The effect of preloading on the liquefaction cyclic strength of mixtures of sand and silt

Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

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The effect of preloading on the liquefaction cyclic strength of mixtures of sand and silt C.A. Stamatopoulos a,n, F. Lopez-Caballero b, A. Modaressi-Farahmand-Razavi b a b

Hellenic Open University; Director, Stamatopoulos and Associates Co, 5 Isavron str, 11471 Athens, Greece Laboratoire MSS-Mat CNRS UMR 8579, Ecole Centrale Paris, Grande Voie des Vignes, 92290 Châtenay-Malabry, France

art ic l e i nf o

a b s t r a c t

Article history: Received 20 January 2015 Received in revised form 3 July 2015 Accepted 8 July 2015

The paper studies the effect of preloading on the liquefaction cyclic strength of silty sands in the free field condition. This effect first is investigated by cyclic shear tests where horizontal shear stress oscillated about a zero mean value. Samples with varying fines content and at varying pre-stress ratios, densities and vertical stresses are tested. Test results show a marked increase of the liquefaction cyclic strength with the pre-stress ratio. The effect is more pronounced for tests with less liquefaction cyclic strength without pre-stress. Using critical state soil mechanics concepts, factors simulating the effect of preloading on the liquefaction cyclic strength are identified and based on the results of the laboratory program an empirical expression is proposed predicting the increase in the liquefaction cyclic strength induced by pre-stress. This expression is validated by numerical simulation of the relevant laboratory tests using an elastoplastic multi-mechanism model. In addition, based on the derived expression, a methodology is proposed predicting the increase in liquefaction cyclic strength as a result of preloading in the field in the case of the free field condition. This methodology is validated by the comparison with field measurements on liquefaction-susceptible soils before and after the field application of preloading. Last but not least, the increase in liquefaction cyclic strength which the proposed methodology predicts for typical soil profiles and embankment preloads is predicted and discussed. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Liquefaction Preloading Cyclic shear laboratory tests Sands Silty sands Silts Numerical analyses Critical state

1. Introduction The liquefaction cyclic strength, as defined hereunder, determines whether or not saturated sandy layers in the free field condition run the risk of earthquake-induced liquefaction [1,2]. The factor of safety against liquefaction below horizontal ground surfaces is defined, versus depth, as the ratio of the in-situ liquefaction cyclic strength by the cyclic stress ratio resulting from the design earthquake [2]. In cases where this factor takes values close to, or less than, unity, soil improvement is an effective way to mitigate the liquefaction risk by increasing the liquefaction cyclic strength. Preloading is a temporary loading, usually an embankment, applied at a construction site to improve subsurface soils primarily by increasing density and horizontal stress [3,4]. Preloading requires simpler equipment than other methods of soil improvement and is often less expensive [3]. The liquefaction cyclic strength of sands has been studied extensively in the laboratory, especially in the triaxial device, but also in the simple-shear and torsional-shear devices. Important factors which affect the liquefaction cyclic strength of sand samples are the void ratio, the consolidation stress and the content of fines:

n

Corresponding author. E-mail address: [email protected] (C.A. Stamatopoulos).

http://dx.doi.org/10.1016/j.soildyn.2015.07.004 0267-7261/& 2015 Elsevier Ltd. All rights reserved.

Tests in the sands of Toyoura [2], Ottawa [5,6] and Monterey [6] show a considerable increase in the cyclic strength as the sand void ratio decreases. In addition, it has been observed that as the consolidation stress increases, the cyclic strength at similar void ratio decreases [7]. Furthermore, laboratory tests show that at similar void ratio and confining stress, the presence of fines up to at least about 25% of the total weight decreases the liquefaction cyclic strength [8,9]. Recently, the effects of both the consolidation stress and void ratio on the liquefaction cyclic strength has been simulated by only one parameter, the state parameter, defined hereunder [10,11]. Furthermore, for sand–silt mixtures it has been observed that the relationship between the state parameter and the liquefaction cyclic strength for soil samples with different fines content tested in the triaxial device (a) is unique for the same sample preparation method [12] and (b) differs for different sample separation methods [13]. In addition to the effect of density, consolidation stress and fines content on the liquefaction cyclic strength of sand samples prepared in a similar manner, described above, the liquefaction cyclic strength measured in the laboratory depends also on the preloading (over-consolidation or pre-stress) of the soil sample before the application of cyclic loading [14–23]. Also, using a totally different technique, dynamic geotechnical centrifuge testing, Adalier and Elgamal [24] studied the preloading effect on

190

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

Nomenclature a1, a2 Dr eo f1, f2, f3, fc

Fitting parameters of Eq. (9) Relative density Initial (prior to consolidation) void ratio f4 Functions Content of fines ¼Weight of fines per total weight of mixture Mea., Pred. Measured, predicted Nf Number of cycles to liquefaction N60 Blow count number measured in the Standard Penetration Test PR Pre-stress ratio of samples prepared in the laboratory (Eq. (2)) PRfield Field pre-stress ratio (Eq. (15b)) p0 Octahedral effective stress p0 o Octahedral effective stress prior to the application of cyclic loading Pa Atmospheric pressure (equals about 100 kPa) qc Resistance measured in the cone penetration test R2 Coefficient of correlation RPR SR15-PR/SR15-1 Rfield SR15-after/SR15-bef SR Stress ratio (¼ τcyc/σ0 vo) under Ko consolidation (Eq. (1))

cyclic liquefaction and subsequent ground subsidence in clean saturated sand deposits based on much larger soil models. Cyclic laboratory tests performed on samples of the same soil, consolidated at different pre-stress ratios, can illustrate the effect of pre-stress on the liquefaction cyclic strength. Based on results of such tests, empirical equations predicting the effect of pre-stress on the liquefaction cyclic strength of sandy soils have been proposed [15,19,20]. However, these equations were based on tests performed on specific sands. As the liquefaction cyclic strength greatly depends on fines content [8,9], the effect of pre-stress on the liquefaction cyclic strength may depend on the fines content of the soil. Sophisticated elasto-plastic analyses predicting the response of liquefaction-susceptible soils under earthquake loadings,, have been developed and extensively validated [25,26]. Such analyses can be used to validate equations predicting the effect of pre-stress on liquefaction-susceptible soils measured in the laboratory. Furthermore, a number of case studies are reported in the bibliography where field data exists, which allows the estimation of the in-situ liquefaction cyclic strength both prior and after the field application of preloading in the free field condition [3,4,27– 30]. These case studies can provide additional data to validate the equations predicting the effect of preloading on liquefactionsusceptible soils measured in the laboratory. The purpose of the paper is to propose and validate a simple equation and associated methodology simulating the effect of preloading on the liquefaction cyclic strength of any liquefactionsusceptible soil in the free field condition. In order to achieve this, the paper below (a) defines the liquefaction cyclic strength of sands and illustrates that the constant-volume cyclic shear tests are suitable to investigate the effect of preloading on this strength, (b) performs in the shear device constant-volume tests in samples where the fines content in combination with the soil density, the consolidation stress and the prestress ratio vary, (c) identifies factors simulating the effect of pre-stress on the liquefaction cyclic strength using critical state soil mechanics concepts and, based on the results of the laboratory program, proposes an empirical expression predicting the increase in the liquefaction cyclic strength

