efektİf gerİlmeler veya kayma mukavemet …efektİf gerİlmeler veya kayma mukavemetİ...

EFEKTİF GERİLMELER VEYA KAYMA MUKAVEMETİ

PROFİLLERİNE DAYALI DOLGU STABİLİTE

ANALİZLERİNIN GÜVENİLİRLİĞİ

RELIABILITY OF STABILITY ANALYSES OF EMBANKMENTS

BASED ON EFFECTIVE STRESSES OR SHEAR STRENGTH PROFILES

Barış TRAK1

ABSTRACT

The paper questions the reliability of stability analyses of embankments based on effective

stresses or shear strength profiles. Effective stress analyses, although theoretically correct,

present problems that may affect the accuracy of the computed factors of safety. Stability

analyses based on shear strength profiles seem to offer a better solution for embankments

built in one stage, or for the short-term stability of cut slopes. Semi-empirical methods are

discussed in relation to the determination of shear strength profiles leading to more reliable

designs.

Keywords: stability, effective stress, shear strength profiles, embankments

ÖZET

Bu bildiri efektif gerilme veya kayma mukavemeti profillerine dayalı dolgu stabilite

analizlerinin güvenilirliğini sorgulamaktadır. Teorik açıdan doğru olmalarına rağmen,

efektif gerilme analizleri güvenilirlik katsayısının hesaplanmasında sorunlar yaratmaktadır.

Kesme mukavemeti profillerine dayalı stabilite analizleri ise, tek etapta inşa edilen

dolguların, ya da şev kazılarının kısa süreli stabilitesi için daha iyi bir çözüm sunmaktadır.

Bildiride daha güvenilir tasarımlar elde etmek icin gereken kayma mukavemeti

profillerinin belirlenmesi için kullanılan yarı ampirik metotlar da tartışılmaktadır.

Anahtar kelime: stabilite, efektif gerilme, kayma mukavemeti profilleri, dolgular

1Technical Director (ret.), Council of Europe Development Bank, Paris, France; e-mail: [email protected]

169

7. Geoteknik Sempozyumu 22-23-24 Kasım 2017, İstanbul

1. INTRODUCTION

Safety analyses of embankments are usually conducted in terms of total stresses (the so-

called φ = 0 method) using the undrained shear strength of the clay foundation, mainly

because it is simpler and easier to determine this parameter by lab or in situ tests. However,

several publications have shown that this approach may lead to unsafe designs (Bjerrum,

1972; Pilot, 1972), and empirical formulas were proposed to correct this anomaly.

When the stability problem is more complex, as in the case of stage-constructed

embankments, and measurements of pore pressures are available, effective stress analyses

are preferred (Pilot et al.,1982). However, as shown by Tavenas et al. (1980), an effective

stress analysis will be correct only if true effective stresses and stress paths, as well as

actual failure mechanisms, are accounted for in the methods of analysis.

The paper discusses the reliability of both methods of stability analysis and makes

recommendations for safer designs.

2. STABILITY ANALYSES BASED ON EFFECTIVE STRESSES

Considering the determining influence of effective stress conditions on soil behaviour

(Terzaghi, 1925; 1936), it would be logical to conduct stability analyses in terms of

effective stresses (Janbu, 1977; Tavenas et al., 1978), particularly in the case of stage-

constructed embankments where variations of pore pressures under loading are properly

monitored. However, Tavenas et al. (1980) have shown that effective stress analyses

present unfortunately major weaknesses, such as:

2.1 Magnitude of assumed effective stresses

Effective normal stresses used in the stability analysis may differ significantly from those

prevailing just prior to failure.

Almost all stability analysis methods are based on the assumption that the vertical stress σv

acting on the failure surface in a slope or an embankment foundation is equal to the weight

of the soil above the considered point. This assumption about σv simplifies the stability

calculation significantly; it was justified by the impossibility of correctly determining exact

stress values in the earth mass.

Fig.1 shows effective normal stresses σn’ obtained from Bishop’s stability analysis and

those determined by La Rochelle (1960) who investigated the stress distribution in an

excavated slope by means of photo-elasticity. Differences between the two stress

distributions are very important, corresponding to a strong overestimation of σn’ and thus,

of the available clay strength along the upper part of the slip surface, and to an even more

important underestimation of σn’ and τ at the toe of the slope (Tavenas et al., 1980).

