Technical manual - v1.1

Introduction

Slake is intended for the complete carry out of quantification analyses of liquefaction hazard of soils subjected to seismic loading in open field by the direct semi-empirical method known as "NCEER" (Youd and Idriss, 2001)⁽¹⁾, using SPT (standard penetration test) and/or CPT(u) (cone penetration test with or without pore-water pressure measurement) data. By simplification, these analyses are called NCEER-SPT and NCEER-CPT in the following.

⁽¹⁾ Slake v1.1 also features an additional calculation option derived from the simplified "NCEER" procedure, taking into account certain adaptations resulting from the recommendations of the AFPS Technical Notebook n°45 "Assessment of the risk of soil liquefaction under the effect of earthquakes - Practical knowledge and applications to geotechnical projects" (NCEER/CT45-AFPS(2020) calculation option).

The determination of factors of safety against liquefaction is coupled with a strain analysis, allowing the estimation of post-liquefaction settlements under a water table according to the correlations proposed by different authors and derived from the charts of Ishihara and Yoshimine (1992).

Notations

Thereafter, the following notations are adopted:

Quantification of liquefaction hazard

Phenomenological reminders

Liquefaction is a characteristic instability of contracting saturated sands (granular materials in a very loose to loose state) that can develop increased pore-water pressures as a result of cyclic stress in undrained conditions. These large pore-water pressures are responsible for the loss of resistance by reducing - or even canceling - the effective stress in the soil. Under certain conditions, this phenomenon can also take place under a monotonous loading, which is called static liquefaction.

The phenomenon of liquefaction is to be distinguished from the cyclic mobility, also characterized by a behaviour at constant or nearly constant volume (saturated materials in undrained conditions) and which affects the dilative sands (granular materials in a moderately dense to very dense state). For the latter, the generation of excess pore-water pressures during loading leads to temporary losses of resistance and significant cyclic deformation, but the tendency to dilatancy that occurs during shear prevents overall loss of resistance and significant permanent deformation.

Soft clays are prone to another phenomenon, cyclic softening, which causes a loss of stiffness due to the potentially large accumulation of irreversible deformations under cyclic loading.

The simplified semi-empirical calculation method implemented in Slake allows the quantification of hazard only within the strict framework of the liquefaction phenomenon as defined above, apart from the instabilities associated with cyclic mobility and cyclic softening.

Soil sensitivity against liquefaction

Soil physical properties were the first characteristics used to estimate their susceptibility against liquefaction. Different criteria for sensitivity analysis have been proposed, historically based on the limits of consistency, granulometric properties and water content, or combinations of these properties.

Slake featers a Liquefaction Sensitivity Assessment Wizard that complies with the current requirements of Eurocode 8 Part 5 (EC8-5). This wizard is independent from the quantitative hazard analysis, which can be examined locally using SPT and/or CPT(u) surveys using different methods. At a specific measurement point, and based on the analysis of the Eurocode criteria detailed below, the Wizard determines "the need for a quantitative hazard analysis", or on the contrary "the possibility of ruling out the risk of liquefaction".

This assessment is based on the examination of one main condition concerning the seismic assumptions of the project, plus three additional criteria derived from laboratory and in situ test results: granulometry, plasticity, normalized SPT test results. The verification of the main condition and, simultaneously, of at least one of the three additional criteria, allows the risk of liquefaction in the sense of EC8-5 to be neglected. If not, the risk of liquefaction must be quantified.

Note: it shall be reminded that there is no consensus on the use of sensitivity criteria, particularly with regard to risk exclusion criteria. Nevertheless, the criteria offer a practical entry point (at least qualitatively) for assessing the sensitivity of soil against liquefaction.

Simplified "NCEER" procedure

Slake is based on the implementation of the simplified analysis procedure formalized by Youd and Idriss in 2001 following the consensus reached at the working seminars on soil resistance to liquefaction organized by the NCEER research in 1996 and jointly NCEER/NSF in 1998 (published in the ASCE journal JGGE in April 2001). By simplification, this methodology is noted as the "NCEER" procedure in the following.

As of 2021, the "NCEER" procedure is still considered to be the international reference for quantifying liquefaction hazard of soils (sole consensus). The methods proposed by Idriss and Boulanger in particular are variations of the latter, but do not reach consensus.

Note: in its current version (v1.1), Slake features an additional calculation option : NCEER/CT45-AFPS(2020).

