It is assumed that the readers are familiar with the HKIE Report and descriptions on the current approach will therefore be skipped.

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2 INTRODUCTION The current approach of design of soil nails for upgrading loose fill slopes is described in the report Soil Nails in Loose Fill Slopes A Preliminary Study published by the HKIE Geotechnical Division (HKIE, 2003), hereafter called the HKIE Report. A summary of this report is also presented in Pappin (2003). In 2009, the Geotechnical Engineering Office (GEO) planned to review the current practice of the design of soil nails in loose fill and invited practitioners to express their views on this topic. This Technical Note is based on the comments made by the Authors to the GEO in June 2009 in response to their invitation for comments. In preparing this Technical Note, the Authors attempt to integrate the theoretical concepts, field observations and practical considerations in reviewing the current practice. It is assumed that the readers are familiar with the HKIE Report and descriptions on the current approach will therefore be skipped. SOIL LIQUEFACTION Before re-examining the design philosophy of the current approach of design of soil nails in loose fill, it is useful to briefly mention the findings of recent research on instability of sands. Static liquefaction is a form of instability prior to the effective stress state reaching the failure surface in the effective stress space (Chu et al, 1992; Leong et al, 2000; Lo, 2007). However, one should not take the occurrence of dq/dε a < 0 as being static liquefaction. For instance, the occurrence of dq/dε a < 0 in a post-failure state for dense sand is clearly not static liquefaction. The occurrence of the condition of dq/dε a < 0 can be caused by experimental aberrations or certain difficulties in conducting the test for loose sands as will be discussed later. Studies by some researchers (Lo, 2007; Take et al, 2004; Chu et al, 2003; Chu et al, 1992,1993) since the early 90s show that static liquefaction will occur when the effective stress state enters an instability zone in an undrained (or near-undrained) condition irrespective of the direction of effective stress path as long as the direction of the effective stress increment or perturbation corresponds to dη > 0, where η = q/p represents the ratio of deviator stress and mean stress (see also definitions of notations at end of this Note). The instability zone is located above the instability line in the q-p. When a strain softening sandy soil is subjected to undrained shearing in triaxial compression under constant cell pressure, a peak point can be identified in the effectives stress path (ESP). Such a peak point is commonly referred to as the onset of strain straining or onset of instability (or less frequently onset of collapse). The locus joining all the instability points, determined from tests commencing at different consolidation stresses, is referred to as the instability line. It is also called the critical stress ratio (CSR) line or collapse line. The instability line may actually be slightly curved. The above concepts are presented in Figure 1. A corresponding line can also be plotted in τ-σ n space for plotting Mohr circles

3 Further discussions on instability behaviours of sands can be obtained from the papers by Lo and his co-workers (Bobei et al 2009; Rahman & Lo 2009, Wanatowski et al, 2008, 2008a; Chu et al 2007; Lo, 2007). GENERAL COMMENTS A two-prong approach The main concern of loose fill is its potential for liquefaction. There are two ways to prevent liquefaction: to prevent the trigger from occurring or to enhance the resistance. The occurrence of trigger requires three prerequisite conditions: i) saturation or near saturation of the loose fill ii) the effective stress state being forced into an instability regime, and iii) occurrence of undrained shearing. At present, the design approach in the HKIE Report is based on enhancing the resistance and assuming that all necessary conditions for triggering static liquefaction of potentially liquefiable loose fill materials will always be met under the credible worst scenario. This will no doubt lead to a very conservative approach. It may be preferable to adopt a two-prong approach that makes the soil nail design less conservative by both suppressing the chance of occurrence of the triggering factors and to increase the intrinsic resistance of the soil. For instance, if we were to construct grillage beams for soil nails (perhaps with some preload) according to the current practice, we can provide a network of drainage pipes and/or additional subsoil drains to discharge the surface runoff quickly away from the slope and to prevent build up of water pressure near the slope surface. We may also use shotcrete surface to prevent infiltration. Failure triggered by sub-surface flow It is well discussed in soil mechanics textbook that for an infinite dry slope, the factor of safety is given by F = tanφ/tanα, where φ is the angle of shearing resistance of soils and α is the inclination of slope. For an infinite slope with seepage flow parallel to the slope, the factor of safety becomes F = (γ /γ) (tanφ/tanα) where γ and γ are the effective and bulk density respectively. As γ /γ is usually of the order of 0.5, this implies that the factor of safety can drop by 50%. For loose fill slopes, a perched water table may be present near the slope surface. If subsurface flow of water parallel to the slope occurs within the loose fill layer, for instance due to layering of soils in end-tipped fill as in the case of the Sau Mau Ping failure (Morgenstern, 1978), the loose fill slope can easily fail due to this reason. It is also possible that the sudden and large volumetric strain that occurs with such a failure trigger may then drive the soil mass into an undrained condition, and the fill materials will then liquefy. This may be considered as having static liquefaction as a result of failure as first proposed by Eckersley (1990) or having static liquefaction triggered in a drained state but evolved into near-undrained shearing (Chu et al, 2007). So, the provision of surface and subsurface drainage is equally if more important than merely concentrating on the design aspects of soil nails in upgrading fill slopes. This is in line with the suggestions made in Take et al (2004)