SR15

Liquefaction cyclic strength, or the cyclic stress ratio (SR) causing liquefaction in 15 uniform cycles of loading. SR15-1 SR15 at PR ¼1 SR15-I SR15 at PR ¼i SR15-bef SR15 in the field before the application of preloading SR15-after SR15 in the field after the application of preloading Vs Shear wave velocity Δσ0 ν The maximum additional effective vertical stress applied during the preload process Γ, λ, ξ Parameters of the critical state line (Eq. (12)) ΔU Excess pore pressure εi (i¼1–3) Principal Strains Ko Earth pressure coefficient at rest θ, A Factors given by Eqs. (A2) and (A3) ν Poisson Ratio σʹh Effective horizontal stress σʹv Effective vertical stress σ0 vo, σ0 ho Effective vertical and horizontal stress prior to the application of cyclic loading σ0 v-c Maximum past effective vertical stress σ0 i (i¼1–3) Principal effective stresses τcyc Cyclic shear stress (half peak-to-peak) φ0 Friction angle at the critical state ψ State parameter defined by Eq. (11)

induced by pre-stress, (d) performs numerical runs simulating the effect of pre-stress on constant-volume cyclic laboratory tests to validate the proposed empirical expression, (e) proposes a methodology predicting the increase in the liquefaction cyclic strength as a result of preloading in the field based on the proposed empirical expression, (f) validates the proposed methodology by the comparison with measurements found in the literature on liquefactionsusceptible soil layers before and after the field application of preloading and (g) discusses the increase in liquefaction cyclic strength which the proposed methodology predicts in the field for typical soil profiles and embankment preloads.

2. Appropriate laboratory tests measuring the effect of preloading on the liquefaction cyclic strengthin the free field condition and previously proposed empirical expressions 2.1. Appropriate laboratory tests As described by Seed and Peacock [14], below horizontal ground surfaces prior to the application of cyclic loading, the shear stresses are zero. In addition, the effective vertical stress, denoted as σʹvo, equals the overburden effective pressure. Furthermore, the ratio of the effective horizontal stress, denoted as σʹho, to σʹvo, is given by the coefficient of earth pressure at rest, Ko. As a result of an earthquake, dynamic loading is applied primary in the horizontal direction. When harmonic shear horizontal loading is applied, a cycle of loading is defined as the complete change of the horizontal shear stress (i) from zero to τcyc, (ii) from τcyc to
τcyc and (iii) from
τcyc back to zero. In this case, the cyclic stress ratio SR is defined as SR ¼ τcyc =σ 0vo

ð1Þ

In addition, cyclic shear strain during a loading cycle is defined as the maximum value of shear strain attained. Permanent strain, or permanent pore water pressure, is the part of the strain, or pore water pressure, which accumulates at the end of each cycle of loading.

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

As a result of cyclic horizontal harmonic loading on Koconsolidated soil about a zero mean shear stress value, permanent pore pressure and cyclic shear strain build up with cycle number, while due to one-dimensional symmetry, considerable permanent horizontal movement does not accumulate [14]. Liquefaction cyclic strength for N cycles of harmonic loading, SRN, is defined as the value of the cyclic stress ratio SR causing liquefaction in N uniform cycles. Liquefaction is the state where the effective stress becomes very small and the cyclic shear strain very large. In a more functional definition this state is defined in the present work, similarly to Ishihara [2], as when the cyclic shear strain exceeds 2.5%. In liquefaction analyses, a reference earthquake of magnitude M ¼7.5, which corresponds to 15 cycles of uniform cyclic loading, is often used [1,2]. For this reason, the liquefaction cyclic strength SR15 will be used below as index to study the effect of preloading on the liquefaction cyclic strength in the free field condition . In addition, it should be noted that it was observed that if the liquefaction cyclic strength is taken as SR10 or SR20 instead of SR15, the results presented below do not change considerably. Preloading is usually applied at horizontal ground and the temporary loading is usually a soil embankment. When the soil embankment is applied, below the embankment and beyond a certain distance from the lateral boundaries of the embankment, lateral displacement is small. At this region, as elastic theory predicts and recent field measurements verify, the horizontal stress increases [29]. After the removal of the surcharge, the new stress σ0 vo may show minor differences as compared to the preloading values, primarily due to increased density, while, as recent field measurements indicate, σ0 ho may have increased considerably [30,31]. The liquefaction cyclic strength is measured in the laboratory by cyclic undrained or constant-volume tests using devices which can be classified into three types: (a) tests on the triaxial device with isotropic consolidation at stress and then oscillation of the vertical stress, (b) tests on the direct-shear or the simple-shear devices where samples are subjected to one-dimensional consolidation at vertical stress σʹvo and then to horizontal oscillation of the shear stress between τcyc and
τcyc and (c) tests on the torsional-shear device where samples are subjected in the triaxial chamber to consolidation at vertical stress σʹvo and any horizontal stress and then to horizontal angular oscillation of the shear stress between τcyc and
τcyc [14,15]. In tests (b) the ratio of the effective horizontal to the effective vertical stress prior to the application of cyclic loading is the coefficient Ko. Without preloading, the cyclic simple-shear and direct-shear tests simulate better field conditions than the cyclic triaxial tests, as only these tests simulate the initial anisotropic state of stress below horizontal ground surfaces , as well as the primary direction of loading under earthquakes [20]. Preloading affects the liquefaction cyclic strength due to (a) the increased density under Ko conditions, (b) the increased horizontal stress and (c) changes in the sand fabric [8,17]. Cyclic direct-shear and simple-shear tests performed by applying different maximum vertical effective stress on samples of the same initial void ratio and final consolidation stress, prior to exerting the same cyclic stress under undrained conditions can simulate all the above effects. Cyclic triaxial tests cannot simulate at least the effect (b) above. As cyclic direct-shear and simple-shear devices (a) without pre-stress simulate better field conditions during earthquakes and (b) with pre-stress simulate all effects of preloading on the field liquefaction cyclic strength, they are preferable for simulating the increase in the liquefaction cyclic strength induced by preloading in the free field condition. 2.2. Form of previous proposed empirical expressions In all the triaxial, simple shear, direct simple shear and tortional shear devices the effect of prestressing on the liquefaction cyclic strength has been studied [8,14–23]. Some of these

191

works attempted to simulate the increase of liquefaction cyclic strength with pre-stress, by empirical equations [15,19,20]. In all these equations: (a) the effect of pre-stress was described by the Prestress Ratio, PR PR ¼ σ 0v
c =σ 0vo

ð2Þ

where σʹv-c is the maximum effective vertical stress exerted during the consolidation process and σ0 vο is the effective vertical stress just prior to the application of cyclic loading and (b) PR is related to the normalized liquefaction cyclic strength RPR defined as RPR ¼ ðSR15
PR =SR15
1 Þ

ð3Þ

where “1” indicates the state at PR ¼ 1 (without pre-stress) of the same soil and at the same consolidation stress and preconsolidation void ratio.