This alternative calculation option takes up the general principles of the "NCEER" procedure (detailed in the following paragraphs) with subtle adaptations/modifications to certain clearly identified calculation parameters, in accordance with the recommendations made in the appendices of Technical Notebook n°45 "Assessment of the risk of soil liquefaction under the effect of earthquakes - Practical knowledge and applications to geotechnical projects" (to be published in 2021) by the "Liquefaction" Sub-Working Group of the AFPS (french association for earthquake engineering).

When applicable, these modifications are detailed in the relevant paragraphs. Wherever this is not expressly stated, the "NCEER/CT45-AFPS(2020)" calculation option strictly follows the calculation assumptions of the "NCEER" procedure.

Domain of validity

The "NCEER" procedure is a deterministic approach, in total stress, and considering the hypothesis of open field site conditions, i.e. assuming the spatially unconstrained liquefiable layers(s) and susceptible(s) to be freely set in motion (one-dimensional calculation).

It was developed empirically from the compilation of field data (in situ tests of different types) and laboratory tests performed on samples composed of siliceous alluvial materials (silty sands), of Holocene age, collected at shallow depths and from sites mainly in California, under level to gently sloping ground having liquefied or not liquefied as a result of earthquakes of magnitude $M_{w}$ equal to 7.5.

The results of any analysis conducted outside the strict assumptions of the "NCEER" method's validity domain must be subject to critical analysis by the geotechnical engineer.

These considerations are also valid when the calculation option "NCEER/CT45-AFPS(2020)" is selected.

Principles of calculation

For an earthquake of any magnitude, the method consists in the assess of a factor of safety against liquefaction defined as the ratio between the normalized cyclic resistance ratio $\text{CR}R_{Mw = 7.5}$ (Cyclic Resistance Ratio) of the ground for a reference earthquake of magnitude of 7.5 on the one hand, and the equivalent normalized shear stress generated by the earthquake $\text{CSR}$(Cyclic Stress Ratio) on the other hand, at any point where measurements are available in the tested soil column.

This ratio is then corrected by the term $\text{MSF}$ (Magnitude Scaling Factor) which allows to take into account the project magnitude of the earthquake under consideration, and by the possibly combined effects of confining pressure ($K_{\sigma}$) and inclination of stress ($K_{\alpha}$).

The factor of safety against liquefaction is then expressed as follows:

Theoretically, liquefaction occurs when the factor of safety is less than the unit (local cancellation of the vertical effective stress). The accumulation of irreversible volumetric strains associated with a significant reduction in vertical effective stress may nevertheless develop due to excess pore-water pressures for computation points with factors of safety greater than the unit.

In Slake, the threshold of the stability domain is thus limited by the limit factor of safety $FS^{\lim}$, while the lower limit of the safety domain is bounded by the targeted factor of safety noted $FS^{\text{target}}$ directly inputed by the user with respect to a pre-defined level of safety.

As an indication, Eurocode 8 currently sets the minimum safety threshold at 1.25, a value associated with the development of excess pore-water pressures in the order of 60% of the vertical effective stress. This threshold can be increased according to the nature and sensitivity of the studied structure, being implicitly understood that beyond this value the earthquake-induced post-liquefaction deformations are negligible and therefore acceptable by the supported structure.

• $FS< FS^{lim}\;=\;1,00$ : liquefaction (total cancellation of vertical effective stress, local loss of total resistance) ;
• $FS^{lim}\leq FS\;< FS^{target}$ : accumulation of significant irreversible deformations (loss of strength) against a predefined level of safety.

Slake presents the FS calculation results only at points where the following double condition is verified:

• The calculation point (ie the test point) is below the project water-table level: $z(i)\leq z_{w,p}$ ;

• The normalized measurement at the point of the test (depending on the type of test) does not exceed the threshold values beyond which it is conventionally accepted that the materials tested are too dense to be liquefiable:

  • Case of SPT surveys: $(N_1)_{60CS}<30$ ;
  • Case of CPT(u) surveys: $(q_{c1N})_{CS}<160$.

In the case of CTP(u) analyses, an additional criterion applies on the basis of the behaviour type index $I_{c}$. Soils with a behaviour type index greater than or equal to a threshold value $I_{c}^{cut - off}$ are considered too fine and/or plastic to liquefy:

• $I_c\leq I_c^{cut-off}$

Usually, $I_{c}^{cut - off}$ is set at 2.6. However, it is possible to change the threshold value on the behaviour type index in Slake. The threshold should then be calibrated from cyclic laboratory tests. Changing this threshold value is strongly discouraged.

The process for safety factor determination against liquefaction is synthesized in the following logigram.