4 SPECIFIC COMMENTS Jelly fill According to the HKIE Report, soil strength will drop to a steady value (c ss ) if static liquefaction under undrained shearing is allowed to take its full course as illustrated in Figure 2. Therefore, soil nails will need to develop its resistance during the course of static liquefaction. However, if the stress state of the soil is prevented from crossing the point of initiation of static liquefaction as represented by point B, either because of a flat slope or the provision of adequately mobilized resisting force, the low steady-state strength (c ss ) of loose fill as represented by point C will not occur. In current approach, one attempts to obtain the lower bound strength by forcing static liquefaction or soil instability to occur and use this measured lower bound value of the steadystate strength for design of soil nails. The steady-state undrained strength c ss is of the order of a few kpa, similar to that of soft jelly. This implies an ineffective and/or hugely conservative approach. Why insist on constant-q tests? The HKIE Report seems to favour the measurement of steady-state strength by constant-q test, probably still influenced by the thinking of Brand (1981). Constant-q tests are often regarded as more realistic in mimicking the process of slope failures triggered by rain infiltration. The presence of matric suction is seen as analogous to a higher effective mean stress, p. Wetting of soils due to rain infiltration and the corresponding loss of matric suction are analogous to a reduction in p in a constant-q test. Eventually all matric suction may possibly be lost and there may possibly even be a build-up of positive porewater pressure. In this process, the effective stress path (ESP) of a soil element in a slope may be approximated by a constant-q path. The data of constant-q tests are often very noisy with a large scatter. One may then be tempted to recommend a design strength based on the lower bound value to cater for the high variability arising from noisy data measured from constant-q tests. This will lead to hugely over-conservative values for design and often embarrassingly long and closely spaced soil nails to achieve the requisite factor of safety. This noisy nature of constant-q test results is due to the intrinsic limitations of constant-q tests and challenges associated with its interpretation. To explain the limitations of constant-q tests, one needs to take a closer look at the nature of the test. 1. A constant q test is conducted by applying a constant axial force (usually using either a pneumatic actuator or a stepper-motor drive actuator). The cell pressure is ramped down at a controlled slow rate. This will lead to shearing as the effective stress ratio η = q/p will be increasing. One does not need advanced theory to recognize that a constant-q test is a stress-controlled test for which the peak value of q or the deviatoric straining softening response may be difficult to obtain. If so, how can we reliably obtain the fully strain-softened resistance (which is assumed to represent steady state) from constant-q tests

