments during some common legs of stress paths. For example, it is common practice to consolidate soil samples anisotropically first by isotropic consolidation to the final confining pressure and then by applying a deviator stress under drained conditions until the desired consolidation stress ratio is reached (stress path OA Bl in Fig. 3n). It may be noted from Table 1 that the strain state of the sample at B is significantly different if the sample was consolidated anisotropically from the beginning (stress path OA2B2 in Fig. 3b). Still another strain state at B will be reached if this stress state was arrived at using stress path OA3B3 (Fig. 3 c).
Further identical stress path loading BC and CD from the stress state B may be seen to yield once again strain increments which are not identical and are obviously linked to the stress path through which stress state B was arrived at. It is thus apparent that the deformation response to a stress increment from a current stress state
I. VOL. 20, 1983
depends not only on the direction of the stress increment but also on the .entire stress path through which the current stress state was arrived at.
Acknowledgements Financial assistance of the Natural Sciences and
Engineering Research Council of Canada is gratefully acknowledged. The assistance of Richard Brun in drafting figures, Mrs. Kelly Lamb in preparation of the manuscript, and Fred Zurkirchen for his expert match- ing is also gratefully acknowledged.
LAMB, T. W. 1964. Methods of estimating settlement. ASCE Journal of the Soil Mechanics and Foundations Division, 90(SM5), pp. 43-68.
LAW, K. T. 1981. A servo system for controlled stress path tests. American Society for Testing and Materials, STP 740, pp. 164-179.
LEWIN, P. I., and BURLAND, J . B. 1970. Stress probe experiments on saturated normally consolidated clay. Geo- technique, 20, pp. 38-56.
MENZIES, B. K., and SUTTON, H. 1980. A control system for programming stress paths in the triaxial cell. Ground Engineering, 13(1), p. 22.
VAID, Y. P., and NEGUSSEY, D. 1982. A critical assessment of membrane penetration in the triaxial test. University of British Columbia, Soil Mechanics Series No. 61, Van- couver, B.C.
WONG, P. K. K., and MITCHELL, R. J. 1975. Yielding and plastic flow of sensitive cemented clay. Geotechnique, 25, pp. 763-782.
Influence factors for settlement estimates of footings on finite layers
BRIAN B. TAYLOR AND ELMER L. MATYAS Department of Civil Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3Gl
Received April 8, 1983 Accepted June 27, 1983
The estimation of the settlements of surface footings on finite layers is examined for values of Poisson's ratio ranging from 0 to 0.5. Influence factors based on Steinbrenner's approximate method were evaluated and were found to be in good agreement with influence factors obtained by a more rigorous and difficult evaluation by Giroud. This study confirms the findings by Christian and Carrier that the Janbu, Bjerrum, and Kjaernsli chart for the estimation of elastic settlements (v = 0.5) is incorrect.
Keywords: elasticity, footings, settlement analysis.
L'Cvaluation des tassements de semelles superficielles sur une couche d'Cpaisseur finie est examinee pour des valeurs du coefficient de Poisson variant de 0 i 0,5. Les facteurs d'influence dCduits de la mCthode approchke de Steinbrenner ont CtC CvaluCs et trouvCs en bonne concordance avec les facteurs d'influence obtenus par la mCthode plus complexe et plus rigoureuse de Giroud. Cette Ctude confirme les rCsultats de Christian et Carrier selon lesquels l'abaque de Janbu, Bjerrum et Kjaernsli pour 1'Cvaluation des tassements Clastiques (v = 0,5) est incorrect.
Mots-cle's: ClasticitC, semelles, analyse des tassements.
Can. Geotech. J. 20, 832-835 (1983)
[Traduit par la revue]
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