Expanded Byrne model for evaluating seismic compression

Abstract

Seismic compression is the accrual of contractive volumetric strain in unsaturated or partially saturated sandy soils during earthquake shaking and has caused significant distress to overlying and nearby structures. The phenomenon can be well characterized by load-dependent, interaction macro-level fatigue theories. Toward this end, the Byrne cyclic shear-volumetric strain coupling model is expanded and calibrated for evaluating seismic compression for several soil types. In addition, the model was transformed to allow it to be implemented in a “simplified” manner, in addition to the original “non-simplified” formulation. Both implementation approaches are used to analyze a site in Japan impacted by the 2007, M_w6.6 Niigata-ken Chuetsu-oki earthquake. The results from the analyses are in general accord with the post-earthquake field observations and highlight the sensitivity of predicted magnitude of the seismic compression to the input variables used and modeling assumptions (e.g. relative density of the soil, magnitude of the volumetric threshold strain, orientation of the ground motions, settlement of soils below the ground water table, and accounting for multidirectional shaking). Although additional studies are needed to further validate the findings presented herein, estimation of relative density and threshold shear strain of the soil and ground motion orientation individually have moderate-to-significant influence on the computed magnitude of seismic compression, but they have a significant influence when taken in combination. Also, the seismic compression models can seemingly be used to predict the settlement in fully saturated sand when the excess pore water pressures are limited. Finally, accounting for multidirectional shaking has a significant influence on the computed magnitude of seismic compression.

Keywords

Seismic compression ground failure soil compaction soil consolidation and settlement unsaturated soil mechanics

Introduction

Seismic compression is the accrual of contractive volumetric strain in unsaturated or partially saturated sandy soils during earthquake shaking (i.e. vibration-induced settlement) (Stewart et al., 2004a). Seismic compression has occurred in several earthquakes and can significantly distress overlying and nearby structures (e.g. Siddharthan and El-Gamal, 1996; Slosson, 1975; Stewart et al., 2004a). Adopting the terminology used for liquefaction triggering procedures, with slight modification, seismic compression evaluation procedures can be broadly classified as “simplified” and “non-simplified.” In the context used herein, simplified approaches use relatively simple ground motion parameterization to characterize the seismic demand (e.g. effective shear strain, γ_eff, and number of equivalent strain cycles, n_eqγ), while non-simplified procedures use a more-detailed characterization of seismic demand (e.g. shear strain, γ, time histories computed using numerical site response analyses).

The majority of the seismic compression evaluation procedures proposed to date are simplified procedures, with an evolved form of the Tokimatsu and Seed (1987) procedure defining the state of practice. Consistent with how simplified liquefaction triggering procedures are defined, the Tokimatsu and Seed (1987) procedure uses a magnitude (M) 7.5 as a reference scenario and quantifies the seismic demand in terms of γ_eff and n_eqγ. Using these seismic demand parameters, volumetric strain is then estimated using correlations derived from the observed trends in laboratory test data.

To the authors’ knowledge, Finn and Byrne (1976) were the first to propose a non-simplified approach for evaluating seismic compression. In their procedure, the seismic demand is quantified in terms of shear strain time histories acting on horizontal planes at various depths within the soil profile, computed by numerical site response analyses. Increments in volumetric strain are then computed using a model proposed by Martin et al. (1975) that relates shear and volumetric strains. As discussed in Green and Lee (2006) and Lasley et al. (2016b), the Martin et al. (1975) model is a load-dependent, interaction macro-level fatigue model, as is the subsequently proposed variant by Byrne (1991) (i.e. the nature of the accumulation of volumetric strain is a function of the amplitude of the load and is influenced by previous loading: Kaechele, 1963). Both the Martin et al. (1975) model and the Byrne (1991) variant were calibrated to the same clean sand data set that was subsequently used by Tokimatsu and Seed (1987). It is difficult to state what defines the state of practice of non-simplified seismic compression evaluation procedures because they have not been widely adopted by practice. However, the non-simplified procedure by Lasley et al. (2016b) is the latest one that has been proposed, at least to the authors’ knowledge.

The main advantage of non-simplified procedures is that they allow for the use of a more-detailed characterization of the seismic demand at all depths in the profile. Most notably, this allows the variation in induced shear strains over the duration of shaking to be accounted for, which influences the resulting volumetric strain in materials that exhibit load-dependent, interaction fatigue behavior. Because performing site response analyses needed for implementing non-simplified procedures has become more commonplace, non-simplified procedures are a viable option for more accurately predicting seismic compression in today’s practice. The disadvantage of using non-simplified procedures is that they require more effort to implement, to include a more-detailed characterization of the site being analyzed, selection of appropriate input ground motions for the site response analysis, and the complexity of implementing the procedure itself.