3. Current program of laboratory tests, results and their consistency The materials used to prepare the mixtures tested were (a) a clean sand and (b) a non-plastic silt. The first material is a subwhite natural sand of quartz from the Egyptian desert. Its quartzy grains are not lustrous but are well rounded, transparent and colorless. There are very few black grains of iron oxides or magnetic oxides of unknown provenance (less than 0.1%), whose existence could be due to manmade contamination. The non-plastic silt was obtained by grinding natural deposits of quartz from the area of Assirou near Thessaloniki, Greece. The materials were provided by Prof. Tika of Aristotle University of Thessaloniki. The measured specific gravity of grains was 2.65 for the sand and 2.64 for the silt. Papadopoulou and Tika [9] and Stamatopoulos [11] give (i) the grain size distribution of the sand and silt materials and (ii) the maximum and minimum void ratio in terms of the fines content of the mixtures. In accordance with the previous discussion, in the current testing program the specimens were subjected to cyclic constant-volume loading oscillating about a mean zero shear stress in samples Table 1 Current laboratory program. States studied (in terms of fc, void ratio before consolidation (eo) and σʹvo) and their liquefaction cyclic strength in terms of PR. No.

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25 S26

Initial conditions fc

eo

σ'vo (kPa)

0.00 0.00 0.00 0.00 0.00 0.00 0.15 0.15 0.25 0.25 0.25 0.25 0.25 0.40 0.40 0.40 0.40 0.60 0.60 0.60 0.80 0.80 1.00 1.00 1.00 1.00

0.79 0.79 0.67 0.67 0.58 0.58 0.67 0.67 0.72 0.72 0.67 0.67 0.67 0.77 0.77 0.50 0.50 0.89 0.89 0.56 1.00 1.00 1.10 1.10 0.90 0.90

50 150 50 150 50 150 50 150 50 150 50 150 150 50 150 50 150 50 150 150 50 150 50 150 50 150

No. of tests

Liquefaction cyclic strength SR15-1

13 17 16 31 6 8 4 11 10 8 8 9 9 20 12 8 7 10 14 4 8 10 14 18 8 9

0.20 0.15 0.20 0.18 0.24 0.19 0.15 0.13 0.14 0.08 0.16 0.09 0.12 0.11 0.11 0.20 0.15 0.12 0.08 0.16 0.10 0.10 0.07 0.08 0.12 0.13

SR15-2

0.18 0.26 0.20

0.14 0.13

0.10

0.10 0.11

SR15-3 0.26 0.21 0.27 0.25 0.33 0.24 0.21 0.18 0.18 0.13 0.21 0.13 0.17 0.15 0.15 0.28 0.19 0.17 0.13 0.22 0.15 0.14 0.13 0.13 0.18 0.19

SR15-4

0.22 0.30 0.27

0.18 0.17

0.15

0.16 0.16

25 20 15 10 5 0 -5 -10 -15 -20 -25

Shear cyclic strain [%]

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

Shear cyclic stress [kPa]

192

0

500

1000

1500

2000

12 10 8 6 4 2 0 -2 -4 -6

0

500

1000

1500

2000

Time [sec]

Time [sec]

0.3

0.3

0.2

0.2

SR [.]

SR [.]

Fig. 1. Constant-volume cyclic shear tests. Typical results of a test. The case of fc¼ 0, eo ¼ 0.67, σʹvo ¼ 150 kPa, PR ¼3 is given.

0.1

0.1

0.0

0.0 1

10

100

Number of cycles to liquefaction [.] PR=1

PR=2

PR=3

PR=4

1

10

100

Number of cycles to liquefaction [.] PR=1

PR=2

PR=3

PR=4

Fig. 2. Constant-volume cyclic shear tests. Typical liquefaction curves in terms of PR. The case with (a) fc ¼ 0, eo ¼ 0.79, σʹvo ¼ 150 kPa and (b) fc ¼ 0.4, eo ¼ 0.77, σʹvo ¼ 50 kPa are given. Linear regression lines of SR versus the logarithm of the number of cycles to liquefaction are also given.

consolidated under Ko conditions in the shear device. Tests were performed on specimens from the mixtures at different (a) initial void ratios (eo), (b) vertical consolidation stresses prior to cyclic load application (σ0 vo), (c) fines content (fc) and (d) pre-stress ratio (PR). Table 1 gives the states studied, defined by (i) the fines content, (ii) eo and (iii) σ0 v-o. Pre-stress was simulated by applying different maximum vertical effective stress on samples of the same initial void ratio and final vertrical effective stress, prior to exerting the same cyclic stress. Specifically, the device used was a Wykeham Farrance directshear apparatus in compliance with British Standard 1377. Undrained response was simulated by maintaining a constant specimen volume during shear by adjusting the vertical stress. The decrease in applied vertical stress during shear is equivalent to the increase in shearinduced pore water pressure which would occur in an undrained test [32]. The device used, the sample preparation method and the procedure of consolidation, pre-stress application and shearing of these tests, was identical to those in the testing program described by the journal publication [20]. Table 1 gives the tests performed per state of initial void ratio, vertical consolidation stress and fines content of soil. It can be observed that in total 288 cyclic tests were performed. Fig. 1 gives typical results of a cyclic shear test. The cyclic shear strain gradually increases until liquefaction occurs. Fig. 2 gives typical liquefaction curves of two states, in terms of PR. All curves are, approximately, parallel to each other and move upwards as PR increases. Table 1 gives the liquefaction cyclic strength obtained, in terms of PR, for all states studied. Figs. 3 and 4 give the measured liquefaction cyclic strength SR15 in terms of the initial void ratio, the vertical consolidation stress and the PR ratio for all mixtures studied. Fig. 5a presents SR15 for σ0 vo ¼50 and 150 kPa, in terms of the void ratio (eo) and the fines content. Fig. 5b gives the measured SR15 for σ0 vo ¼ 150 kPa for eo approximately equal to 0.65, in terms of the fines content. Fig. 6a plots the measured SR15 in terms of the PR value. Only states where more than two PR values were studied are considered. As in all the soil states considered in the present study the

