Magnitude scaling factors (MSFs) account for the influence of the duration of ground motion shaking on liquefaction triggering in stress-based simplified models. The basis for MSFs stems from metal fatigue theories wherein loadings of different durations with cycles of varying amplitudes can be converted to an equivalently damaging loading of uniform cycles (i.e., a given number of equivalent cycles having a constant amplitude). Central to this conversion are factors that relate loading characteristics to induced damage. In metal fatigue analyses, this relationship is typically expressed in terms of S-N curves, where S = amplitude of the applied loading, and N = number of cycles to induce failure corresponding to S. Analogously, in liquefaction analyses this relationship is typically expressed in terms of cyclic stress ratio ðCSRÞ-N L curves, where CSR = amplitude of the applied loading, and N L = number of cycles required to trigger liquefaction corresponding to CSR. Based on empirical evidence (and mathematical expediency), both S-N and CSR-N L curves are commonly assumed to plot as straight lines in log-log space, with the line having a slope of -b. However, for both S-N and CSR-N L curves, the straight-line trend only holds for values of S that exceed the fatigue limit of the metal or for CSR values exceeding the value corresponding to the volumetric threshold strain of the soil.
For historical reasons, MSFs quantify the influence of the duration of shaking on liquefaction relative to the duration of a moment magnitude 7.5 (M7.5) event, where the durations of events having moment magnitudes M and 7.5 are expressed in terms of their corresponding numbers of equivalent stress cycles, N eq and N eq M7.5 , respectively. As noted previously, procedures for computing N eq may also rely on b-values (e.g., Liu et al. 2001;Green and Terri 2005;Hancock and Bommer 2005;Stafford and Bommer 2009;Lasley et al. 2017). For example, if a CSR-N L curve plots as a straight line in log-log space, then the Seed et al. (1975) procedure for computing N eq can be expressed as
where a i = ith amplitude (absolute value) of acceleration pulses in an acceleration time history acting in a horizontal direction recorded at the surface of a soil profile during an earthquake having magnitude M; n i = number of acceleration pulses in an acceleration time history having amplitude a i , where i ranges from 1 to the number of different amplitudes of the acceleration pulses in the acceleration time history that are greater than 0.3 • a max ; and a max = maximum value of a i . Once values for N eq and N eq M7.5 are determined, these values can be related to MSF using the following equation:
As may be surmised from Eqs. ( 1) and ( 2), the b-value is central to computing MSF and, as such, has a significant influence on computed normalized seismic demand in simplified liquefaction evaluations.
The authors have found general acceptance among practitioners and researchers that the b-value is strongly influenced by the relative density (D r ) of the soil (e.g., Boulanger and Idriss 2014). However, as detailed subsequently in this paper, many factors can affect b-values; often, the influence of these factors is ignored or not fully considered in studies that recommend b-values for liquefaction studies (Ulmer et al. 2018).
The objective of this study was to propose a b-value(s) for computing N eq and MSF for evaluating liquefaction triggering in sandy soils. Toward this end, a critical review of published laboratory data is presented assessing the dependency of the b-value on D r and other factors, such as the liquefaction triggering criterion used to analyze laboratory data, laboratory testing protocols, and quality acceptance criteria for laboratory data. With regard to quality acceptance criteria, the authors analyzed a large set of constantvolume cyclic direct simple shear (CV-CDSS) tests performed as part of this study. Then, using an alternative approach, the authors interpreted laboratory test data to determine b-values; this was an energy-based approach. This approach was extended to determine b-values using modulus reduction and damping curves, with the derived b-value being more generalized and less test-and soilspecific. The recommended b-value was then compared to other values proposed in the literature. The implications of the recommended value on MSF are also presented and discussed.
As mentioned previously, the authors found general acceptance among practitioners and researchers that the b-value is strongly influenced by the relative density (D r ) of the soil. As a result, Ulmer et al. (2018) performed a detailed review of data from numerous published laboratory studies to explore the relationships (or lack thereof) between b-values and D r . The studies that Ulmer et al. (2018) reviewed included cyclic direct simple shear (CDSS), cyclic triaxial (CTRX), and cyclic torsional shear (CTS) test data. The review showed that trends between b-values and D r are more ambiguous than is widely believed. This is due in part to the nonlinearity in the CSR versus N L curves in log-log space exhibited by some soils, particularly in the lower value range of N L for denser soils. If a straight line is fit to all data, an apparent relationship between the b-value and D r may be observed, with the b-value increasing as D r increases. However, the b-value for the linear portion of the CSR versus N L curve is relatively constant for a wide range of D r values.