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Fig.2 shows a comparison between normal stresses obtained from a FEM analysis using a

nonlinear elasto-plastic stress-strain relationship and those computed in Bishop’s stability

analysis of an embankment. Effective stresses presumed to act on the section of the slip

surface beyond the toe of the embankment are very small, as compared with the well-

defined values in the elasto-plastic solution. Inversely, under the center part of the

embankment, the normal stresses computed in the Bishop’s method are quite in excess of

the elasto-plastic solution (Tavenas et al., 1980).

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2.2 Assumed mobilization of the clay strength

The degree of mobilization of the clay strength is expressed mathematically in a way

that implies a very specific stress path up to failure, seldom encountered in practical

problems. As a result the local value of F is most likely in error.

In all methods of analysis, the local factor of safety at any point in a stable clay mass is

obtained by comparing the applied shear stress τ and the normal effective stress

σ’ to the shear stress at failure τf under the same normal effective stress:

F = τf / τ = (c’ + σ’ tanφ’) / τ Eq. (1)

This definition implies a very specific stress path up to failure (Fig.3). If effective stress

paths in the investigated problem correspond to that shown in Fig.3, then Eq. (1) will

actually represent the degree of mobilization of clay strength.

However, in most practical problems, effective stress paths followed up to failure will

be different, as shown below.

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Fig.4 shows a point in a clay mass under an embankment (subscript e) and behind a

slope (subscript s) at a computed factor of safety F and submitted to two effective stress

conditions Xe and Xs theoretically satisfying Eq. (1). In order for F to be the true picture

of the degree of mobilization of the clay strength, stress paths followed from Xe and Xs

to failure must correspond to 1e or 1s.

In the case of an embankment under construction, Leroueil et al. (1978) have shown that

during the last stages of construction, the effective stress path beyond Xe corresponds to

σv’ = Cst, i.e. a line such as 2. The strength at failure is (σ1 – σ3)fe and the local factor of

safety is thus:

Fe = (σ1 – σ3)fe / (σ1 – σ3)Xe Eq. (2)

This is clearly different than the one implied in Eq. (1).

If the clay has strain-softening characteristics, the stress condition (fe) corresponds to

the onset of the strain-softening process, developing along the Mohr-Coulomb envelope

down to the critical state CSe.

In the case of the slope of an excavation in overconsolidated clay, the effective stress

path from Xs up to failure could be such as 3. The strength available at initial failure (fs)

or at the critical state (CSs) is much in excess of that implied in Eq. (1), so that this

equation leads to a significant underestimation of the true factor of safety (Tavenas et

al., 1980).

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2.3 Equation of local and overall stability conditions

The equation between local and overall values of F is valid only under particular

circumstances that are generally impossible to meet.

In all methods of stability analysis, the degree of mobilization of the local shear strength

of the clay is assumed to be identical at all locations along the potential failure surface

and equal to the overall stability condition. In order for this assumption to be

acceptable, one of the following sets of conditions should be met:

1) A computed factor of safety, F = 1.0, in perfectly plastic materials. In this case,

where no strain-softening occurs, the shear strength of the clay can be mobilized

simultaneously at all points along the failure surface;

2) If F > 1.0, then a perfectly uniform stress state and a unique stress path up to

failure must prevail at all points in the considered clay mass; otherwise, the degree of

mobilization of the clay strength would vary from point to point along the potential

failure surface! The first condition is very seldom satisfied and the second one, clearly

impossible in all practical cases (Tavenas et al., 1980). Wright et al. (1973) determined

the local factor of safety along the failure surface in a slope with an overall factor of

safety of 1.0, using an elastoplastic finite-element analysis. A typical result is shown in

Fig.5, indicating systematic and large differences between the local and overall factors

of safety.

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3. STABILITY ANALYSES BASED ON STRENGTH PROFILES

The concept of critical state is applicable to natural clays (Trak, 1980); this implies that

a clay loaded up to failure will necessarily achieve its critical state either immediately

(perfectly plastic clays) or after some strains (strain-softening clays). This critical state

strength can be used to characterize the minimum strength available under

embankments at failure (Trak et al., 1980), or behind unstable slopes. As shown above,

the critical state stress condition represents the imposed termination of all effective

stress paths. It is located on the Mohr-Coulomb envelope of the normally consolidated

clay and its position is a unique function of the void ratio or the preconsolidation

pressure of the clay.

In a clay deposit characterized by its preconsolidation (σp’) profile and the

corresponding void ratio (e) profile, the critical state strength is a unique function of the

depth. It can therefore be used in the same manner as an undrained shear strength

parameter in a so-called φ = 0 analysis. Such an analysis is applicable to all cases where

the void ratio of the clay can be assumed constant. This is essentially true for

embankments built in one stage or for the short-term stability of cut slopes or

excavations. These stability analyses are simpler than effective stress analyses and many

of the weaknesses discussed above in relation to effective stresses are automatically

eliminated.