preload cases PR¼1 and PR¼ 3 were examined (Table 1), Fig. 6b plots the measured factor R3 (defined by Eq. (3)) in terms of the factor SR15-1. First the consistency of the results of the cyclic tests is checked. It can be observed that, similarly to previous studies [8,9,12,20], in Figs. 3 and 4 the liquefaction cyclic strength (a) for PR¼1 (i) increases as the void ratio decreases and (ii) increases as the confining stress decreases and (b) increases as the PR value increases. In addition, from Fig. 6b it can be observed that the factor R3 varies between 1.2 and 1.9 in all states, in general agreement with the range of 1.5– 2.4 measured in previous laboratory programs [19,20]. Finally, Fig. 5b compares SR15 for PR¼1, σ0 vo ¼ 150 kPa and eo approximately equal to 0.65, in terms of the fines content of the present study with measurements on the triaxial device on the same mixtures by Papadopoulou and Tika [9]. It can be noted that, assuming Ko¼0.5, in the shear tests with σʹvo ¼150 kPa, the effective octahedral stress prior to cyclic loading application (p0 o), equals 100 kPa, and the thus two test programs in the different devices had similar consolidation stress. Fig. 5b illustrates that (1) in both test programs, SR15 decreases as fc increases until an fc value of about 0.3 and (2) the cyclic strength measured in the shear device is systematically less than that measured in the triaxial device. Item (2) is in agreement with observations of researchers that the liquefaction cyclic strength measured in shear devices is systematically less than that measured in the triaxial device [14,33].

4. Development of empirical equation

Ιt was stated in Section 1 that the relationship between the state parameter and the liquefaction cyclic strength for sand–silt mixtures (a) is unique for the same sample preparation method and (b) differs for the different sample separation method. In our case, as indicated in Section 3 above, the sample preparation method is dictated by the PR value. These statements, which are validated by

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

PR=1 - σ'vo=150kPa

0.35

0.25 0.20

PR=1 - σ'vo=150kPa 3 - 150kPa 1 - 50 kPa 3 - 50 kPa

0.30

SR15 [.]

0.30

SR15 [.]

0.35

2 - 150kPa 3 - 150kPa 4 - 150kPa 1 - 50 kPa 2 - 50kPa 3 - 50 kPa 4 - 50 kPa

0.15 0.10

0.25 0.20 0.15 0.10

0.05

0.05

fc=0

0.00

0.5

0.7

0.9

fc=0.15

0.00

1.1

0.5

0.7

PR=1 - σ'vo=150kPa 3 - 150kPa 1 - 50 kPa 3 - 50 kPa

0.35 0.30

1.1

PR=1 - σ'vo=150kPa

0.35

2 - 150kPa 3 - 150kPa 4 - 150kPa 1 - 50 kPa 2 - 50kPa 3 - 50 kPa 4 - 50 kPa

0.30

SR15 [.]

0.25

0.9

eo [.]

eo [.]

SR15

193

0.20 0.15 0.10

0.25 0.20 0.15 0.10

0.05

fc=0.25

0.00 0.5

0.05 0.7

0.9

1.1

0.00

fc=0.4 0.5

eo [.]

0.7

eo [.]

0.9

1.1

Fig. 3. Constant-volume cyclic shear tests. Liquefaction cyclic strength in terms of eo, σʹvo and PR for sand–silt mixtures with fc¼ 0, 0.15, 0.25, and 0.4.

PR=1 - σ'vo=150kPa 3 - 150kPa 1 - 50 kPa

0.35

0.35

0.25

0.25 0.20 0.15

0.15

0.05

fc=0.6

0.00

0.00 0.5

0.7

eo [.]

PR=1 - σ'vo=150kPa 3 - 150kPa 1 - 50 kPa 3 - 50 kPa

0.35 0.30

SR15 [.]

0.20

0.10

0.10 0.05

PR=1 - σ'vo=150kPa 3 - 150kPa 1 - 50 kPa 3 - 50 kPa

0.30

SR15

SR15 [.]

0.30

2 - 150kPa 4 - 150kPa 3 - 50 kPa

0.25

0.9

1.1

fc=0.8 0.5

0.7

0.9

1.1

eo [.]

2 - 150kPa 4 - 150kPa 2 - 50kPa 4 - 50 kPa

0.20 0.15 0.10 0.05 0.00

fc=1 0.5

0.7

eo [.]

0.9

1.1

Fig. 4. Constant-volume cyclic shear tests. Liquefaction cyclic strength in terms of eo, σʹvo and PR for sand–silt mixtures with fc ¼ 0.6, 0.8, and 1.0.

the current laboratory program in Section 5 below, indicate that SR15
PR ¼ f1ðPR; ψ Þ

ð4Þ

where f1 represents a function and ψ is the state parameter defined below. Eq. (4) predicts that SR15
1 ¼ f1 ð1; ψ Þ

ð5Þ

Thus, Eq. (4) can be re-written as SR15
PR ¼ f2ðPR; SR15
1 Þ

ð6Þ

where f2 represents a different function. As indicated in Section 2.2, SR15-PR is usually normalized by the liquefaction cyclic strength SR15-1.

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

fc=0 =0.25 =0.6 =1 Trend fc=0.25

SR15-1 (σ'vo=50kPa) [.]

0.30 0.25

=0.15 =0.4 =0.8 Trend fc=0 Trend fc=0.4

0.20 0.15 0.10 0.05

fc=0 =0.25 =0.6 =1 Trend fc=0.25 Trend fc=1

0.30

SR15-1 (σ'vo=150kPa) [.]

194

0.25 0.20

=0.15 =0.4 =0.8 Trend fc=0 Trend fc=0.4

0.15 0.10 0.05 0.00

0.00 0.5

0.6

0.7

0.8

0.9

1.0

0.5

1.1

0.6

0.7

0.8

0.9

1.0

1.1

e o [.]

e o [.] 0.3

SR15-1 [.]

0.25 0.2 0.15 0.1 Current study- σ'vo=150kPa Triaxial tests - P'o=100kPa

0.05 0 0

0.2

0.4

0.6

0.8

1

1.2

fc [.] Fig. 5. Constant-volume cyclic shear tests. (a) Effect of fc and eo on the liquefaction cyclic strength at PR ¼ 1 for σʹvo ¼50 kPa and σʹvo ¼ 150 kPa and (b) SR15-1 for σʹvo ¼150 kPa for eo approximately equal to 0.65, in terms of the fines content (fc) from (a) and comparison with results of the previous study in the triaxial device by Papadopoulou and Tika [9] on similar soil-silt mixtures and confining stress. It can be noted that, assuming Ko ¼ 0.5, in the shear tests with σʹvo ¼150 kPa, p0 o, equals 100 kPa.