The objective of the study presented herein is to expand the Byrne cyclic shear-volumetric strain coupling model to accurately predict seismic compression for several soil types and to provide both simplified and non-simplified versions of the model. Achievement of this objective will further the overall goal of providing a framework that can accurately evaluate seismic compression, that is scalable based on available data and the importance of the project, and that overcomes some of the complexity issues with existing models in implementing non-simplified procedures.

In the following, additional information is provided regarding simplified and non-simplified seismic compression evaluation procedures. Next, the Byrne model is expanded to better account for observed volumetric strain behavior in laboratory test data and is calibrated for different soil types using data from literature and using published simplified procedures. The proposed simplified and non-simplified forms of the expanded Byrne model are then used to analyze a well-documented case history from the 2007, M_w6.6 Niigata-ken Chuetsu-oki earthquake. The sensitivity of predicted magnitude of the seismic compression is assessed based on the input variables used and modeling assumptions (e.g. relative density of the soil, magnitude of the volumetric threshold strain, orientation of the ground motions, settlement of soils below the ground water table, and accounting for multidirectional shaking).

Background

Simplified procedures

The first simplified seismic compression procedure was proposed by Seed and Silver (1972). In this procedure, the shear strain time histories acting on horizontal planes at various depths within a profile are computed using a numerical site response analysis, from which both γ_eff and n_eqγ are computed as a function of depth within the profile. Drained cyclic direct simple shear tests performed on samples representative of in situ soil and state are used to develop relationships among γ_eff, relative density (Dr), n_eqγ, and volumetric strain (ε_v), where Seed and Silver (1972) present such relationships for Crystal Silica No. 20 sand (i.e. a uniform angular quartz sand having D₁₀∼ 0.5 mm and a uniformity coefficient of ∼1.5; D₁₀ is the effective soil particle diameter corresponding to 10% passing on the grain size distribution curve).

Tokimatsu and Seed (1987) furthered the simplified framework put forward by Seed and Silver (1972) in several ways. In the Tokimatsu and Seed (1987) procedure, γ_eff is estimated using an expression that was derived similarly to the one used to compute cyclic stress ratio (CSR) in simplified liquefaction evaluation procedures (Dobry et al., 1982):

γ_{eff} = \frac{τ_{av}}{G_{γ_{eff}}} \approx \frac{0.65 \cdot \frac{a_{\max}}{g} \cdot σ_{v} \cdot r_{d}}{G_{\max} \cdot {(\frac{G}{G_{\max}})}_{γ_{eff}}}

(1)

where τ_av is the “average” cyclic stress imposed on the soil at given depth in the profile over the duration of strong ground shaking; G_γeff is the secant shear modulus corresponding to γ_eff and having the same units as τ_av; a_max is the peak horizontal ground acceleration at the surface of the soil profile; g is the acceleration due to gravity in the same units as a_max; σ_v is the total vertical stress at the depth of interest; r_d is the dimensionless depth-stress reduction factor that accounts for the nonlinear (NL) response of the profile during earthquake shaking; G_max is the small-strain (γ < 10⁻⁴%) secant shear modulus in the same units as σ_v; and (G/G_max)_γeff is the ratio of G_γeff and G_max. Because γ_eff is a function of a G_γeff (or G_max·(G/G_max)_γeff), which in turn is a function of γ_eff, Equation 1 needs to be solved iteratively or using the chart solution proposed by Tokimatsu and Seed (1987), or similar ones.

The resulting γ_eff value is used in conjunction with n_eqγ, which is estimated using a correlation that relates n_eqγ to M, a_max, site-to-source distance, and/or other parameters, to define the seismic demand imposed on the soil at a given depth in the profile. Estimation of γ_eff and n_eqγ using this approach avoids the need to perform numerical site response analyses and the associated efforts of performing detailed site characterization and selecting appropriate input ground motions for the site response analysis.

Using the laboratory data for Crystal Silica Sand No. 20 from Silver and Seed (1971) and Seed and Silver (1972), Tokimatsu and Seed (1987) developed the relationships shown in Figure 1, one relating ε_v for n_eqγ = 15 (i.e. ε_v,₁₅), relative density (Dr) of the soil, and γ_eff, and the other relating the volumetric strain ratio (C_N), which is the ratio ε_v for a given value of n_eqγ to ε_v_,15 (i.e. C_N = ε_v_,n/ε_v_,15), and n_eqγ. The basis for using n_eqγ = 15 as a reference condition was likely to provide consistency with the simplified liquefaction triggering procedures which use M7.5 as the reference condition, with early correlations relating M and number of equivalent stress cycles (n_eqτ) predicting n_eqτ = 15 for M7.5 (e.g. Seed et al., 1975).