For illustration, Fig. 1 displays the results of anisotropically consolidated CTS tests performed by Tatsuoka et al. (1986) on air-pluviated samples of Sengenyama sand. In Fig. 1 Based on their critical review, Ulmer et al. (2018) found that while some data show that b-values increase as D r increases (as is commonly assumed), other data show that b-values remain constant, decrease, or have no clear trend. However, Ulmer et al. (2018) found that the b-value can be affected by soil type, sample preparation method, effective confining stress, liquefaction triggering criterion, and laboratory testing protocol. In addition, Ulmer et al. (2019) found that quality acceptance criteria of laboratory data also had a significant influence on the b-value. Some of these factors are detailed in the following.
As mentioned previously, there are several factors that can affect the b-values computed from tests (Ulmer et al. 2018(Ulmer et al. , 2019)). A few of these factors are highlighted in the following.
A liquefaction triggering criterion is required to estimate N L from cyclic tests, although this criterion is somewhat ambiguous and inconsistent in published studies (Wu et al. 2004). Liquefaction is initiated (or triggered) when the vertical effective stress (σ 0 v ) reduces to zero (i.e., the complete transfer of the overburden stress to the pore water). This is often expressed in terms of excess pore water pressure ratio (i.e., r u ¼ 1), where r u is defined as excess pore water pressure, Δu, divided by initial vertical effective stress, σ 0 vo . However, due to the dilative tendencies of medium-dense to dense soils, σ 0 v ¼ 0 or r u ¼ 1 is not always achievable. As a result, strain-based liquefaction triggering criteria are commonly used, although defining liquefaction triggering in this way entails judgement, particularly for medium dense to dense soils. Common thresholds for CDSS and CTS tests range from 3% to 4% for singleamplitude shear strain (γ SA ) and 1.5% to 15% for double-amplitude shear strain (γ DA ), and common thresholds for CTRX tests range from 2% to 10% for double-amplitude axial strain (ε DA ) (e.g., Tatsuoka and Silver 1981;Tatsuoka et al. 1986;Mandokhail et al. 2017).
In addition to the ambiguity regarding which liquefaction triggering criterion to use to determine N L , there are also several issues surrounding the most commonly used criteria. For example, ambiguity exists in how the r u -based criterion is interpreted in determining liquefaction triggering in cyclic tests. As stated previously, liquefaction is unambiguously initiated when σ 0 v ¼ 0, which corresponds to r u ¼ 1, but only when the applied loading is zero. In this case, r u is referred to as residual r u (i.e., r u;Residual ) and is the value of r u in cyclic tests at times when the applied cyclic deviatoric stress (for CTRX tests) or cyclic shear stress (for CDSS or CTS tests) equals zero (i.e., r u ¼ r u;Residual two times during a loading cycle for a specimen that does not have an imposed static stress). In lieu of using r u;Residual ¼ 1 to define liquefaction, r u;Transient ¼ 1 is often used, where r u;Transient ¼ r u at any time during a cycle of loading; note that r u;Transient ¼ 1 does not necessarily correspond to σ 0 v ¼ 0. It is likely that most published studies that purport to define liquefaction as r u ¼ 1 are defining it as r u;Transient ¼ 1.0, unless it is explicitly stated that r u;Residual ¼ 1.0 is used as the liquefaction triggering criterion. The reason for this is that dense sands and silty sands often stop accumulating excess pore pressure before reaching r u;Residual ¼ 1.0, despite accumulating large strains during cyclic loading-but they may reach a state where r u;Transient ¼ 1 (Wu et al. 2004). Also, pore pressures in a soil specimen during a cyclic test can be somewhat difficult to measure accurately, particularly when excess pore pressures do not rapidly and evenly distribute throughout the soil specimen-for example, in silty sands (Casagrande 1976;Wu et al. 2004). Regarding strain-based triggering criteria, the value of N L in medium-dense to dense sands can be very sensitive to the assumed strain threshold, while the value for loose sands is not as sensitive (Wu et al. 2004;El Mohtar 2009).
Given that judgement is used in selecting the strain threshold, the value of N L for medium-dense to dense sands is inherently subjective.
To illustrate the effect of liquefaction triggering criterion, b-values and the associated standard errors (ϵ b ) were computed using published data from a number of studies in which CTRX, CDSS, and CTS tests were performed (Ulmer et al. 2018). A large ϵ b indicates a high degree of uncertainty in the associated b-value, which may result from a CSR versus N L curve that is not linear in log-log space, significant scatter in the test data, and/or the available data representing a small range of N L . In general, increasing the threshold value of a given liquefaction criterion or changing the criterion from one type (e.g., based on r u ) to another (e.g., based on strain) has an effect on the b-value and should be considered when determining b-values. For example, Fig. 2 shows that b-values from r u;Residual -based criteria can be significantly different from b-values from strain-based criteria. These differences are more pronounced in denser sands than in looser sands. Furthermore, Fig. 3 shows that sample preparation method can have varying influences on b-values determined using different liquefaction triggering criteria. If samples were prepared using air pluviation, then b-values were less affected by the choice of ε DA than if the samples were prepared using wet vibration.