5. Validation of Eq. (4)

In this respect, Eq. (6) can be re-written as RPR ¼ SR15
PR =SR15
1 ¼ f3ðPR; SR15
1 Þ

ð7Þ

where f3 represents another function. Referring to Eq. (7), first the effect of the factor PR on RPR on Eq. (7) is investigated. From Fig. 6a it can be observed that SR15 increases, approximately in a logarithmic manner, with PR. Thus, Eq. (7) may be rewritten as RPR ¼ PRf 4ðSR15
1 Þ

ð8Þ

where f4 indicates a function. In order to investigate the form of the function f4, Fig. 6b which plots the measured factor R3 in terms of the factor SR15-1 is considered. It illustrates that R3 decreases as SR15-1 increases. Furthermore, R3 is, and must always be, greater or equal to unity. These indicate that the factor f4 must decrease as SR15-1 increases and must always be greater or equal to zero. An expression for the function f4 consistent to the above is: f4 ¼ ða1=SR15
1 Þa2

ð9Þ

where a1 and a2 are positive fitting parameters. Regression of all tests of Table 1 gave that for best-fit results, a1¼0.04 and a2¼1.00. Combining Eqs. (8) and (9), the following expression is proposed predicting the effect of pre-stress on the liquefaction cyclic strength measured in the laboratory: RPR ¼ PR0:04=SR15
1

ð10Þ

For Eq. (10), the difference of all predicted and measured values of RPR of Table 1, normalized by the measured value, has a mean value of 0.00 and a standard deviation of 0.07. As the mean value is zero and the standard deviation value is small, it is inferred that Eq. (10) is adequate to predict the measured RPR values. Last but not least, the error of Eq. (10) was analyzed in terms of (i) the soil density, (ii) the vertical consolidation stress and (iii) the fines content. No significant trend was observed: The coefficient of correlation of the error in terms of all these parameters was less than 0.3. Therefore, adding these parameters in Eq. (10) will not increase considerably the accuracy of the predictions.

Eq. (4) assumes that the relationship between the liquefaction cyclic strength and the state parameter for the given sample preparation method and PR value is not affected by the consolidation stress, the void ratio and the fines content of the sand–silt mixture. Fig. 7, which plots SR15-1 and SR15-3 against the state parameter of the states of the sand–silt mixtures tested, given in Table 2, confirms this proposition. Indeed, the coefficient of correlation of the data of these figures, if it is represented, similarly to [13], by an exponential function, takes a value reasonably near unity: 0.78 and 0.71 for PR ¼1 and 3 respectively. In Fig. 7 and Table 2 the state parameter ψ is defined, similarly to [12], as

ψ ¼ e
ecs

ð11Þ

where e is the void ratio during the cyclic test and
ξ ecs ¼ Γ
λ p0o
cs =Pa

ð12Þ

where p0 o-cs is the effective octahedral stress at the critical state prior to the cyclic loading application, Pa is the atmospheric pressure and Γ, λ, ξ are fitting parameters of the critical state line. The critical state lines for the sand–silt mixtures of the present study were measured by Papadopoulou and Tika [9] (Fig. 8i). Based on these, the parameters Γ, λ, ξ were obtained in terms of fc, as given in Fig. 8ii. The obtained critical state curves predict well the measured ones (Fig. 8i) and the correlation coefficient of the expressions giving the factors Γ, λ and ξ in terms of fc in all cases is greater than 0.88 (Fig. 8i). These illustrate the accuracy of the approximation and, thus, Eq. (12) was expressed in terms of the fines content of the mixture as For fc o 0:35

0

ecs ¼
1:1 fc þ 0:81 þ ð0:1fc
0:043Þ po
cs =Pa


0:95f c þ 0:46

ð13Þ For fc 4 0:35



ecs ¼ 0:57 fc þ 0:23 þ ð
0:054 fc þ 0:011Þ


0:52f c þ 0:96 o
cs =Pa

p'

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

fc=0 fc=0.15 fc=0.25 fc=0.4 f=0.6 fc=0.8 fc=1 Proposed equation (10)

2.6

2.0 1.8 1.6 1.4 1.2 1.0 1

10

PR [.]

fc=0, eo=0.79, σ'vo=150kPa fc=0, eo=0.67, σ'vo=150kPa fc=0.4, eo=0.77, σ'vo=150kPa fc=1, eo=1.04, σ'vo=50kPa

SR15-3 / SR15-1 [.]

SR15 / SR15-1 [.]

2.2 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.05

fc=0, eo=0.67, σ'vo=50kPa fc=0.4, eo=0.77, σ'vo=50kPa fc=0.6, eo=0.89, σ'vo=150kPa fc=1, eo=1.04, σ'vo=150kPa

195

0.10

0.15

0.20

0.25

SR15 - PR=1 [.]

Fig. 6. Constant-volume cyclic shear tests. (a) Effect of the PR ratio on the liquefaction cyclic strength. (b) Ratio of the liquefaction cyclic strength at PR ¼ 1 and 3 in terms of the liquefaction cyclic strength at PR ¼ 1 and the fines content. The corresponding predictions of Eq. (10) are also given.

0.4

=0.15

=0.25

=0.4

=0.6

=0.8

0.4

=1

fc=0 =0.4 =1

0.3

SR15-3 [.]

SR15-1 [.]

0.3

fc=0

0.2

0.1

=0.15 =0.6

=0.25 =0.8

0.2

0.1

0.0 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 -0.2

0.0

State parameter [.]

0.2

0.4

0.6

0.8

1.0

State parameter [.]

Fig. 7. Constant-volume cyclic shear tests. Liquefaction cyclic strength in terms of the state parameter and the fines content for (a) PR ¼ 1 and (b) PR ¼ 3.

Table 2 Current laboratory program. Average void ratio after consolidation (e) at PR ¼ 1 and PR ¼ 3 and corresponding state parameter (ψ) for all states studied( given in Table 1). No.