Normalized equivalent cyclic shear stress

The cyclic shear stress induced by the earthquake $\tau_{e}$ normalized by the vertical effective stress is assessed from the following equation (Seed & Idriss, 1971):

The coefficient $r_{d}$ is a coefficient of stress reduction based on the depth resulting from soil flexibility; it is defined in Slake according to Blake's formula (1996):

Where z is the depth. This formulation of the $r_{d}$ term is strictly valid only up to 15 m deep, and is no longer applicable beyond 20 m. It is implicitly accepted in Slake that the user provides in situ test results to depths deemed compatible with the development and surface expansion of liquefaction effects. Attention is drawn to the fact that the $r_{d}$ factor is therefore not limited and is extrapolated to any depth following this formulation by the software.

Normalized cyclic resistance ratio

The term $CRR_{Mw = 7.5}$ (Cyclic Resistance Ratio), thus the normalized cyclic resistance ratio offered by the examined soils, is estimated by post-processing of the results of in situ SPT and/or CPT(u) tests. The definition of the $CRR_{\text{Mw} = 7.5}$ term is detailed in the paragraphs relating to the computation of each test category.

General assumptions of calculation

Stress states in the soil

Liquefaction analyses are conducted by defining two distinct multilayer geotechnical models to distinguish the calculation of stress states relating to the cyclic resistance term (CRR) and the term of earthquake action (CSR). The first one is representative of the context prevailing at the time of the surveys, while the second one must be representative of the conditions projected in the operational phase and concurrent with the considered seismic actions.

These models take into account:

• The possible change in the topographical conditions of the site after the completion of the surveys: filling or excavation on a large area of the site in the operational phase compared to the intial conditions. It should be noted that these variations in the overall appearance of the site are to be distinguish from the localized overloads (point loads) brought by the structures and which should not be taken into account in the case of open field analyses;
• Variations in water-table levels between piezometric measurements in initial conditions at the time of in situ testing and project water-table levels concomitant with seismic actions.

Taking into account project seismic magnitudes

The "NCEER" procedure recommends a range for estimating the correcting factor $\text{MSF}$ based on seismic magnitudes $M_{w}$. The latter are to be distinguished from the magnitudes of surface waves ($\text{Ms}$). As an indication, it is shown below the transformation of a magnitude measure to another that can be made according to the bilinear relations of Scordilis (2006), valid for shallow earthquakes (less than 70 km).

It is commonly accepted that below a magnitude $M_{w}$ of 5.0 the energy released by seismic waves is not sufficient to initiate the phenomenon of liquefaction (lack of counter-examples referenced as of 2019). It should be noted that Slake determines MSF values by extrapolation beyond the limits defined by the "NCEER" procedure ($M_{w} < 5.0$ and $M_{w} > 8.0$).

The MSF range recommended by the "NCEER" procedure as well as the equations of the lower ($\text{MS}F^{N -}$) and upper ($\text{MS}F^{N +}$) bounds are shown below.

Note 1: Slake provides an intermediate option for the determination of the magnitude scaling factor directly corresponding to the arithmetic mean, for a given magnitude of seismic moment, between the values associated with the lower and upper bounds of the recommended range : $MSF^{(N,mean)}=mean[{MSF^{(N-)} ; MSF^{(N+)}}]$.

Note 2: it is possible to manually insert into Slake any value of $\text{MSF}$. However, it is recalled here that the $\text{MSF}$ and $r_{d}$ parameters are strictly linked; it is therefore not recommended to input $\text{MSF}$ correction factors outside of the proposed range as part of analyses conducted strictly according to the "NCEER" procedure (nor the “NCEER/CT45-AFPS(2020)” option).

Additional corrections

The equation (1) involves two complementary correction factors $K_{\sigma}$ and $K_{\alpha}$ which are outside the framework of the simplified "NCEER" procedure.

$K_{\sigma}$ correction factor

Laboratory cyclic loading tests indicate that liquefaction resistance increases with increasing confining stress, but not linearly (the normalized resistance CRR decreases as the effective confining stress increases). In order to extrapolate the normalized resistance CRR (obtained by simplified method for a vertical stress of 100 kPa) to layers where the effective stress are greater than 100 kPa taking into account this non-linearity, Seed (1983) introduced a corrective $K_{\sigma}$ factor. The $K_{\sigma}$ values were obtained from cyclic triaxial tests on clean-sands with various relative densities and consolidated under stress between 100 kPa and 600 kPa.

The use of this correction factor may be justified in potentially liquefiable sandy soils located at a deep depth, or in the case of a deep water-table.

In the 2001 paper, the formulation of Hynes and Olsen (1999) defines the parameter $K_{\sigma}$ as the vertical effective stress $\sigma'_{v0}$ normalized by atmospheric pressure $P_{a}$ (approximately 100 kPa) and then raised to the power $f$ depending on site conditions: relative density, age of deposits, level of overconsolidation.