5 2. In order to mimic the effective stress path during infiltration of rainfall, a constant-q test is conducted with drainage permitted at either the top and/or bottom platen(s), one may wonder how static liquefaction (which is an undrained behaviour) can occur under drained shearing. This is somewhat related to how a constant-q test is conducted. The recent work of Chu and co-workers (Chu et al, 2007; Wanatowski et al, 2008a) showed that in a constant-q test, the core of the specimen could reach undrained instability even though the boundaries are drained, and even if the cell pressure were reducing at a very low rate of 1 kpa/min. Referring to Figure 2, the path B C is largely undrained and the continued reduction in p is achieved by generation of porewater pressure at the core of the specimen. This is because the rate of shearing as represented by dη/dt will increase at an ever increasing rate although the cell pressure is being reduced at a constant and slow rate. Thus, a large volumetric strain is generated suddenly and at a rate high enough to drive the core of specimen into undrained mode despite the presence of drained boundaries at one or both ends of the test specimen. Some simple mathematics showing the possibility of this happening is given in Appendix 1. In such case, p peak may or may not have any meaning of physical behaviour. It may simply be related to the specimen core switching into an undrained condition, leading to onset of strain softening. If a constant q-test is inherently difficult to be controlled properly, the data on p peak and c ss are difficult to be defined and determined in the laboratory and are prone to errors, resulting in large scatter of data. The ratio of two error prone values will lead to even larger scatter of data. This may perhaps explain the large scatter of results for the ratio of c ss / p peak presented in the HKIE Report. According to the law of probability, the lower bound decreases with the number of test data. If one continues to adopt a lower bound approach, the design approach for soil nail in loose fill will only become increasingly more conservative when more suspect and scattering data are available and included in the database. 3. It is also possible that apparent strain softening may be observed even if the core of the specimen does not switch to undrained shearing. This can occur in two ways. a. If the effective stress state hits the failure surface, further reduction in cell pressure will simply force the stress state to track downwards along the failure surface (Take et al, 2004). As the specimen is not perfectly uniform, a downward bend in the actual stress path will occur prior to reaching the failure surface, giving rise to apparent strain softening. This simple mechanism is illustrated in Figure 3. The observed p peak and the residual q value so observed are not related to soil instability. b. Although the rate of vertical strain rate, dε a /dt may be low initially, it will become high eventually partly because dη/dt is being increased at an ever increasing rate and partly because of the non-linear behaviour of soils. The actuator for applying the supposedly constant q value may not be able to keep up with the high axial strain rate, dε a /dt, associated with the constant-q condition and therefore q has to drop. The q-ε a response will then appear to strain soften as clearly illustrated by some of the very early constant-q tests performed by Lo (1985). This is not static liquefaction, but should be considered as experimental aberration. It is important not to formulate design theories based on experimental aberration

6 Uniqueness of effective stress path in undrained shearing An undrained condition means that the volumetric strain increment, dε vol, MUST be zero. One can easily demonstrate theoretically that the ESP during undrained shearing of a normally consolidated specimen in a given shearing mode is independent of direction of the total stress path as defined by dq/dp (see Appendix 2). This is a well-established concept in the area of constitutive modelling. A constant-q test simply implies dq = 0 and hence dq/dp must be zero! The path B C in Figure 2 and the steady-state strength c ss can be inferred from a constantcell-pressure undrained triaxial compression test (ie ICU or K o CU) conducted in a conventional displacement controlled mode. Such tests are much easier to control than the constant q-tests and it is much more likely to produce results with less data scatter. A soil can exhibit limited flow which is a temporary unstable behaviour (Lo et al, 2008). A constant-q test, by its nature, cannot distinguish the phenomenon of limited flow from flow behaviour of static liquefaction. The limited flow behaviour may then be wrongly interpreted as a flow phenomenon and the data are wrongly extrapolated to obtain the steady-state strength. This incorrect interpretation of flow behaviour will introduce further conservatism in the design process in determining the design strength. The HKIE report uses the term c ss and p' peak. In the HKIE Report, p' peak is defined as the stress point B in Figure 2. In a constant-cell-pressure undrained triaxial compression test, a peak point is more easily identifiable in the test result if the specimen manifests flow liquefaction. The recent work of Chu and co-workers suggested that the point B occur shortly after the effective stress state crossed the instability line obtained from constant-cell-pressure undrained triaxial compression tests. In a constant-q test, there is no distinct peak although point B may signify the onset of the core of the specimen attaining the mode of undrained shearing. The peak point is ill-defined in the ESP of a constant-q test. What exactly are p peak and c ss measured in constant-q tests From the above discussions, one can conclude that p peak and c ss may be a result of several things: soil behaviour, testing conditions, and experimental aberrations. Under some circumstances, the contribution may be dominated by the latter two factors. One must admit that a design guideline established based on a lower bound value of c ss or the c ss -p peak correlation, such as those in the HKIE Report, should lead to a very conservative outcome even if the reliability of the data is questionable and the physical meaning and nature of the database used to establish the lower bound values are not clearly established. While the theoretical concepts of c ss and p' peak and the typical range of c ss and p' peak for loose fill may be very interesting and important to researchers (if measured reliability!), practising engineers are at a loss as they cannot relate these parameters to some parameters which they are more familiar with. If we were to promote correct and wider use of a sound approach for design of soil nails in loose fill, one should be mindful that one must establish some simple albeit slightly conservative rules such that practicing engineers can relate the complicated parameters, such as p' peak, to some more easily quantifiable parameters, such as overburden pressure