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24 S25 S26

Average void ratio

State parameter

e(PR ¼1)

e(PR ¼3)

ψ(PR ¼ 1)

ψ(PR ¼3)

0.73 0.70 0.61 0.58 0.52 0.49 0.61 0.58 0.66 0.63 0.61 0.58 0.58 0.71 0.68 0.44 0.41 0.83 0.80 0.47 0.94 0.91 1.04 1.01 0.84 0.81

0.70 0.66 0.58 0.54 0.49 0.45 0.58 0.54 0.63 0.59 0.58 0.54 0.54 0.68 0.64 0.41 0.37 0.80 0.76 0.43 0.91 0.87 1.01 0.97 0.81 0.77

0.19 0.32 0.07 0.20
0.02 0.11 0.29 0.55 0.41 0.71 0.36 0.66 0.66 0.49 0.75 0.22 0.48 0.57 0.85 0.52 0.56 0.76 0.50 0.62 0.30 0.42

0.15 0.28 0.03 0.16
0.06 0.07 0.25 0.51 0.38 0.67 0.33 0.62 0.62 0.46 0.71 0.19 0.44 0.53 0.81 0.48 0.52 0.72 0.46 0.58 0.26 0.38

Furthermore, the octahedral stress p0 o-cs was obtained from the consolidation effective vertical stress of the states tested in the shear device (σʹvo in Table 1) as p0o
cs ¼ σ 0vo ð1 þ AÞ ð1 þvÞ=½3 ðA U cos 2 θ þ sin 2 θÞ

ð14aÞ

where

θ ¼ 45o þ φ0 =2

ð14bÞ

A ¼ tan 2 θ

ð14cÞ

and φ is the friction angle at the critical state, reported for the sand–silt mixtures tested by [9], and ν is the Poisson Ratio, which typically equals 0.3 for soils. The derivation of Eq. (14) is given in Appendix A. 0

6. Application procedure of Eq. (10) Based on Eq. (10), a procedure predicting the increase in liquefaction cyclic strength induced by preloading in the field in the case of the free field condition can be proposed. For this purpose, Eq. (10) is rewritten as 0:04=SR15
bef

SR15
after ¼ SR15
bef PRfield

ð15aÞ

where SR15-bef and SR15-after are the in-situ liquefaction cyclic strength before and after the field application of preloading and PRfield equals
PRfield ¼ σ 0vo þ Δσ 0ν =σ 0vo ð15bÞ where σʹvo is the overburden effective vertical stress and (Δσʹν) is the maximum additional effective vertical stress applied during the preload estimated versus depth. Eq. (15) should be applied versus depth. The factor SR15-bef is usually obtained from results of in-situ tests: the blow count (N60) measured in Standard Penetration Test, the cone resistance (qc) measured in Cone Penetration Tests and the shear wave velocity (Vs) [1,2,31]. Estimation of the overburden effective vertical stress (σʹvo) is straightforward, while the stress Δσʹν is usually obtained with linear elastic theory [3].

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Fig. 8. The critical state lines for (a) fc o 0.35 and (b) 0.35fc40.3. (i) Data by Papadopoulou and Tika [9] and approximation using Eq. (12) and (ii) the parameters Γ, λ, ξ in terms of fc, used to obtain the approximations given in (i).

Fig. 9. Numerical simulations for the sand model. Simulated (a) isotropic consolidation prior to application of cyclic loading and (b) liquefaction curves obtained in the triaxial path. The Seed and Idriss [36] liquefaction curves are also given for comparison.

7. Validation of Eq. (10) by numerical simulations The elastoplastic multi-mechanism model developed by Ecole Centrale Paris, known as ECP model, is used to represent the soil behavior. This model can take into account the soil behavior in a large range of deformations. The model is written in terms of effective stress. The representation of all irreversible phenomena is made by four coupled elementary plastic mechanisms: three planestrain deviatoric plastic deformation mechanisms in three orthogonal planes and an isotropic one. The model uses a Coulomb type failure criterion and the critical state concept. The evolution of hardening is based on the plastic strain (deviatoric and volumetric strain for the deviatoric mechanisms and volumetric strain for the isotropic one). To take into account the cyclic behavior a kinematical hardening based on the state variables at the last load reversal is used. The soil behavior is decomposed into pseudo-elastic, hysteretic and mobilized domains. Refer to Aubry et al. [34] and Hujeux [35] for further details about the ECP model. In order to assess the ability of the soil behavior model to simulate liquefaction cyclic strength, several liquefaction laboratory tests models were selected. Three sets of model parameters

which represent three kinds of sandy soil with different relative density (Dr) were used. The model parameters of these three soils are obtained using the methodology suggested by Lopez-Caballero et al. [25]. Fig. 9 shows the stress–strain response during the consolidation process and the obtained liquefaction curves during cyclic loading in triaxial paths with isotropic consolidation at p0 o ¼30, 50 and 100 kPa. The modeled liquefaction curves test results are compared with the reference curves given by Seed and Idriss [36] for sands at different densities (i.e. N60 values) and good agreement is observed. First of all, it is important to study the effect of the loading path, both in the consolidation phase and the cyclic loading, on the response of the laboratory test simulation. To this intent, four loading paths were tested (i) isotropic consolidation and cyclic triaxial loading (IC þCTx), (ii) isotropic consolidation and cyclic 1D (no lateral strain) simple shear loading (IC þSCOed), (iii) isotropic consolidation up to 20% of total load then oedometric consolidation up to total load and cyclic 1D shear loading (IC þ OedC þSCOed), (iv) isotropic consolidation up to 10% of total load then oedometric consolidation up to total load and cyclic 1D shear loading (ICþOedC þSCOed). Fig. 10 shows the model prediction for

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

197

Fig. 10. Simulated undrained stress controlled cyclic shear tests. (a) Shear stress versus effective vertical stress paths and (b) evolution of excess pore pressure (ΔU) with computation step.

Fig. 11. Simulated liquefaction curves. (a) Effect of the loading path. (b)–(d) Effect of prestress ratio (PR) on the three sands for loading path (iv). The Seed and Idriss [36] liquefaction curves are also given for comparison. In this figure, four loading paths were tested (i) isotropic consolidation and cyclic triaxial loading (ICþ CTx), (ii) isotropic consolidation and cyclic 1D (no lateral strain) simple shear loading (ICþSCOed), (iii) isotropic consolidation up to 20% of total load then oedometric consolidation up to total load and cyclic 1D shear loading (ICþ OedCþSCOed), (iv) isotropic consolidation up to 10% of total load then oedometric consolidation up to total load and cyclic 1D shear loading (ICþ OedCþSCOed).

the variation of shear stress versus the effective vertical stress and the excess pore pressure generation during test simulation of a shear test for one of the set parameters. According to the results (Fig. 11a), for PR ¼1, a higher liquefaction curve (i.e. cyclic strength) is found when the loading path (ii) is used. However, by comparing the path (i) and (iii) small difference was found. Finally, no difference was found between path (iii) and (iv). In order to study the effect of pre-stressing on the liquefaction cyclic strength, four levels of pre-stress ratio (PR) were studied (i.e. 1, 2, 3 and 4). The loading path (iv) was used. Fig. 11b–d displays liquefaction curves obtained for the three soils at one initial stress and three PR

values. It is noted that for similar confining stresses, the SR value increases as PR increases, as shown in the laboratory tests. Based on all the results of the current numerical analyses, the ratio SR15/SR15-1 is plotted versus the PR value in Fig. 12a and a comparison between the SR15/SR15-1 ratio obtained by simulations and estimated with Eq. (10) is provided in Fig. 12b. It is interesting to note that all computed values are found in a range of 720% of the predicted values. Furthermore, analysis illustrated that the difference of the all predicted and computed values, normalized by the measured values, has a mean value of 0.03 and a standard deviation of 0.09. As both the mean and the standard deviation values are near zero, the predictions are satisfactory.