In practice, the definition of the $f$ exponent is reduced to a correlation based on relative density $D_{r}$. The values chosen for $f(D_{r})$ are summarized in the table below.

Precaution: The relative density is a notion only relevant to granular materials, with a fine content ¹ strictly less than 15%. To correct this problematic definition, because it is not independent of the nature of the soils, Slake does not take into account by default the correcting factor $K_{\sigma}$ ($K_{\sigma} = 1.0$). However, the user can force the calculation of this correction term, if he wishes, by activating the option in the advanced calculation parameters. If so, the $K_{\sigma}$ calculation is systematically limited to a unit value:

¹ Strictly speaking, the fine content ($\text{FC}$) used to establish these correlations must be consistent with the USCS, and therefore corresponds to the percentage of passing by weight through the 75 µm sieve. This sieve is not used in the French standards of sieve analyses (in the old NF P 94-056 standard, the fine content is defined by the percentage of passing by weight through a sieve of 80 µm; in the new NF P 94-512-4 standard it is defined by the passing through a 63 µm sieve). It is therefore important that the definition of chosen $\text{FC}$ is compatible with the strict application of the NCEER-SPT procedure, i.e. a passing through a 75 µm sieve, or, failing that, 80 µm sieve.

$K_\sigma$ for NCEER-SPT

In the case of an NCEER-SPT analysis, the user can choose to impose the correction factor $K_{\sigma}$ calculation by activating the dedicated switch in the advanced calculation settings. If so, the following should be provided:

• The threshold value of the fine content $FC^{\lim}$ below which a relative density value can be estimated (by default, $FC^{\lim} = 15\%$);
• The layer or layers for which the calculation of $K_{\sigma}$ is imposed by the user.

At calculation points included in the interval (or intervals) defined by the selected stratigraphic layer(s) where the condition on the fine content is verified ($FC \leq FC^{\lim}$), the relative density is estimated thanks to the Skempton (1986) correlation:

At calculation points where $FC > FC^{\lim}$, no relative density is calculated and the $K_{\sigma}$ factor is automatically taken equal to $K_{\sigma} = 1.0$.

The correction factor $K_{\sigma}$ is then calculated from the equation (4).

Important note: Skempton's correlation was established for medium to coarse, clean quartz sands of relative densities between 0.35 and 0.85, under stress between 50 kPa and 250 kPa. Outside the validity domain of this correlation, and in general, the user is asked to control the calculation results. In the result tables, Slake highlights the calculation results where the relative density estimated by correlation is outside the range $0.35 \leq D_{r} \leq 0.85$.

Case of the "NCEER/CT45-AFPS(2020)" calculation option

When selected, the definition of $K_\sigma$ correction factor for SPT based analysis remains an "advanced" calculation option to be set manually. However, the calculation conditions differ on the following points:

• The default value of the fines content threshold $FC^{lim}$ below which the $K_\sigma$ calculation is performed is arbitrarily raised to 100% (although it can be modified by the user), with the effect of "forcing" (a priori) the calculation at any measurement point of a layer where the option has been selected;

• The upper bound of $K_\sigma (K_\sigma^{lim}$, equation (5)) is no longer locked to the unit but is to be filled in by the user; by default its value is taken to be equal to $K_\sigma^{lim}=1.1$ ;

• The value of the exponent f is no longer directly calculated by correlation with the relative density $D_r$, and is considered constant even in the case of a multilayer. It must be filled in by the user. By default, the exponent $f$ is taken equal to 0.7 (conservative approach);

• At the calculation points contained in the interval(s) defined by the selected stratigraphic layer(s) where the condition on the fine fraction is verified $(FC\leq FC^{lim})$, equation (4) is replaced by the following expression :

  $$K_\sigma=min[K_\sigma^{lim}; (\frac{\sigma'_{v0}}{P_a})^{[f-1]}]$$

This approach aims at simplifying the calculation of the correction factor $K_\sigma$ without setting conditions on the relative density $D_r$ of the soils on a flat-rate basis.

$K_\sigma$ for NCEER-CPT

In the case of an NCEER-CPT analysis, the user can choose to impose the correction factor $K_{\sigma}$ calculation by activating the dedicated switch in the advanced calculation settings. If so, the following should be provided:

• The threshold value of the soil behaviour type index $I_{c}^{\lim}$ below which the tested soil is deemed to be granular and therefore a value of relative density can be estimated (by default, $I_{c}^{\lim} = 1.64$) ;
• The layer(s) for which the calculation of $K_{\sigma}$ is imposed by the user.