The influence of testing protocol on b-values is illustrated by the results of a cooperative testing program that involved multiple laboratories performing CTRX tests on reconstituted samples from the same batch of Toyoura sand (Toki et al. 1986). As shown in Fig. 4, the results from this study yielded a range of b-values from the various laboratories, presumably due to differences in the testing protocols among the laboratories. While we did not attempt to assess the appropriateness of the testing protocols used by the various laboratories, it was clear that differences in operators, apparatuses, specimen dimensions, and potentially other unidentified factors likely affected the b-values. Note that the data in this figure also illustrate the influence of the liquefaction triggering criterion on the b-value, with the spread of the individual b-values increasing as the value of the ε DA used to define liquefaction initiation increased.
To examine how quality acceptance criteria for laboratory test data can affect b-values, Ulmer et al. (2019) performed a series of CV-CDSS tests on Monterey 0/30 sand specimens (the Supplemental Materials provide additional details regarding this testing). Monterey 0/30 sand is poorly graded clean sand with subangular to rounded particles. The index properties and grain-size distribution of this sand are shown in Table 1 and Fig. 5, respectively. The specimens were prepared using air pluviation (Vaid and Negussey 1988) to different D r values and confined to σ 0 vo values of 60, 100, or 250 kPa. The specimens were subjected to stress-controlled, sinusoidal horizontal loading at their bases, having a predetermined CSR. The tests performed by Ulmer et al. ( 2019) maintained constant volume during the cyclic phase using passive control (PC). PC maintains constant volume via a physical locking mechanism designed to minimize axial deformation. The cyclic phase of the tests continued until the horizontal linear variable differential transformer (LVDT) reached its limit, and liquefaction initiation was defined as γ SA ¼ 3.5%.
Ulmer et al. ( 2019) developed and applied acceptance criteria for the PC CV-CDSS tests. The criteria included acceptable levels of shear stress (τ ) and shear strain (γ) during consolidation (i.e., prior to cyclic loading), acceptable levels of axial strain (ε) during the cyclic loading phase, and the presence of several anomalies in the τ versus σ 0 v path (see Supplemental Materials). For each criterion, a grade of A-D was assigned, and each grade was associated with a numerical score (Table 2). Tests that earned a grade of D in any of the categories or that manifested irregular spacing or a bias in the τ versus σ 0 v path (i.e., asymmetrical shape in the stress paths around τ ¼ 0) were automatically excluded. For each remaining test, a total score was then computed as the sum of all numerical scores associated with the letter grades. It was observed, however, that soil samples with D r ¼ 25% were able to achieve higher scores more easily than samples with D r ¼ 60% or 80% (the maximum possible score was 10; a very low score was lower than -1). Therefore, the minimum total scores for the quality acceptance criteria were adjusted based on D r values of 8.5, 8.0, and 6.5 for D r ¼ 25%, 60%, and 80%, respectively, in order to have sufficient tests in each D r bin. Note that all test data that passed the quality acceptance threshold scores and many of the tests that did not would almost assuredly have been included in previously published studies (i.e., the use of the threshold quality acceptance criteria proposed herein results in an exceptionally high-quality test data set).
To investigate the effect of removing tests that did not pass the PC CV-CDSS quality acceptance criteria, with focus on σ 0 vo ¼ 100 kPa and using γ SA ¼ 3.5% to define liquefaction, Ulmer et al. (2019) computed the b-values for two scenarios: (1) only tests that passed the acceptance criteria and met the minimum total score were included; and (2) a random sample of all tests was selected. The random sample included the number of cyclic tests performed for a typical liquefaction resistance study (e.g., 4 to 5 tests). The b-values using only those tests that passed the acceptance criteria were 0.17, 0.16, and 0.19 for D r ¼ 25%, 60%, and 80%, respectively. The b-values obtained using 1,000 random samples of 4 to 5 tests from the entire data set (i.e., all available data for tests having σ 0 vo ¼ 100 kPa, including those which did not pass the new acceptance criteria but which may have generally been acceptable in other studies) for each D r ranged from 0.05 to 0.65, 0.02 to 0.39, and 0.09 to 0.31 for D r ¼ 25%, 60%, and 80%, respectively. This shows that the use of the quality acceptance criteria affected the possible range of b-values, particularly when only a few tests were performed for a given D r .