7 Implication of instability line GEO is understandably nervous about loose fill. We have come across GEO engineers unwilling to approve cut slope formed in fill materials even if the proposed slope gradient is very gentle. For a retaining wall retaining loose fill, GEO seldom raise the concern of static liquefaction of the loose fill behind the wall, probably because geotechnical engineers have people have never seen such a phenomenon, except perhaps during an earthquake. Referring to Figure 4(a), the real explanation may be attributed to the fact that the stress state of the fill behind the wall, as indicated by the Mohr circle, is prevented from crossing, or even moving close to the instability line when the base resistance of the retaining can be mobilised rapidly. For a loose fill slope, the stress state as indicated by the Mohr circle may be close to the instability line (Figure 4(b)) and that is why geotechnical engineers are so nervous about steep loose fill slopes. There must be an intermediate scenario as perhaps shown in Figure 4(c) in which one can construct a tied-back retaining wall to reduce the gradient of fill slope low enough to reduce the risk of liquefaction to an acceptably low limit. With this approach, one should be less concerned about the liquefaction potential of fill behind the retaining wall because (a) the backfill is compacted and (b) the soil nail is in the right inclination to mobilize constrained dilatancy more effectively. The only concern is that the thickness of compacted fill may be less than 3m, which is against the well-established norm for treatment of loose fill slopes as recommended in the Geotechnical Manual for Slopes (GCO, 1984). The creation of gentler slope is in itself a robust design option of changing the stress state of the fill to ensure that the Mohr circle is not close to the instability line. If this is the case, it is no longer necessary to keep the conventional prescriptive rule of minimum 3m of recompaction. Failure kinematics The HKIE Report requires designer to basically perform a water tank design and the soil nails have to resist the "hydrostatic pressure" of the jelly fill (see Figure 5). This is a direct consequence of assigning a low jelly strength to fill material. If the material is so weak in strength, the loose fill simply behave like a heavy water. The grillage beam is in effect the structural wall of a water tank. As the hydrostatic pressure due to the jelly fill is high, the tieback force required to resist the uplift will also be high. This perhaps explains why the HKIE Report permits soil nails with steep inclinations to be used for anchoring the grillage beams. Although the water tank assumption may be conservative in terms of determining the disturbing force, it may not reflect the real kinematics of potential failure. If the actual failure kinematics is a downward flow/slip failure, the soil nails at such steep angle may not be effective in mobilizing the nail tension (Lo & Xu, 1992). The explains with Geoguide 7 (GEO, 2008) suggests typical inclinations to be between 5 o and 20 o. Grillage beam construction: the reality The report recommends the use of grillage beam. We have seen grillage beam constructed in the field. The construction workers dig trenches for constructing the grillage beams, making a terrible mess on the slope. The soils are loosened even further in the process. It is difficult to ensure proper compaction of the fill materials when backfilling around the trenches after completion of the grillage beam