At calculation points included in the interval (or intervals) defined by the selected stratigraphic layer(s) where the condition on the soil behaviour type index is verified ($I_{c} \leq I_{c}^{\lim}$), the relative density is estimated thanks to the Baldi (1986) correlation:

At the calculation points where $I_{c} > I_{c}^{\lim}$ no relative density is calculated and the $K_{\sigma}$ factor is automatically taken equal to $K_{\sigma} = 1.0$.

The correction factor $K_{\sigma}$ is then calculated from the equation (4).

Important note: Baldi's correlation is strictly applicable only in the case of fine to medium clean siliceous sands, homogeneous and uncemented, normally consolidated and moderately compressible. Moreover this relationship is not valid at shallow depth of penetration: the calculation of in Slake is therefore fixed to carry out only beyond 3.0 m deep. Outside the validity domain of this correlation, and in general, the user is asked to control the calculation results. In the result tables, Slake highlights the calculation results where the relative density estimated by correlation is outside the range.

Case of the "NCEER/CT45-AFPS(2020)" calculation option

When selected, the definition of $K_\sigma$ correction factor for CPT(u) based analysis remains an "advanced" calculation option to be set manually. However, the calculation conditions differ on the following points:

• The default value of the fines content threshold $I_c^{lim}$ below which the $K_\sigma$ calculation is performed is arbitrarily raised to 2.6 (although it can be modified by the user), with the effect of "forcing" (a priori) the calculation at any measurement point of a layer where the option has been selected;
• The upper bound of $K_\sigma (K_\sigma^{lim}$, equation (5)) is no longer locked to the unit but is to be filled in by the user; by default its value is taken to be equal to $K_\sigma^{lim}=1.1$ ;
• The value of the exponent f is no longer directly calculated by correlation with the relative density $D_r$, and is considered constant even in the case of a multilayer. It must be filled in by the user. By default, the exponent $f$ is taken equal to 0.7 (conservative approach);
• At the calculation points contained in the interval(s) defined by the selected stratigraphic layer(s) where the condition on the fine fraction is verified $(FC\leq FC^{lim})$, equation (4) is replaced by equation (7).

This approach aims at simplifying the calculation of the correction factor $K_\sigma$ without setting conditions on the relative density $D_r$ of the soils on a flat-rate basis.

$K_{\alpha}$ Correction Factor

Theoretically, the development of static shear stress $\tau_{\text{st}}$ in a sloping soil mass influences its ability to withstand liquefaction. This influence - positive or negative - depends in particular on the nature, density and confining state of soil. Some authors propose to express it using a specific correction factor,$\ K_{\alpha}$, without the latter being the subject of a consensus as of the date (2019).

Slake does not allow the correction factor $K_{\alpha}$ to be taken into account in the execution of routine liquefaction analyses ($K_{\alpha} = 1.0$).

Evaluation of the cyclic resistance ratio

Assessment based on standard penetration tests (SPT)

Principle of the SPT test

The Standard Penetration Test (SPT) is a standardized dynamic in situ test that also allows the collect of disturbed samples. It is framed by the American standard ASTM-D-1586-11, and partially included in the French application standard NF EN ISO 22476-3.

In a pre-drilled borehole, the test involves pushing a casing following a standardized driving device and counting in successive stages the number of blows needed for a specific penetration of the casing into the ground. Conventionally, and in the absence of refusal, the "SPT resistance" corresponds to the number of blows needed to penetrate 15 cm of the second ($N_{1}$) and third ($N_{2}$) driving increments after a first priming penetration ($N_{0}$) of 15 cm:

Ideally, the energy actually delivered to the rod string by the driving device is measured with an analyzer, via an instrumented rod.

As part of the liquefaction analyses, the disturbed sample taken from each test is subject to sieve analysis by sieving (NF P 94-056), or at least the measurement of the 80 µm sieve passing.

Input parameters - SPT

Conducting liquefaction analyses from the computation of SPT tests requires the following parameters.

¹ Strictly speaking, the fine content (FC) given for the implementation of the "NCEER" method complies with the requirements of the ASTM, and therefore corresponds to the percentage by weight of passing through the 75 µm sieve. This sieve is not used in the French standard of granulometric analyses (NF P 94-056), the closest is the 80 µm sieve.

² The sampling method depends on the type of core used: truncated cone cores (variable inner diameter 35 to 38 mm), cylindrical cores (constant inner diameter of 35 or 38 mm) that may or may not be equipped with sampling liners

Formulation of $\text{CR}R_{Mw = 7.5}$ based on the SPT tests

The cyclic resistance ratio term is calculated sequentially. The procedure is schematized in the logigram below.