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Fig. 12. Simulated liquefaction curves. (a) Ratio SR15/SR15-1 in terms of prestress ratio (PR) and (b) Comparison between simulated and estimated by Eq. (10) SR15/SR15-1 ratios.

Table 3 Brief description of cases found in the literature where soil improvement by preloading was applied on liquefaction-susceptible sites in the field and measurements which allow the estimation of the liquefaction cyclic strength both before and after the application of preloading exist: The location, the soil profile characteristics, the preload height and relevant references are given. Case study 1 no

2

3

4

5

Location

Derveni village, Macedonia, Greece

Derveni village, Macedonia, Greece

Langadas town, Macedonia, Greece

Successive random deposits of low plasticity clay and silty sand soft layers 10 m

Successive random deposits of low plasticity clay and silty sand soft layers 10 m

Successive random deposits of low plasticity clay and silty sand soft layers 9m

Porto Romano, Abanian cost, Albania 0–3.5 clay 3.5–7 m silty sand below 7 m silt 9m

[4]

[4]

[28,29]

[30]

Soil profile Successive random deposits of sandy and clayey soft layers Preload 12 m height References [27]

Table 4 Detailed description of the liquefaction cyclic strength measured in the case studies given in Table 3. No. Case study

Depth range

1 2 3 4 5 6 7 8 9 10

1 2 2 2 3 3 4 4 4 5

6–10 0–4 4–10 10–15 9–12 12–15 0–2.5 10–13 13–15.5 3–8

11

5

8–15

fc

25 25 25 25 30 30 33 27 60 21

Field measur.

N60 N60 N60 N60 N60 N60 N60 N60 N60 N60, qc, Vs 69 N60

SR15- SR15bef

PRfield SR15-after/SR15-bef

after

Mea. Pred. (a) (b)

(b)/ (a) 0.92 1.07 0.77 0.93 0.85 0.91 1.07 0.90 1.05 1.00

0.13 0.24 0.29 0.22 0.30 0.27 0.28 0.27 0.33 0.43

0.22 0.31 0.47 0.27 0.39 0.33 0.37 0.31 0.35 0.52

4.2 7.2 3.2 1.9 2.1 1.8 9.8 1.7 1.5 4.1

1.70 1.30 1.62 1.20 1.30 1.20 1.30 1.20 1.00 1.14

1.56 1.39 1.25 1.12 1.10 1.09 1.39 1.08 1.05 1.14

0.45

0.49

1.9

1.13

1.06 0.94 Ave 0.95 Stdev 0.10

8. Field measurements and comparison with Eq. (15a) predictions Four studies were found in the literature where preloading was applied on liquefaction-susceptible sites in the field and field measurements, which allow the estimation of the liquefaction cyclic strength both before and after the application of preloading, are available. Table 3 gives the location, the soil profile characteristics, the preload height and the relevant references of these cases. Regarding the soil layers in these case studies which are susceptible to liquefaction, Table 4 gives their depth, their fines content, the available type of

2

SR15-after /SR15-bef - Predicted [.]

Almiros town, Thessaly, Greece

Data Pred=Meas -30% +7%

1.8

1.6

1.4

1.2

1 1

1.2

1.4

1.6

1.8

SR15-after /SR15-bef - Measured [.] Fig. 13. Preloading cases of Table 4. Comparison between estimated based on field measurements (meas)and predicted by Eq. (15) (pred) increase in liquefaction cyclic strength induced by preloading.

field measurements which allow the estimation of the liquefaction cyclic strength, the average estimated liquefaction cyclic strength before (SR15-bef) and after (SR15-after) the application of preloading and the corresponding ratio Rfield(¼SR15-after/ SR15-bef). The values of SR15bef and SR15-after were estimated (a) for sands and silty sands based on the N60 and qc measurements using the state-of-the-art procedures described by [37], (b) for sands based on VS measurement using the relationship given by [1] and (c) for non-plastic silt layers, based on the N60 value according to [31]. Table 4 also gives the factors PRfield, at mid-depth of each soil layer which liquefies. They were obtained using Eq. (15b) and linear elastic theory. The obtained (PRfield, Rfield) pairs were compared with predictions of Eq. (15a). As illustrated in Table 4, the difference of the all predicted and measured values, normalized by the measured

C.A. Stamatopoulos et al. / Soil Dynamics and Earthquake Engineering 78 (2015) 189–200

0.6

=0.2

=0.3

=0.4

0.6 0.5

SR15-after [.]

SR15-after

0.5

SR15-bef=0.1

0.4 0.3 0.2 0.1

199

SR15-bef=0.1

=0.2

=0.3

=0.4

0.4 0.3 0.2 0.1

0

0 0

2

4

6

8

10

12

14

16

0

2

4

Depth (m)

6

8

10

12

14

16

Depth [m]

Fig. 14. Predictions by Eq. (15) of the post-improvement liquefaction cyclic strength (SR15-after) versus depth in terms of the pre-improvement liquefaction cyclic strength (SR15-bef) for the typical cases of preload embankment height of (a) 6 m and (b) 12 m. The depth of the water table is taken at 2 m and the unit weight of both the preload embankment and the underlying soil is assumed 18 kNt/m3.

values, has a mean value of
0.05 and a standard deviation of 0.10. This accuracy can be considered satisfactory given the uncertainty which exists in the measurement of the liquefaction cyclic strength in the field, especially due to the erratic nature of the soil deposits considered in the present study. In addition, as illustrated in Fig. 13, all measured values are found in a range of þ30% to
7% of the predicted values. It can be noted that the large deviations are in the conservative side, while there is no case where the predicted value exceeds the measured one by more than 7%. Based on these, it is inferred that caveats in the procedure of Section 6 to account for the error of the above comparisons with field measurements are not needed.