In addition to the PC CV-CDSS tests performed by Ulmer et al. (2019), the authors also performed a series of active control (AC) CV-CDSS tests as part of the present study (the Supplemental Materials provide additional details about this testing). These tests were also performed using Monterey 0/30 sand specimens prepared using air pluviation to different D r values and confined to σ 0 vo values of 60, 100, or 250 kPa. AC uses a feedback loop between the vertical actuator and an LVDT close to the top of the specimen to reduce vertical deformations (generally, axial strains were within AE0.05% before liquefaction initiated). A similar set of quality acceptance criteria to those chosen for the PC CV-CDSS tests (Table 2), with a few modifications, was developed as part of the present study and applied to the AC CV-CDSS tests. Because the AC CV-CDSS tests used an active feedback loop to maintain constant volume, the axial strains were more effectively restricted than they would have been by using physical means (e.g., PC); for nearly all of the AC tests, ε was less than 0.05% during the cyclic phase. Therefore, the AC tests achieved higher total scores in general, and the minimum acceptable total scores adopted for the PC tests did not result in the exclusion of many of the AC tests due to excessive vertical strains. However, the presence of a biased τ versus σ 0 v path (i.e., asymmetrical shape in the stress paths around τ ¼ 0) was more prevalent in the AC tests than in the PC tests. AC tests with a bias in the τ versus σ 0 v path were allowed but received a reduction in their total score of -1. Tests that did not have this bias received an increase in their total score of þ1. Each test was assessed using the acceptance criteria, and a sensitivity analysis was performed to identify the minimum total score below which the b-values would be noticeably influenced.
When developing the quality acceptance criteria for the PC tests, the focus was on tests performed with σ 0 vo ¼ 100 kPa, because there were relatively few tests performed with σ 0 vo ¼ 60 and 250 kPa. Therefore, the minimum threshold scores for the PC tests were not dependent on σ 0 vo . When developing the quality acceptance criteria for the AC tests performed as part of the present study, sufficient AC tests were performed with σ 0 vo ¼ 60 and 250 kPa to observe a relationship between the mean total score and initial vertical effective confining stress. Tests performed with σ 0 vo ¼ 250 kPa tended to be more prone to slightly higher values of γ during ramp-up, consolidation
ε ≤ 0.05% for 80% of the c.p. or until r u ¼ 0.75
a Modification for AC tests: AC tests with a bias in the τ versus σ 0 v path (i.e., asymmetrical shape in the stress paths around τ ¼ 0) received a reduction in their total score of -1, whereas tests that did not have this bias received an increase in their total score of þ1.
τ during the ramp-up and consolidation stages and more often had a biased τ versus σ 0 v path. Both of these tendencies led to a lower mean total score. However, D r did not have a significant effect on the total scores of the AC tests, mainly because the AC method was better able to maintain constant volume conditions (an issue for high D r samples in PC tests). Therefore, the identified minimum acceptable scores for the AC CV-CDSS tests were not dependent on D r but were dependent on σ 0 vo and were assigned as 7.0, 7.0, and 6.0 for σ 0 vo ¼ 60, 100, and 250 kPa, respectively, for all D r values. Applying this screening criterion resulted in the removal of approximately 20% of the AC CV-CDSS tests.
Using the AC CV-CDSS tests that passed the acceptance criteria and had a minimum total score of 10.0, the b-values were 0.19, 0.29, and 0.36 for D r ¼ 25%, 60%, and 80%, respectively (σ 0 vo ¼ 100 kPa, γ SA ¼ 3.5%). The b-values obtained using 1,000 random samples of four tests from the entire data set for each D r ranged from 0.12 to 0.32, 0.08 to 0.40, and 0.22 to 0.71 for D r ¼ 25%, 60%, and 80%, respectively. Note that the four random tests were selected such that they were not clustered around a narrow range of Alternative Approach to b-Value Determination: Dissipated Energy So far, it is clear that there is significant uncertainty in b-values determined using the foregoing approaches, due in large part to the selection of a liquefaction triggering criterion and the quality acceptance criteria of laboratory tests. It is also clear that a consistent liquefaction criterion to apply to laboratory tests that results in more confident estimates of b-values is desirable. Toward this end, the authors propose the use of an energy-based criterion, as detailed in the following.
The basis for using energy-based liquefaction triggering criteria is the relationship between dissipated energy in a unit volume of soil and the generation of excess pore pressures leading to liquefaction (Nemat-Nasser and Shokooh 1979; Green et al. 2000;Polito et al. 2008). Energy is primarily dissipated in sands due to friction developed from relative movement between sand grains as the soil skeleton breaks down under cyclic loading. In addition, dissipated energy underlies the commonly used approaches to compute N eq for earthquake motions, whether explicitly (e.g., Green and Terri 2005;Lasley et al. 2017) or implicitly (e.g., Seed et al. 1975;Boulanger and Idriss 2015). These fundamental relationships make a strong case for using an energy-based criterion to identify liquefaction initiation in laboratory tests and thereby define b-values.