Step n°1: Normalizing the number of $N_{\text{SPT}}$ blows

The evaluation criterion of cyclic resistance ratio through analysis based on the computation of SPT results rests on a normalization of the number of $N_{\text{SPT}}$ blows:

The nature and detail of the corrections made are described in the following table.

⁰ For the definition of the "type" of correction, "Automatic" means that it is directly applied by the software, "Manual" means that the correction depends on specific input parameters to be provided by the user.

¹ The normalization of the number of SPT blows relative to the stress level is calculated using the formulation of Liao and Whitman (1986). This correction term is automatically limited to $C_{N} \leq C_{N}^{\lim} = 1.7$. This correction applies to the states of stress prevailing in the studied soil at the time of the execution of the tests; they are normalized by a reference pressure taken equal to the atmospheric pressure $P_{a}$ (~100 kPa).

² The driving impact energy is expressed in terms of the energy ratio between the energy actually transmitted to the string of rods (measured with an analyzer, $E_{mesurée}$) and the theoretical maximum potential energy ($E_{\text{th}} = \rho gh$), i.e. $ER = E_{mesurée}/E_{\text{th}}$. This ratio should, of course, be positive and less than 100%; it depends on both the type of hammer/driving cap system and the hammer release device. The energy ratio$\text{\ ER}$ is reduced, via the factor $C_{E}$, to a reference value of 60%, which represents in the first approximation the value of restored energy in American practice:

In case the restored energy is not directly measured with an analyzer, i.e. where the $\text{ER}$ ratio is not a direct entry data into the software, average values of $C_{E}$ are proposed based on the driving system used for the execution of the tests (Robertson and Wride, 1998). Note that if the energy ratios $\text{ER}$ are provided as well as the driving impact device, Slake prioritizes direct measurement.

³ The total length of the rod string must strictly take into account the above-ground part to the cap of the driving device in addition to the depth of the test:

The length of the above-ground rod string varies (depending on the size of the drill rods). Slake considers by default a metric length above ground: $L_{\text{rod}}^{above-ground} = 1.0 m$, however, it can be adjusted in advanced options.

In the unavailability of accurate data on this subject, it is recommended to neglect this overlength of the rod string and consider only the depth of the test (conservative hypothesis). Note that, strictly speaking, the depth of the test should be increased by 15 cm to take into account the priming penetration.

⁴ The use of truncated cone cores (a variable inner diameter of 35 to 38 mm) or cylindrical cores with split spoons for a liner, but used without a liner, results in a significant decrease in the number of $N_{\text{SPT}}$ blows (+10/+30%). If one of these sampling methods is used, Slake applies by default a correction of 20% on the number of SPT blows. On the other hand, no correction is applied for a sampling carried out with cylindrical cores without splits (constant inner diameter of 35 mm), or with splits but equipped with liners.

Case of the "NCEER/SGT-LIQ-AFPS" calculation option

When this calculation option is selected, the determination of the correction factors $C_N$, $C_R$ and $C_S$ is modified as follows:

⁰ For the definition of the "type" of correction, "Automatic" means that it is directly applied by the software, "Manual" means that the correction depends on specific input parameters to be provided by the user.

¹ The normalization of the number of SPT blows relative to the stress level is calculated using the formula of Liao and Whitman (1986) for overburden pressures less than 200 kPa, and usig the formula of Kayen et al. (1992) for higer overburden pressures. This correction term is automatically limited to $C_{N} \leq C_{N}^{\lim} = 1.7$. This correction applies to the states of stress prevailing in the studied soil at the time of the execution of the tests; they are normalized by a reference pressure taken equal to the atmospheric pressure $P_{a}$ (~100 kPa).

² The total rod string length $L$ is determined as specified by equation (12). The correction factor for the total rod string length is taken from NF EN ISO 22476-3 (AFNOR, 2005).

³ The use of truncated cone cores (a variable inner diameter of 35 to 38 mm) or cylindrical cores with split spoons for a liner, but used without a liner, results in a significant decrease in the number of $N_{\text{SPT}}$ blows (+10/+30%). If one of these sampling methods is used, a correction of up to +30% on the number of SPT blow count should be applied. By default this value is set to +15%, but can be set manually. On the other hand, no correction is applied for sampling with cylindrical core barrels without reservations (constant internal diameter of 35 mm), or with reservations but equipped with liners.

Step n°2: Correction for fines content to get a "clean-sand" equivalent

The cyclic resistance ratio increases with increased fine content. The normalized number of blows is reduced to a "clean sand" equivalent (CS = clean sand) according to the correlations of Idriss and Seed:

The coefficients $\alpha$ and $\beta$ are determined from the detailed relationships in the following table.