 Based on critical state concepts, Eq. (4) is proposed illustrating 

 

 9. Typical predictions of the effect of preloading on liquefaction cyclic strength In order preloading to be efficient and cost-effective, the preloading embankment must have height about 6–12 m [3]. For the cases of preload embankment height of 6 and 12 m, Fig. 14 gives the postimprovement liquefaction cyclic strength predicted by Eq. (15) in terms of initial liquefaction cyclic strength, versus depth. The depth of the water table is taken at 2 m and the unit weight of both the preload embankment and the underlying soil is assumed 18 kNt/m3. Moderate soil improvement is observed. The maximum acceleration value which some design codes require may correspond to a SR exceeding 0.4 at small depths in some regions. For these cases the soil improvement of Fig. 14 may not be enough to produce the required in codes factor of safety of 1.2 for earthquakes of large magnitude at some depths. However, this may not be a concern actually, as in the case of rare very severe events, limited liquefaction may be beneficial for the built-environment, as it decreases inertia loads applied in overlying structures [26].



the factors affecting the increase of liquefaction cyclic strength induced by pre-stress in the free field condition. Based on the test results, (a) the empirical expression (10) predicting the increase in the liquefaction cyclic strength induced by pre-stress in the free field condition is proposed, and (b) Eq. (4) is validated. Eq. (10) was validated by numerical simulation of the relevant laboratory tests using the elastoplastic multi-mechanism model of Ecole Centrale Paris. Based on Eq. (10), Eq. (15) and an associated methodology predicting the increase in the liquefaction cyclic strength induced by preloading in the field in the case of the free field condition, are proposed. Eq. (15) predictions are compared with field measurements on liquefaction-susceptible soil before and after the field application of preloading. The difference of all predicted and measured values, normalized by the measured values, has a mean value of
0.05 and a standard deviation of 0.10. This accuracy can be considered satisfactory given the uncertainty which exists in the measurement of the liquefaction cyclic strength in the field, especially due to the erratic nature of the soil deposits considered in the present study. Fig. 14 gives the increase in liquefaction cyclic strength which the proposed methodology predicts for typical soil profiles and embankment preloads.

Acknowledgment The work was funded by the Seventh Framework Programme of the European Community, European Commission Research Executive Agency under grant Agreement FP7-SME-2010-1-262161PREMISERI. Mr Petros Petridis assisted in the analysis of the field case studies. Mr Fragiskos Stratis, Mr Dimitris Stefanis, Mrs Eleni Stavroyanopoulou and Mrs Lydia Balla assisted in the performance and analysis of the laboratory tests.

10. Conclusions The main conclusions drawn from this study are as follows:

 Cyclic shear tests simulate the main boundary and loading



conditions relevant to the effect of preloading on the liquefaction cyclic strength below horizontal ground surfaces in the field. Cyclic shear tests performed on samples with varying fines contents and at varying pre-stress ratios, densities and vertical stresses showed a marked increase of the liquefaction cyclic strength with the pre-stress ratio (PR).

APPENDIX A. Octahedral stress at the critical state at the shear device In the shear device, plane stain conditions exist. Denoting the horizontal lateral direction as “2”, ε2 ¼0 and thus, according to Hook's law

σ 02 ¼ vðσ 01 þ σ 03 Þ

ðA1Þ

where v is the Poisson Ratio. According to the Mohr–Coulomb law, at horizontal failure, the vertical stress, which corresponds to the

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normal stress at the slip surface, equals

σ ¼σ 0 v

0 1U

cos θ þ σ 2

0 3U

sin θ 2

ðA2aÞ

where

θ ¼ 45o þ φ0 =2

ðA2bÞ

where φ0 is the effective friction angle at the critical state. Also, according to the Mohr–Coulomb law

σ 01 ¼ Aσ 03

ðA3aÞ

where A ¼ tan 2 ð45o þ φ0 =2Þ

ðA3bÞ

Finally, Eqs. (A1)–(A3) predict that at failure, the effective octahedral stress equals p’ ¼ σ 03 ð1 þ AÞ ð1 þ vÞ=3 ¼ σ 0v ð1 þ AÞ ð1 þvÞ=½3 ðA U cos 2 θ þ sin 2 θÞ ðA4Þ References [1] European Prestandard. Eurocode 8 – design provisions of earthquake resistance of structures – Part 5: foundations, retaining structures and geotechnical aspects; 1994. [2] Ishihara. Soil behaviour in earthquake geotechnics. Oxford engineering science series; 46, 1996. [3] Stamatopoulos AC, Kotzias PC. Soil improvement by preloading. New York: John Wiley & Sons; 1985 261 pp. [4] Petridis P, Stamatopoulos C and Stamatopoulos A. Soil Improvement by preloading of two erratic sites. In: Proceedings of the international conference on geotechnical and geological engineering (GeoEng2000), Melbourne, Australia; 2000 (in CD-ROM). [5] Finn WD, Pickering DJ, Bransby PL. Sand liquefaction in triaxial and simple shear tests. J Geotech Eng Div ASCE 1971;97(4):639–60. [6] De Alba P, Seed HB, Chan CK. Sand liquefaction in large-scale simple shear tests. J Geotech Eng Div ASCE 1976;102(9):909–28. [7] Vaid YP, Thomas J. Liquefaction and postliquefaction behavior of sand. J Geotech Eng ASCE 1995;121(2):163–73. [8] Xenaki VC, Athanasopoulos GA. Liquefaction resistance of sand–silt mixtures: an experimental investigation of the effect of fines. Soil Dyn Earthq Eng 2003;23:183–94. [9] Papadopoulou A, Tika T. The effect of fines on critical state and liquefaction resistance characteristics of non-plastic silty sands 2008;48(No. 5):713–25Soils Found 2008;48(No. 5):713–25. [10] Chen YC and Liao TS. Studies of the state parameter and liquefaction resistance of sand. In: Proceedings of the 2nd international conference on earthquake geotechnical engineering, Lisbon Portugal; 1999. p. 513–18. [11] Stamatopoulos C, Stamatopoulos A, Balla L. Cyclic strength of sands in terms of the state parameter. In: Proceedings of the 11th international conference on soil dynamics and earthquake engineering (11thICSD) and the third international conference on geotechnical earthquake engineering; 2004 (on CD). [12] Stamatopoulos CA. An experimental study of the liquefaction strength of silty sands in terms of the state parameter. Soil Dyn Earthq Eng 2010;30(Issue 8):662–78. [13] Qadimia A, Mohammadi A. Evaluation of state indices in predicting the cyclic and monotonic strength of sands with different fines contents. Soil Dyn Earthq Eng 2014;66:443–58. [14] Seed HB, Peacock WH. Test procedures for measuring soil liquefaction characteristics. J Geotech Eng Div ASCE 1971;97(No. 8):1099–119.

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