Dissipated energy in a unit volume of soil (ΔW) in laboratory tests is the cumulative area bound by shear stress-shear strain hysteresis loops, such as those shown in Fig. 6, and can be computed using the trapezoidal rule (Green 2001):
where τ k = kth increment of shear stress; γ k = kth increment of shear strain (in decimal form); and n = total number of digitized points in the shear stress and shear strain time histories. A threshold value of ΔW can be used to designate when liquefaction initiates during a laboratory test, just as threshold values for shear strain, axial strain, r u;Residual , or r u;Transient can be used. To account for variations in σ 0 vo , the energy-based threshold is defined as a normalized value, ΔW=σ 0 vo . In both laboratory tests and appropriate numerical analyses, normalized dissipated energy (i.e., ΔW=σ 0 vo ) can be determined within either an effective stress or a total stress framework; an effective stress framework includes the influence of the reduction in soil stiffness on energy dissipation due to excess pore water pressures, and a total stress framework does not. Inherently, computing ΔW=σ 0 vo by directly integrating the shear stress-shear strain hysteresis loops from CV-CDSS laboratory tests (e.g., Fig. 6) results in a value of dissipated energy that includes the influence of the reduction in soil stiffness due to excess pore water pressures (i.e., ΔW eff =σ 0 vo ). This is indicated by the increased size of the hysteresis loops due to excess pore water pressure generation as cyclic loading progresses. However, in the early phases of loading, the hysteresis loops are relatively constant in shape; the size of the loops increases in later phases of loading as excess pore water pressures increase. This is illustrated by the constant slope of ΔW eff =σ 0 vo from a CV-CDSS test plotted as a function of the number of cycles of applied loading (N cyc ) (e.g., Fig. 7), where ΔW eff =σ 0 vo is plotted every half-cycle of loading at times when the applied shear stress is equal to zero (i.e., τ ¼ 0). The influence of the reduction in soil stiffness due to excess pore water pressures results in the plot of ΔW eff =σ 0 vo versus N cyc deviating from a straight line; this occurred at approximately 6.5 cycles for the CV-CDSS test data shown in Fig. 7.
Based on the trends illustrated in Fig. 7, the influence of the reduction in soil stiffness due to excess pore water pressures on dissipated energy can be removed by extrapolating the straight-line Fig. 6. Shear stress-strain hysteresis loops of a CDSS test on Monterey 0/30 sand (D r ¼ 62%, CSR ¼ 0.156, σ 0 vo ¼ 100 kPa).
Fig. 7. Illustration of effective and total normalized dissipated energy (ΔW eff =σ 0 vo and ΔW total =σ 0 vo , respectively) for the same CDSS test on Monterey 0/30 sand represented in Fig. 6. portion of the plot of ΔW eff =σ 0 vo versus N cyc from the early phase of loading to the later phase of loading. This is illustrated in Fig. 7; the extrapolated line represents ΔW=σ 0 vo determined within a total stress framework (i.e., ΔW total =σ 0 vo ). Note that ΔW total =σ 0 vo is comparable to values of ΔW=σ 0 vo determined using equivalent-linear site response analyses where damping and shear modulus remain unchanged throughout the application of the loading time history (Green 2001). Threshold values of ΔW eff =σ 0 vo or ΔW total =σ 0 vo can be used to define the initiation of liquefaction in a laboratory test, with the corresponding value of N cyc being designated as N L . The threshold value of ΔW total =σ 0 vo or ΔW eff =σ 0 vo used as the liquefaction triggering criterion is somewhat subjective (similar to selecting a threshold value of shear or axial strain), but ΔW total = σ 0 vo ¼ 0.001 or ΔW eff =σ 0 vo ¼ 0.005 were used herein because they are closely associated with the point at which r u;Residual reached 1.0 or its maximum value in the tests, as shown in Fig. 8.
The AC CV-CDSS tests performed as part of this study that met the quality acceptance criteria (Table 2) and met the minimum scores (7.0, 7.0, and 6.0 for σ 0 vo ¼ 60, 100, and 250 kPa, respectively) were analyzed using the energy-based criteria. The value of N L was estimated for each test, and b-values were computed using a best-fit regression of the data using a power-law function (i.e., a straight line in log-log space). The value of ϵ b was also obtained as part of the curve-fitting algorithm and represents the uncertainty in the b-values.