8 Alternative options From a practical point of view, the use of steel reinforced shotcrete can provide a robust structural membrane for retaining the loose fill although it may be against the government policy of slope greening. The Authors have used steel reinforced shotcrete membrane reinforced by rebars in a private development project in Hong Kong. This is easier to construct, with less disturbance to soil. More importantly, it provides better protection against infiltration and behaves more convincing as a water tank. One can use available techniques of greening concreted slopes (GEO, 2000) or to use stone pitching to make the shotcrete surface look better and greener. If one is stuck with the only choice of treating loose fill as jelly and adopting a water tank approach for design of soil nails in loose fill, the use of soil nails is not an effective option of slope stabilization because if the soil nails are gentle, it is not effective in providing a sufficiently large force component to tie back the grillage beams. If the soil nails are steep, the actual kinematics may not ensure the development of nail tension and the nails may, in reality, fail in shear (which automatically implies also bending). The pullout resistance may not develop properly because failure in soil nails may be controlled by shear-controlled failure mode. A possible approach is to design the soil nails as micro-piles. Soil nails are installed in pairs as shown in Figure 7 to provide the truss action to resist lateral and/or upward loads more effectively than soil nails. Raking micro-piles have been commonly used and designed by the Authors for providing the lateral resistance of rockfall barriers in Hong Kong. A potentially liquefiable soil only liquefies when a number of necessary conditions are met. An alternative design paradigm is to prevent liquefaction of a potentially liquefiable soil, rather than using steady-state strength for design assuming static liquefaction to be the unavoidable design event. One can either provide drainage and vegetation to guard against saturation or provide enough mobilized resistance to ensure that the stress state will not approach point B in Figure 2. If one wants to be conservative, he can even adopt the twoprong approach discussed above, i.e. preventing both saturation and inhibiting the stress state from approaching point B. This may still be less conservative than current design paradigm. If one were to follow the current design approach of constructing grillage beams, will it be acceptable to provide a network of drainage pipes within the grillage beams and/or install additional subsoil drains to discharge the water quickly enough to prevent build up of water pressure near the surface? Will it also be acceptable to use shotcrete surface to prevent infiltration instead of grillage beams to reduce infiltration of rain water? If grillage beams are to be used, some backfilling will invariably be required to backfill the trenches after completion of grillage beams. It is recommended that soil cement be used for backfilling as a robust measure to minimize the adverse effect of poor workmanship. In early 1980s, Professor Peter Lumb initiated two final-year projects of soil cement fill (Woo, 1981; Li, 1983). The findings were that a small percentage of cement (less than 2%) could significantly enhance the strength of soils even for loosely compacted soils (Figure 8). Soil cement, particularly cement stabilized completely decomposed granite, would fail in a dilatant mode even if the relative compaction of the soils was as low as 75% (Figure 9). This is to be expected as the cement transforms the effective grading of the soil. A few percent of cement will do the trick and is an economic means to providing another safeguard for static liquefaction

9 FINAL REMARKS It is high time that the current design practice of soil nails in loose fill should be reviewed. The current approach is too costly and sometimes embarrassingly conservative, but not necessarily effective. Similar to dealing with natural terrain hazards, it is not possible to achieve zero risk. Geotechnical engineers are all about providing economic options with acceptably small risk and consequence of failure. The current approach is not failure-proof and there should be other acceptable options available. REFERENCES Bobei, D., Lo, S.R., Wanatowski, D., Gnanendran, C.T. and Rahman, M. (2009). A modified state parameter for characterizing static liquefaction of sand with fines, Canadian Geotechnical Journal, Vol.46, [3], Brand, E.W. (1981). Some thoughts on rain-induced slope failures, Proceedings of 10 th ICSMFE, Stockholm, Vol.3, Chu, J., Leroueil, S. and Leong, W.K. (2003). Unstable behaviour of sand and its implication for slope instability, Canadian Geotechnical Journal, Vol.40, Chu, J., Leong, W.K., Balasubramaniam, A., and Lo, S.R. (2007). Identification of possible failure mechanisms of tailing slope, Proc 10th ANZ Conference in Geomechanics, Oct 2007, Vol. 2, Chu, J, Lo, S-C.R. and Lee, I.K. (1993). Instability of granular soils under strain path testing, Journal of geotechnical Engineering, ASCE, Vol. 119, [5], Chu J., Lo, S-C.R. and Lee, I.K. (1992). Strain softening of granular soil in strain path testing, Journal of Geotechnical Engineering, ASCE, Vol. 118, [2], Eckersley, J.D. (1990). Instrumented laboratory flowslides, Geotechnique, Vol.40, [3], Geotechnical Control Office (GCO) (1984). Geotechnical Manuals for Slopes. Geotechnical Engineering Office (GEO) (2000). Technical Guidelines on Landscape Treatment and Bio-engineering for Man-made Slopes and Retaining Walls, GEO Publication No. 1/2000. Geotechnical Engineering Office (GEO) (2008). Geoguide 7 Guide to Soil Nail Design and Construction. Hong Kong Institution of Engineers (HKIE) (2003). Soil Nails in Loose Fill Slopes A Preliminary Study, HKIE Geotechnical Division. Leong, W. K., Chu, J., and Teh, C. I. (2000). Liquefaction and instability of a granular fill material, Geotechnical Testing Journal, ASTM, Vol. 23, No, 2,