Several laboratory studies have built on the Tokimatsu and Seed (1987) framework by examining the effect of saturation (S), Dr, fines content (FC), mineralogy, fabric, overconsolidation ratio (OCR), plasticity index (PI), effective overburden stress $(σ'_{v})$ , and multidirectional shaking on seismic compression (e.g. Carter et al., 2016; Chu and Vucetic, 1992; Duku et al., 2008; Hsu and Vucetic, 2004; Pyke et al., 1975; Sawada et al., 2006; Stewart et al., 2004a; Whang, 2001; Whang et al., 2004; Yee et al., 2014). Also, additional studies have developed revised number of equivalent cycle correlations (e.g. Green and Lee, 2006; Hancock and Bommer, 2005; Lasley et al., 2017; Lee and Green, 2017; Liu et al., 2001; Stafford and Bommer, 2009). As a result, the overall simplified framework proposed by Tokimatsu and Seed (1987) has evolved, but still underlies the state of practice for evaluating seismic compression.

As part of the evolution of the Tokimatsu and Seed (1987) procedure, Stewart et al. (2004b) and Duku et al. (2008) performed extensive laboratory tests on 16 different types of clean sands and proposed the following relationships for ε_v_,15 and C_N:

ε_{v, 15} = a \cdot {(γ_{eff} - γ_{tv})}^{b}

(2a)

C_{N} = R \cdot \ln (n_{eq γ}) + c

(2b)

where a and b are material-specific constants, γ_eff is in percent, γ_tv is the volumetric threshold strain in percent (0.01%–0.03% for sand; Hsu and Vucetic, 2004), R is the slope of the line fit through C_N versus log(n_eqγ) data, and c = 1 − (ln(15)·R). Duku et al. (2008) found that the material-specific constant a varied as a function of Dr and $σ'_{v}$ , and proposed the following relationship for a for σ′_v = 1 atm (i.e. a_1 atm):

a_{1 atm} = 5.38 \cdot \exp (- 0.023 \cdot Dr)

(2c)

where Dr is in percent.

To compute a for different effective overburden stresses (i.e. a_σ), Equation 2c is multiplied by following overburden correction factor:

K_{σ, ε} = \frac{a_{σ}}{a_{1 atm}} = {(\frac{{σ'}_{v}}{P_{a}})}^{- 0.29}

(2d)

where P_a is atmospheric pressure and has the same units as $σ'_{v}$ .

Duku et al. (2008) found that b and R could be treated as constants for the clean sands tested when used in conjunction with the above expression for a: b = 1.2 and R = 0.29. Furthermore, they found that mean grain size, uniformity coefficient, particle angularity, soil fabric, mineralogy, and void ratio “breadth” (i.e. void ratio minus minimum void ratio: e − e_min), S, and age do not significantly influence the seismic compression response of clean sands. Accordingly, the resulting simplified expression used to compute volumetric strain is:

ε_{v} = K_{σ, ε} \cdot a_{1 atm} \cdot {(γ_{eff} - γ_{tv})}^{b} \cdot C_{N}

(2e)

where γ_eff and γ_tv are in percent.

Yee et al. (2014) continued the work of Duku et al. (2008) by testing non-plastic-to-moderately-plastic silty sands/sandy silts (i.e. PI ≤ 10), with FC ranging from 0% to 60%. In contrast to clean sands, Yee et al. (2014) found that FC and S influence the volumetric strain behavior of the silty sands/sandy silts tested. For consistency, Yee et al. (2014) used the same functional form of the equations proposed by Duku et al. (2008), but proposed the following “correction” factors for FC and S.

Fines content (FC):

K_{FC} = \frac{a_{FC}}{a_{FC = 0}} = {\begin{matrix} 1 & if 0 \leq FC \leq 10 % \\ e^{- 0.042 \cdot (FC - 10)} & if 10 % < FC < ~ 35 % \\ 0.35 & if FC \geq ~ 35 % \end{matrix}

(3a)

Saturation (S):

K_{S} = \frac{a_{S}}{a_{S = 0}} = {\begin{matrix} - 0.017 \cdot S + 1 & if S < 30 % \\ 0.5 & if 30 % \leq S < 50 % \\ 0.05 \cdot S - 2 & if 50 % \leq S < 60 % \\ 1 & if S \geq 60 % \end{matrix}

(3b)

where FC and S are in percent.