Fig. 9 shows the relationship between b-values (with error bars representing AEϵ b ) and D r using the energy-based criteria of ΔW total =σ 0 vo ¼ 0.001 and ΔW eff =σ 0 vo ¼ 0.005. Only AC CV-CDSS tests that passed the quality acceptance criteria set forth in Table 2 were included in this figure (the Supplemental Materials provide additional details about these tests). The figure shows that, in general, b-values were relatively constant with increasing D r when ΔW total =σ 0 vo ¼ 0.001 was used as the liquefaction triggering criterion; they were a little less so when ΔW eff =σ 0 vo ¼ 0.005 was used. In addition, the b-values were relatively insensitive to the selection of the threshold value of ΔW total =σ 0 vo to define liquefaction, as long as the selected ΔW total =σ 0 vo threshold values were reasonable. For comparison purposes, the same test data were analyzed using more traditional liquefaction triggering criteria, γ SA ¼ 3.5% and r u;Residual ¼ 0.98. These latter results are plotted in Fig. 10 to a specific soil by using contours of constant dissipated energy computed using modulus reduction and damping (MRD) curves, as mentioned in Green et al. (2019). Assuming that the relationship between CSR and N L is a contour of constant dissipated energy, the b-value representing this relationship can be computed by estimating CSR for a range of N L values from the dissipated energy per unit volume of soil for a M7.5 event (ΔW M7.5 ). Toward this end, the same approach used in equivalent-linear site response analysis can be used to compute ΔW M7.5 : using a viscoelastic constitutive model in conjunction with MRD curves (e.g., Ishibashi and Zhang 1993;Darendeli 2001). This approach is represented with the following equation:
where N eq M7.5 is assumed to be 14 cycles (e.g., Green et al. 2019), per Approach 2 of Lasley et al. (2017) for shallow crustal events in active tectonic regimes (e.g., western US), where Approach 2 accounts for both horizontal components of motion; CRR M7.5 is the cyclic resistance ratio normalized to the shaking duration of a M7.5 event determined from a liquefaction triggering curve (e.g., Green et al. 2019); K γ accounts for the overburden per Green et al. (2022) and is analogous to the overburden correction factor, K σ ; G max is the small-strain shear modulus; and D γ and ðG=G max Þ γ are the damping and shear modulus ratios, respectively, associated with a given value of γ.
Because the relationship between CSR versus N L is assumed to be a contour of constant dissipated energy, the remaining portions of the curve can be computed for different amplitudes of loading (i.e., CSR) by simply computing the number of cycles for the assumed loading amplitude required for the dissipated energy to equal ΔW M7.5 . In this approach, the ΔW for one cycle of loading (i.e., ΔW 1 ) having amplitude CSR is computed as:
and the N L corresponding to this CSR amplitude is ΔW M7.5 =ΔW 1 . Note that in this approach for determining the CSR versus N L curve, ΔW M7.5 and ΔW 1 are analogous to total dissipated energy (ΔW total ) in laboratory tests. Once the CSR versus N L curve that represents a contour of constant dissipated energy is developed, the b-value is estimated following the same approach as is used in determining the b-value from laboratory test data, using a power-law function to fit the CSR versus N L relationship.
To estimate ΔW M7.5 and ΔW 1 , G max is computed as
s where ρ is mass density of the soil and V s is small-strain shear-wave velocity. The value of ρ can be reasonably assumed to be 1,988.45 kg=m 3 (19.5 kN=m 3 , e.g., Moss et al. 2006;Green et al. 2014), and V s can be estimated from normalized cone tip resistance from the cone penetration test (CPT) corrected for fines, q c1Ncs (Boulanger and Idriss 2014), using a correlation developed by Ulmer et al. (2020) specifically for liquefaction evaluations:
where P a is atmospheric pressure in the same units as σ 0 vo . Although an estimate of V s (or G max ) is necessary to compute ΔW M7.5 or ΔW 1 individually, it appears in both the numerator and the denominator when computing N L ¼ ΔW M7.5 =ΔW 1 , and it is cancelled out. Therefore, the value of N L (and b-value, as will be shown subsequently) computed from MRD curves in this manner is not contingent on the chosen V s or G max correlation.
Values of D γ and ðG=G max Þ γ can be determined using any published, applicable MRD curves. In this study, both the Ishibashi and Zhang (1993) and Darendeli and Stokoe (Darendeli 2001) curves are used, hereafter denoted IZ and DS, respectively. These MRD curves are dependent on the initial mean effective stress, σ 0 mo , and soil type or plasticity index, PI. For the purposes of this study, PI was assumed to equal zero and σ 0 mo was computed as a function of the at-rest lateral earth pressure coefficient, K o , which is assumed to equal 0.5. Fig. 11 shows an example set of CSR versus N L curves developed using this method in conjunction with the IZ MRD curves.