10 Li, K.S. (1983). Strength of Soil-cement: Unconfined Compression and Indirect Tension Tests on Soil-cement Mixtures. Unpublished Final Year Project Report, Department of Civil Engineering, The University of Hong Kong.. Lo, S.R. (2007). Static liquefaction: confusion, myths and possibilities. Proc. Southeast Asian Geotechnical Conference, Lo, S. R., Rahman, M. and Bobei, D. (2008). Limited flow behaviour of sand with fines under monotonic and cyclic loading, Proc.2nd International Conference on Geotechnical Engineering for Disaster Mitigation and Rehabilitation, Nanjing. Lo, S-C.R (1985). Constitutive Relationships for Granular Soils, PhD thesis, The University of New South Wales. Lo, S-C.R. and Chu, J. (1991). Discussion on Instability of granular materials with nonassociated flow, Journal of Engineering Mechanics, ASCE, Vol.117, [4], 930. Lo, S-C.R. and Xu, D-W. (1992). A strain based design method for the collapse limit state of reinforced soil wall or slope. Canadian Geotechnical Journal, Vol. 29, [4], Morgenstern, N.R. (1978). Mobile soil and rock flows. Geotechnical Engineering, Vol.9, Pappin, J.W. (2003). Soil nails in loose fill slope the HKIE Report, Proc. International Conference on Slope Engineering, HKU, Rahman, M. and Lo, S. R. (2009). Equivalent granular state parameter and undrained responses for sand with fines, Proc. 17 th International Conference on Soil Mechanics and Geotechnical Engineering, Alexandria, Oct Take, W.A., Bolton, M.D., Wong, P.C.P. and Yeung, F.J. (2004). Evaluation of landslide triggering mechanisms in model fill slopes, Landslide, Vol.1, Wanatowski, D., Chu, J., and Lo, R.S-C. (2008). Strain softening behaviour of sand in strain path testing under plane strain conditions. Acta Geotechnical, Vol. 3, [2]: Wanatowski, D., Chu, J. and Lo, S.R. (2008a). Types of flowslide failures and possible failure mechanisms, Proc. 2nd International Conference on Geotechnical Engineering for Disaster Mitigation and Rehabilitation, Nanjing, Springer, Woo, P.C. (1981). The Strength of Soil-cement. Unpublished Final Year Project, Department of Civil Engineering, The University of Hong Kong

11 NOTATIONS AND DEFINITIONS denotes effective stress triaxial compression is defined as corresponding to Lode angle = 0 superscript e denotes elastic component superscript p denotes plastic component subscript ss denotes steady state σ 1 = major principal stress σ 3 = minor principal stress q = σ 1 σ 3 p = ( σ 1 + 2σ 3 )/3 η = q/p ε 1 = major principal strain = axial strain, ε a, for triaxial compression. ε vol = volumetric strain

12 Figure 1 Concept of instability line (reproduced from Take et al, 2004)

13 Figure 2 Behaviour of soil element on failure surface (reproduced from HKIE, 2003)

14 Figure 3 Constant q test

15 Figure 4 Stress state of soil element in loose fill (a) behind retaining wall (b) on steep slope and (c) on gentle slope