Accordingly, the resulting simplified expression to compute volumetric strain is:

ε_{v} = K_{FC} \cdot K_{S} \cdot K_{σ, ε} \cdot a_{1 atm & FC = 0 & S = 0} \cdot {(γ_{eff} - γ_{tv})}^{b} \cdot C_{N}

(3c)

where γ_eff and γ_tv are in percent.

Although Yee et al. (2014) found that it was reasonable to assume that b can be treated as a constant having the same values as determined by Duku et al. (2008) for clean sands (i.e. b = 1.2), they found that R cannot be treated as a constant. Rather, Yee et al. (2014) found that R varied as a function of the imposed shear strain:

R = - 0.026 \cdot \ln (γ_{eff} - γ_{tv}) + 0.26

(3d)

where γ_eff and γ_tv are in percent.

Non-simplified procedures

As mentioned in the Introduction, only a few non-simplified procedures have been proposed for evaluating seismic compression (e.g. Byrne, 1991; Finn and Byrne, 1976; Lasley et al., 2016b; Martin et al., 1975; Nasim and Wartman, 2006). Most significantly, Byrne (1991) proposed the following variant of the Martin et al. (1975) non-simplified model to estimate volumetric strains in dry sands:

ε_{v} = \sum_{i} {(Δ ε_{v, 1 / 2})}_{i}

(4a)

where ε_v is the accumulated volumetric strain in percent at the end of loading; and (Δε_v_,1/2)_i is the increment in volumetric strain in percent at the end of the ith half-shear strain cycle of loading having an amplitude γ_i. For earthquake loading, γ_i is typically taken as the peak shear strain between two zero crossings in the shear strain time history (e.g. Green and Terri, 2005). (Δε_v_,1/2)_i is computed as:

{(Δ ε_{v, 1 / 2})}_{i} = 0.5 \cdot (γ_{i} - γ_{tv}) \cdot C_{1} \cdot \exp [- C_{2} \frac{ε_{vi}}{(γ_{i} - γ_{tv})}]

(4b)

where C₁ and C₂ are material-specific parameters; ε_vi is the volumetric strain in percent at the beginning of the ith load increment; where γ_i and γ_tv are in percent. Based on the analysis of the laboratory data for Crystal Silica Sand No. 20 from Silver and Seed (1971) and Seed and Silver (1972) (i.e. the same data used by Tokimatsu and Seed, 1987), Byrne (1991) provided expressions to estimate C₁ and C₂:

C_{1} = 7, 600 \cdot D r^{- 2.5}

(4c)

C_{2} = \frac{0.4}{C_{1}}

(4d)

where Dr is in percent.

Although neither Martin et al. (1975) nor Byrne (1991) make reference to fatigue theories, their models are inherently load-dependent, interaction macro-level fatigue models in which ε_v is used as the damage metric (e.g. Kaechele, 1963). This means that the nature of the accumulation of volumetric strain is a function of the amplitude of the load and is influenced by previous loading (i.e. sequencing of the pulses in a loading history influences the resulting volumetric strain) (e.g. Green and Lee, 2006; Lasley et al., 2016b, 2017). The basis for this type of model comes directly from the observed volumetric strain behavior in laboratory tests, with this behavior not being accounted for by the procedures used to develop many of the existing n_eqτ and n_eqγ relationships (more details on this are provided in Green and Terri (2005) and Green and Lee (2006)).

Lasley et al. (2016b) proposed a variant of the macro-level fatigue model by Richart-Newmark (1948) (i.e. the R-N model) for evaluating seismic compression. The R-N model has the general form:

D = H^{r}

(5a)

where D is the accumulated “damage” (e.g. ε_v); H is the cycle ratio (i.e. the ratio of number of applied cycles having a given amplitude to the number of cycles of that amplitude required to cause “failure” in the material: H = n/N); and r is a material-specific parameter that varies as a function of the amplitude of loading (e.g. γ). For evaluating seismic compression due to earthquake loading, Equation 5a expands to:

{ε_{v}}_{i} = {[{({ε_{v}}_{i - 1})}^{1 / r_{i}} + (\frac{n_{i}}{N_{i}})]}^{r_{i}}

(5b)

where ε_vi is the volumetric strain at the end of the ith load increment. To introduce the interaction behavior to the model, Lasley et al. (2016b) made r a function of H (i.e. the nature of the accumulation of volumetric strain is influence by previous loading). Unfortunately, this makes the model somewhat difficult to implement (e.g. Yee and Stewart, 2018), which is a significant impediment to its use. Because it is easier to implement than the modified R-N model proposed by Lasley et al. (2016b) and yields similar results, the Byrne (1991) model is used as the basis for advancing non-simplified seismic compression procedures herein.