Using this procedure, b-values were regressed for a range of q c1Ncs and σ 0 vo values (Table 3; additional details provided in Table S2). Values of q c1Ncs were correlated to D r using the following expression (Salgado et al. 1997a, b;Idriss and Boulanger 2008):
where D r is in decimal. These same b-values are plotted as a function of D r in Fig. 12 along with the b-values from the AC CV-CDSS tests shown previously using energy-based liquefaction triggering criteria: ΔW total =σ 0 vo ¼ 0.001. In general, b-values remain relatively constant with increasing D r and σ 0 vo . The b-values computed using IZ MRD curves are higher than the b-values from using the DS MRD curves for all combinations of D r and σ 0 vo . Although either of the two MRD curves used in this study (among others) could be used to estimate b-values, the authors prefer the IZ MRD curves because they are valid for the full range of shear strains of interest and have been shown to yield consistent trends with liquefaction test data (Green et al. 2022). An average b-value from the IZ MRD curves for σ 0 vo ¼ 100 kPa is 0.28. Note that although the b-values showed some sensitivity to changes in D r and σ 0 vo , the ranges of b-values from the IZ and DS MRD curves shown in this study (IZ: 0.25-0.31; DS: 0.18-0.23) have only a mild impact on MSF. Finally, it is noted that although the range of b-values for the DS MRD curves better match the b-values from the laboratory tests performed as part of this study, the range of b-values for the IZ MRD curves better represent the range from laboratory tests presented in literature for varying soils, test conditions, etc.
Although it does not provide comprehensive coverage of every study on N eq and MSF, this section presents b-values used in some recent and/or prominent studies for computing N eq and/or MSF and compares them to the b-value recommended herein (i.e., b ¼ 0.28). The previous studies included in this comparison are Arango (1996) [A96], Idriss (1997) [I97], Liu et al. (2001) [Lea01], Boulanger and Idriss (2014) [BI14], and Green et al. (2019) [Gea19]. A brief summary of the bases for the b-values used in these studies is presented, but particular attention is given to BI14 because it is one of the most recent studies on the list and also because it is the only study on the list that proposes a D r -dependent b-value.
Similar to the present study, Arango (1996) developed a b-value for a proposed MSF relationship based on energy principles. However, the Arango (1996) b-value was based on stored energy in a cycle of loading as opposed to dissipated energy in a cycle of loading. As a result, his derived b-value (i.e., b ¼ 0.5) inherently assumes that the soil's response is constrained to the elastic range, which is inconsistent with the liquefaction phenomenon. I97 repeated the N eq study performed by Seed et al. (1975) but used a b-value that he derived from cyclic triaxial test data from Yoshimi et al. (1984), b ¼ 0.34. Based on an extensive review of published CDSS data (i.e., De Alba et al. 1976;Ishihara and Yamazaki 1980;Tatsuoka and Silver 1981;Boulanger and Seed 1995), Lea01 used b-values ranging from 0.33 to 0.39 for sand having 45% ≤ D r ≤ 70% to develop their N eq relationship, with an approximate average value of b ¼ 0.36. In addition, Lea01 also used the b-value derived by Arango (1996) (i.e., b ¼ 0.5) in developing their N eq relationship. From a limited parametric study using the IZ modulus reduction and damping (MRD) curves in conjunction with the dissipated energy approach detailed in this paper, Gea19 used b ¼ 0.34 for their proposed MSF relationship. Because this value was consistent with the value used by I97 based on laboratory data, Gea19 did not perform a more extensive parametric study. Similar to the present study, all of these past studies proposed D r -independent b-values; BI14 proposed a new MSF relationship as part of their revised stress-based liquefaction evaluation model, which entailed a D r -dependent b-value relationship. BI14 assumed that motions for M ≤ 5.25 can be reasonably represented by three-fourths cycles having amplitude a max . Per Eq. (1), N min (i.e., N eq for M ≤ 5.25) for these motions can be computed as follows:
where b varies as a function of soil density as shown in Fig. 13; this was back-calculated from expressions given in Boulanger and Idriss (2014), and Eq. ( 7) was used to relate q c1Ncs and D r . However, based on a parametric analysis, BI14 determined the N eq M7.5 ≈ 15 cycles, independent of D r . Using these values of N min and N eq M7.5 in conjunction with Eq. ( 2) for MSF and the b-value relationship shown in Fig. 13, BI14 developed an MSF relationship by interpolating/extrapolating in logarithmic space for other magnitude values. As a result, the dependency of the BI14 MSF relationship on soil density is most pronounced for M ≤ 5.25 and inherently becomes less dependent on soil density as M approaches 7.5 (and is independent of soil density for M7.5).