Trak et al. (1980) proposed an approach to the stability analysis of embankments based

on an interpretation of Bjerrum’s (1972) data made by Mesri (1975) who observed that

the mobilized shear strength at failure (cu(mob)) under an embankment is nearly

independent of the plasticity index and is a direct function of the preconsolidation

pressure (σp’) of the clay deposit. This relationship was defined by the following

formula:

cu(mob) = 0.22 σp’ Eq. (3)

By analyzing a large number of failure cases, Trak et al. (1980) gave evidence that

almost all of the methods proposed to determine the stability of clay deposits, namely

Bjerrum’s (1972), SHANSEP (Ladd and Foott, 1974) and the USALS (La Rochelle et

al., 1974) methods make use of a practically constant shear strength value given by

Eq. (3). It is interesting to note that the above given expression for cu(mob) for

overconsolidated clays is very similar to the empirical “rule of thumb” given by Puzrin

et al. (2010): cu = 0.21 σvo’ for normally consolidated clays, where σvo’ is the effective

overburden pressure.

4. DISCUSSION

Effective stress methods of stability analysis present major weaknesses, because they

are based on assumptions that do not correspond to actual stress conditions prevailing in

situ. Stability analyses based on a clay strength independent of the applied effective

stress present fewer sources of error than the effective stress methods, and appear to be

better suited for design purposes.

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While such analyses are usually referred to as the φ = 0 method, with an implied

reference to undrained conditions, it is preferable that their use should be based on the

critical state strength, the fixed and predictable termination of all effective stress paths.

As shown above, this strength corresponds to the minimum value of the shear strength

of the clay, mobilized at complete failure.

The mobilized shear strength value is a direct function of the preconsolidation pressure,

which is probably the most important characteristic of a clayey soil. It is therefore

essential that its value be determined by careful testing on good quality samples.

However, when the determination of the preconsolidation profile is difficult or too

costly, empirical relationships such as the one proposed by Kootahi and Mayne (2016)

between index properties and σp’ could be used, but the former should preferably be

determined by objective, reliable and repeatable tests, such as the fall cone (Hansbo,

1957; Karlsson, 1961; Garneau and LeBihan, 1977), instead of the classical Casagrande

methods (Trak, 2017a).

Using the mobilized strength profile, it must be recognized that this value represents the

minimum value of the available strength of the foundation clay, achieved at complete

failure (F = 1.0). For design purposes, it is therefore recommended to carry out the

φ = 0 stability analysis using the mobilized shear strength to determine the embankment

height Hf corresponding to the failure condition (F = 1.0), and choose the embankment

height H corresponding to the desired performance level, conserving the same

geometry. Then, if needed, a “safety factor” can be defined as F = Hf / H, since the

geometry is the same (Trak, 2017b).

5. CONCLUSIONS

The critical review of stability analysis methods based on effective stresses or shear

strength profiles leads to the following main conclusions:

1) Although theoretically correct, the methods of stability analysis in terms of

effective stresses are based on questionable assumptions that contradict the fundamental

aspects of shear strength mobilization in clay foundations under loading. They should

therefore be used with caution.

2) Stability analyses based on shear strength profiles, using the φ = 0 method and

the critical state strength, appear to offer a better solution, because they are free from

most of the errors associated with effective stress analyses.

3) Using the mobilized strength profile, it must be recognized that this value

represents the minimum value of the available strength of the foundation clay, achieved

at complete failure.

4) The mobilized shear strength value is a direct function of the preconsolidation

pressure, which is probably the most important characteristic of a clayey soil. It is

therefore essential that its value be determined carefully in any project involving clay

foundations under embankment loading.

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Conference on Performance of Earth and Earth-supported Structures, Purdue

University, Lafayette, 2, pp 1-54.

[2] Pilot, G. (1972), “Study of five embankment failures on soft soils”, Proceedings,

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[3] Pilot, G., Trak, B. and La Rochelle, P. (1982), “Effective stress analysis of the

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[4] Tavenas, F, Trak, B. and Leroueil, S. (1980), “Remarks on the validity of stability

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[17] Puzrin, A. M., Alonso, E. E. and Pinyol, N. M. (2010), “Geomechanics of

Failures”, Springer Dordrecht, Heidelberg, London, New York.

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