16 (a) Design surface pressure on slope (b) Layout of surface grillage & soil nails Figure 5 Design of soil nails in loose fill based on HKIE report (reproduced from HKIE, 2003)

17 Figure 6 Support of grillage beam / shotcrete by micro piles

18 Figure 7 Unconfined compression strength against cement content for decomposed granite (reproduced from Li, 1983)

19 Figure 8 Critical dry density versus cement content for Hong Kong soils (reproduced from Woo, 1981)

20 Appendix 1 : rate of shearing in a constant-q test Both the axial and volumetric strain behaviour of a soil are governed by η and dη where η = q/p, q = σ 1 σ 3 and p = (σ 1 + 2σ 3 )/3. In general, we have (simply by differentiation) the following relation: dq q dq η dη = dp = dp p 2 p p ( p ) η For constant-q test, dq=0 and therefore: dη = dp p Therefore, shearing as represented by increase in η is achieved by ramping down the cell pressure. The rate of shearing is thus given by: dη = dt η dp p dt where dp /dt = rate of ramping down the cell pressure. Since dp < 0 and dη > 0 in a constantq test, dη/dt will occur at an ever increasing rate even if dp /dt is maintained at a constant small value. For example, for an initial stress state of major and minor principal stresses of 200 and 100 respectively and if matric suction increases the equivalent effective stress by 75 kpa, q A = 100 and p A = Even with a very slow ramping rate of 1 kpa/min, dη/dt will reach (i.e. ~5.5 times the initial dη/dt) and η = (taking a mobilised friction angle of ~28 o ) after 120 minutes. This is a fast rate of shearing for a stress-controlled test at high η-value. This concept is further explained in Figure A1. The ramping of cell pressure is performed at discrete steps and with each step corresponding to either the accuracy of the pressure transducer and/or the control, which is generally 0.5 kpa. This means the actual dη/dt is much higher than the calculated value based on an ideal and smooth ramping rate of 1 kpa/min. In another word, the specimen will eventually be subjected to pulses of very fast shearing. Before the core of the specimen attain an undrained behaviour, i.e. the whole specimen is drained. But the very high rate of strain generated may drive the core of the specimen into an undrained state

21 Note: c ss = q ss /2 Path Constant q test Field A A A or (A B) B B C start of ramping down start of wetting of cell pressure drained throughout gradual loss of matric specimen suction Ill-defined state jumping into undrained shearing because dη /dt is high and η is high. core of specimen in - undrained state Figure A1-20 -

22 Appendix 2 : Uniqueness of effective stress path in undrained shearing An undrained condition is mathematically defined by dε vol = 0. For a soil element under shearing, the strain increment can be decomposed into an elastic and a plastic component. Therefore, undrained shearing MUST obey: d ε vol = dε e vol + dε p vol = 0 where superscripts e and p denote elastic and plastic components respectively. Theoretically, there is a subtle approximation here. We neglect the so-called elasto-plastic coupling which can be important if many cycles of load repetitions are involved. If we accept effective stress principle: dε dε e vol p vol = = [ ] D e [ ] D p dq dp dq dp Undrained shearing can be expressed as: dε vol = dq dp dq dp e p [ D] + [ D] = 0 Simplifying: dε vol = e p dq {[ D] + [ D] } = 0 dp Although the terms of [D] e and [D] p are complicated functions of effective stress state (q, p, and Lode angle), they are independent of dq/dp. For triaxial condition, [D] e and [D] p are 2 2 matrices. For any triaxial compression mode (ie, 1 major and two minor principal stresses) and Lode angle = 0, the above equation can be solved by stepping dp or dq. The solution is the effective stress path in undrained shearing. Re-iterating that [D] e and [D] p are independent of dq/dp, the mathematics ensures that for a given initial stress state, the solution will generate a single curve. To put it in another way, whether it is conventional triaxial compression (dq/dp = 1/3), or other tests with any value of dq/dp (including the case of zero for constant-q test), we have the same answer. This concept is further explained in Figure A

23 Effective stress path BC is approximately same as effective stress path B C if we configure the test so that B and B corresponds to each other. Hence q ss /p is the same for both tests. p is can also be related to initial effective stress. Figure A2-22 -