To illustrate the effect on MSF of using a D r -independent b-value versus a D r -dependent b-value, the b-value recommended herein (i.e., b ¼ 0.28) and the b-value relationship proposed by BI14 (i.e., Fig. 13) were used to compute MSF for case histories in the BI14 CPT-based case history database. Because BI14 did not propose an N eq relationship per se, the N eq relationship proposed by Lasley et al. (2017) for shallow crustal events in active tectonic regimes (e.g., western US) was used in conjunction with Eq. ( 2) for MSF for this illustration:
Eq. ( 9) expresses N eq as a function of M and a max (in units of g). Based on a review of liquefaction case history databases, a max for the reference M7.5 event was assumed to be 0.35g. This resulted in N eq M7.5 ¼ 14 cycles (Green et al. 2019). Therefore, Eq. ( 2) reduces to
The resulting MSF relationships are shown in Fig. 14. Per Eq. ( 9) the MSF varies as a function of a max and M (i.e., MSF ¼ 1 for M7.5 and a max ¼ 0.35g), and the effect of the b-value's dependency on D r is noticeable. For example, when the D r -dependent b-value is used, the MSF ranged from 1.09 to 1.21 for M6.0 and a max ¼ 0.5g for D r ¼ 20% to 80%. Also plotted in Fig. 14 are the MSFs for the liquefaction/no-liquefaction case histories in the BI14 CPT database, computed using both the D r -independent and D r -dependent b-values. Nearly all of the case histories were associated with 5.9 ≤ M ≤ 7.7, and only one event was outside this range (2011 Tohoku, Japan: M9). In addition, the majority of the critical layers identified in this case history database had a D r between 30% and 70%. Given this relatively confined range, the b-value D r -dependency may seem to have a relatively minor impact when the resulting MSFs are compared directly, as shown in Fig. 15. However, for forward analyses cases where D r , a max , and/ or M approach their lower or upper extremes, such as evaluating liquefaction during induced events (e.g., Green et al. 2020), the difference between D r -independent and D r -dependent MSFs can be significant.
An important assumption inherent to Eq. ( 1) (for computing N eq for earthquake ground motions) and to Eq. ( 2) (for MSF) is that the CSR versus N L relationship is linear in log-log space. However, several studies show that this relationship is not always linear, or at least it is not linear over the full range of N L (Mandokhail et al. 2017;Ulmer et al. 2018). This nonlinearity is particularly pronounced in denser soils compared to looser soils and in the lower N L range and may skew the observed relationship between b-values and D r , leading to the common assumption that b-values increase with increasing D r . This, along with the dependency of b-values on the liquefaction triggering criterion and quality acceptance criteria in laboratory tests, provided an impetus to devise an alternative approach for determining the b-value. The merits of an The energy-based criterion proposed herein is consistent with how N eq is determined using fatigue theories (Green and Terri 2005) and correlates well with pore pressure generation (Green et al. 2000;Polito et al. 2008). In this study, the laboratory testing results showed that b-values derived from using ΔW total =σ 0 vo ¼ 0.001 as the liquefaction triggering criterion were relatively insensitive to changes in D r and σ 0 vo , which justifies the use of a constant b-value rather than a b-value that is a function of D r and σ 0 vo . The b-values based on MRD curves are computed using contours of constant ΔW, are also relatively insensitive to changes in D r and σ 0 vo , and represent a wider range of soils than b-values determined from cyclic laboratory tests performed on a single soil type or on a few different soils. Therefore, b ¼ 0.28 is recommended; this value was determined using the IZ MRD curves.
Although the recommended b-value of 0.28 is higher than the values obtained from the AC CV-CDSS tests using energy-based criteria and performed as part of this study (0.15-0.25), it is within the reasonable range of b-values from other studies (i.e., Fig. 13
The objective of this study was to recommend a b-value(s) to compute N eq and/or MSF as part of simplified, stress-based liquefaction triggering frameworks. Some of the conclusions drawn from this study are as follows:
1. An analysis in which b-values were computed using results of cyclic laboratory tests in the literature showed that b-values can be sensitive to many parameters, including soil type, sample preparation method, confining stress, liquefaction triggering criterion, and acceptance criteria in laboratory tests. The relationship between b-value and D r is more ambiguous than
J. Geotech. Geoenviron.Eng., 2022, 148(12): 04022113 Downloaded from ascelibrary.org by VIRGINIA TECH UNIVERSTIY on 12/15/22. Copyright ASCE. For personal use only; all rights reserved.
© ASCE 04022113-4 J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 2022, 148(12): 04022113 Downloaded from ascelibrary.org by VIRGINIA TECH UNIVERSTIY on 12/15/22. Copyright ASCE. For personal use only; all rights reserved.
© ASCE 04022113-9 J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 2022, 148(12): 04022113 Downloaded from ascelibrary.org by VIRGINIA TECH UNIVERSTIY on 12/15/22. Copyright ASCE. For personal use only; all rights reserved.
© ASCE 04022113-11 J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 2022, 148(12): 04022113 Downloaded from ascelibrary.org by VIRGINIA TECH UNIVERSTIY on 12/15/22. Copyright ASCE. For personal use only; all rights reserved.
© ASCE 04022113-12 J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 2022, 148(12): 04022113 Downloaded from ascelibrary.org by VIRGINIA TECH UNIVERSTIY on 12/15/22. Copyright ASCE. For personal use only; all rights reserved.