¹Nota: Strictly speaking, the fine content (FC) used to establish these correlations is in accordance with ASTM, and therefore corresponds to the percentage by weight of passing through the 75µm sieve. This sieve is not used in the French standard of granulometric analyses (NF P 94-056), the closest is the 80 µm sieve.

Step n°3: Expression of cyclic resistance ratio

For a "clean sand" equivalent $(N_{1})_{60CS}$ and for an earthquake of $M_{w}$ 7.5 magnitude, the expression of normalized cyclic resistance ratio extracted from the liquefaction curves of Seed et al (1985), is given by Rauch (1998):

This formulation is valid for a normalized number of blows $(N_{1})_{60CS} < 30$; beyond this threshold the materials are considered too dense to liquefy and no factors of safety are calculated.

Assessment based on cone penetration tests (CPT et CPTu)

Principle of the CPT(u) test

The cone penetration test is framed by the French standard NF P 94-113. It consists generally of driving a probe into the ground at controlled speed (low and constant), up to a target depth or obtaining the refusal. During the driving, are recorded almost continuously (with a variable measurement step, usually of a centimetre order for electrical tips and decimetre for mechanical tips) the parameters of apparent resistance at the tip and friction around the probe's sleeve, which is widened from the rod string.

The cone penetration test with piezocone (CPTu) differs from the cone penetration test (CPT) by the additional measurement of pore-water pressure. This can be measured in different positions of the probe ($u1$, $u2$, $u3$). In the context of using tests with piezocone for implementation of the "NCEER" method, it is the measurement of pore-water pressure at the pressure sensor $u2$ (located just above the shoulder of the cone) that should be recorded.

Input parameters - CPT(u)

The conduct of liquefaction analyses from the CPT or CPT(u) tests requires information on the following parameters from the test minutes.

¹ A correction on the measured tip resistance $q_{c}$ is made necessary as the $q_{c}$ and the pressure $u2$ at the back of the cone apply to different sections. This correction is all the more important when the pore-water pressure is high, for example in soft clays (Reiffsteck, Lossy and Benedict).

The $a$ parameter is expressed as the ratio of the sections of the probe identified in the previous figure:

Providing this parameter leads to the automatic correction of the static tip resistance regarding the pore-water pressure (the correction on lateral friction measurements is negligible):

In practice, the $a$ parameter that depends on the particularities of the used probes is data that must be provided by the sounder and included on the test minutes. Its value is generally between 0.5 and 0.9 and should not be less than 0.5.

Formulation of $\text{CR}R_{Mw = 7.5}$ based on CPT(u) tests

The cyclic resistance ratio term is calculated sequentially. The procedure is schematized in the logigram below.

Step n°1: Normalization of Cone Penetration parameters CPT(u)

The criterion for evaluating the cyclic resistance ratio through analysis based on the use of cone penetration test implies the normalization of the tip resistance term:

This normalization involves a reference pressure ($P_{a}$, equal to atmospheric pressure) and an exponent on the stress ratio, $n$, which depends on the nature of the soil tested and varies between 0.5 (granular soils) and 1.0 (cohesive soils). The normalization procedure is detailed in the following figure.

For CPT(u) tests, without the possibility of sampling, the qualification of the examined soils is based on the introduction of the "soil behaviour type index" $I_{c}$ term (Robertson and Wride, 1998):

This expression depends on the normalized parameters of tip resistance $Q_{\text{tn}}$ and friction ratio $F_{R}$:

In the case of testing with piezocone (CPTu), the static tip resistance considered in the equations (16), (18) and (19) is the corrected term for pore-water pressures $q_{t}$ (14) instead of $q_{c}$.

The bi-logarithmic scale diagram $Q_{\text{tn}} = f(F_{R})$ allows, thanks to the computation of a large database, to estimate the nature and state of the tested soils. Robertson's classification of soils from the analysis of normalized parameters of in situ measurements is recalled in the following figure.

Step n°2: Correction for fines content to get a "clean sand" equivalent

The cyclic resistance ratio increases with increased fine content. Normalized tip resistance $q_{c1N}$ is reduced to a "clean sand" equivalent (CS = clean sand), according to Robertson and Wride correlations:

The correction factor $K_{c}$ is determined, as a function of the soil behaviour type index $I_{c}$, according to the criteria presented in the following table.

Step n°3: Expression of cyclic resistance ratio

For a "clean sand" equivalent $(q_{c1N})_{\text{CS}}$ and for an earthquake magnitude of 7.5, the expression of the cyclic resistance ratio characterizing the empirical liquefaction curve of a clean sand is given below (Robertson and Wride.1998):

This formulation is valid for a normalized tip resistance $(q_{c1N})_{\text{CS}}$ less than 160; beyond this threshold the materials are considered too dense to liquefy and no factors of safety are calculated.

Additional notes on the soil behaviour type index $I_{c}$

Because of the approximate nature of the relationship between the soil behaviour type index $I_{c}$, and the actual nature of the soils tested, the user is cautioned about:

• Soils with $I_{c}$ values above 2.4 should be sampled and laboratory-tested to confirm their nature (especially their fine content defined as the 80 µm sieve passing and their plasticity index).
• The "NCEER" method introduces a threshold value of the soil behaviour type index $I_{c}^{\text{cut} - \text{off}}$ beyond which the tested material is deemed too clay-rich or too plastic to liquefy. This default value is set at $I_{c}^{\text{cut} - \text{off}} = 2.6$ , but can be changed in Slake by the user in advanced options (not recommended).
• Soils with a soil behaviour type index $I_{c}$ greater than 2.6 but associated with a normalized friction ratio $F_{R}$ less than 1% should be sampled and tested in the laboratory. The latter are described by the authors of the "NCEER" procedure as very sensitive to cyclic softening (a phenomenon not quantified by the "NCEER" procedure).
• For materials with low to very low corrected tip resistances $q_{t}$ (less than 3 MPa), the interpretation of the soil behaviour type index $I_{c}$ may become questionable.

Evaluation of earthquake-induced settlements

Principles of calculation

The methods implemented in Slake are derived from the Ishihara and Yoshimine (1992) curves originally linking the volumetric strain resulting from the dissipation of excess pore-water pressures in shear tests correlated with the factor of safety regarding the liquefaction hazard and the relative density of clean sands, the latter itself being correlated to the normalized $(N_{1})_{60CS}$ and $(q_{c1N})_{\text{CS}}$ parameters. These curves have been validated by measurements of post-liquefaction settlements performed in situ, at different sites with a generally flat and sub-horizontal topography, by using the hypothesis of volumetric strains ($\epsilon_{v}$) roughly equal to principal axial strains ($\epsilon_{1}$) (one-dimensional calculation with zero lateral strain). Post-liquefaction volumetric strain rates are then linked to normalized resistance parameters for a given level of safety:

With $R^{\text{norm.}} = (q_{c1N})_{\text{CS}}$ in the case of CPT(u) tests, and $R^{\text{norm.}} = (N_{1})_{60CS}$ in the case of SPT tests.

Settlements are then determined by summing local settlement increments on all measurement points:

By default, the integration step $\bigtriangleup z_{i}$ of volumetric strains around a measurement point is given by the interval between the two measures surrounding the test point (geometric increment). However, this height can be arbitrarily bounded by a maximum integration step $L_{\max}^{\varepsilon_{v}}$:

The introduction of this limit is particularly useful in the case of SPT analyses, where discrete measurements between consecutive tests can be very spaced without an actually liquefiable layer being locally intercepted, having however a vertical extension equal to this geometric increment. The maximum integration step limit $L_{\max}^{\varepsilon_{v}}$ is by default set at 1.00 m but can be modified by the user in advanced options.

Important notes: the hypothesis of construction of post-liquefaction settlements under water-table by continuous integration of volumetric strains calculated at each measurement point of the examined soil column implicitly assumes that the settlement in depth generated by a liquefied level is fully reflected on the surface, regardless of the stratification conditions of the site and of the development of arch effects in the case of liquefied soil lenses. The volumetric strains estimated by this method are considered only by considering a fine grained matrix, without taking into account the possible presence of coarse elements of more or less large diameter (incompressible) and whose rearrangement would limit necessarily settlement on the surface.

In addition, in the case of an unconfined water-table, a sandy material located less than one meter below the water-table surface will not liquefy because it will be drained, and the post-liquefaction settlements calculated there will be overestimated. The attention is therefore focused on the fact that the settlements estimated by this method systematically correspond to a higher bound. On the other hand, estimated settlements between different survey points cannot be directly interpreted in terms of differential settlements.

Estimation of volumetric strains

Zhang, Robertson and Brachman (2002) Correlations

From CPT(u) tests

For a previously calculated factor of safety $\text{FS}$, the empirical relationship between post-liquefaction volumetric strain and normalized tip resistance for a clean-sand equivalent $(q_{C1N})_{\text{CS}}$ is given by equations shown hereafter (Zhang, Robertson and Brachman.2002), based on the curves of Ishihara and Yoshimine (1992).