Expanded Byrne (1991) model

Simplified form of the Byrne model

As detailed in Supplemental Material, the Byrne model can be written in the alternative form:

{ε_{v}}_{i} = - \ln (\underset{i}{Π} t_{i}) \cdot \frac{(γ_{i} - γ_{tv})}{C_{2}}

(6a)

where:

t_{i} = {\begin{matrix} e^{- 0.5 \cdot C_{1} \cdot C_{2}} & if i = 1 \\ {(t_{i - 1})}^{t_{i - 1}} & if i > 1 \end{matrix}

(6b)

and ε_vi is the volumetric strain in percent at the end of the ith load increment having amplitude γ_i (γ_i and γ_tv are both in percent). If the seismic demand is expressed in terms of γ_eff and n_eqγ, Equation 6a can be written in simplified form:

ε_{v} = - \ln (Π_{i = 1}^{2 \cdot n_{eq γ}} t_{i}) \cdot \frac{(γ_{eff} - γ_{tv})}{C_{2}}

(6c)

where ε_v is the volumetric strain at the end of shaking.

Figure 2a shows the computed values of ε_v_,15 as a function of γ_eff using Equation 6c for two different values of γ_tv, plotted in the same form as the relationship proposed by Tokimatsu and Seed (1987) (Figure 1a). Recall that both the Tokimatsu and Seed (1987) and Byrne (1991) models were calibrated using the same clean sand data from Silver and Seed (1971) and Seed and Silver (1972), with this data also shown in Figure 2a. As may be observed from Figure 2a, Equation 6c predicts ε_v_,15 values for a given Dr that deviate from a straight line on log-log scale as the γ_eff approaches the γ_tv, when γ_tv > 0. This deviation is supported to some extent by the laboratory test data shown.

Equation 6c can be used to compute C_N as a function of n_eqγ by computing the ratio of ε_v for a given value of n_eqγ and for n_eqγ = 15 for the same γ_eff: Recall that i is the number of half cycles, so i = 30 corresponds to n_eqγ = 15 cycles.

C_{N} = \frac{{ε_{v}}_{i}}{ε_{v, 15}} = \frac{\ln (\underset{i}{Π} t_{i})}{\ln (Π_{j = 1}^{30} t_{j})}

(7)

As shown in Figure 2b, the predicted values of C_N fall well within the range of values from the Silver and Seed (1971) and Seed and Silver (1972) data.

Calibration of the expanded Byrne model

Comparison of Equations 2e and 6c implies that:

\frac{- \ln (Π_{i = 1}^{2 \cdot n_{eq γ}} t_{i})}{C_{2}} = K_{σ, ε} \cdot a_{1 atm} \cdot C_{N}

and

b = 1

This forms the basis for expanding and calibrating the Byrne model to evaluate seismic compression in soils other than just clean sands. Specifically, to account for soils that exhibit seismic compression behavior for b ≠ 1, the simplified form of the Byrne model can be expanded to:

ε_{v} = - \ln (Π_{i = 1}^{2 \cdot n_{eq γ}} t_{i}) \cdot \frac{{(γ_{eff} - γ_{tv})}^{C_{3}}}{C_{2}}

(8a)

{(Δ ε_{v, 1 / 2})}_{i} = 0.5 \cdot {(γ_{i} - γ_{tv})}^{C_{3}} \cdot C_{1} \cdot \exp [- C_{2} \frac{ε_{vi}}{{(γ_{i} - γ_{tv})}^{C_{3}}}]

(8b)

for the non-simplified form. Equation 8b was actually proposed in a recent and independent study by Chen et al. (2019) based on the analysis of excess pore water generation in undrained cyclic triaxial test samples, giving further credence to this expanded form.

Calibrating Equation 8a using the data and model from Duku et al. (2008) for clean sands:

C_{1} = \frac{1}{2.8001} \cdot K_{σ, ε} \cdot a_{1 atm}

(9a)

C_{2} = \frac{1}{C_{1}}

(9b)

C_{3} = 1.2

(9c)

and using the data and model from Yee et al. (2014) for non-plastic-to-moderately-plastic silty sands/sandy silts (i.e. PI ≤ 10), with FC ranging from 0% to 60%:

C_{1} = \frac{1}{F_{P} (γ)} \cdot K_{FC} \cdot K_{S} \cdot K_{σ, ε} \cdot a_{1 atm & FC = 0 & S = 0}

(10a)

F_{P} (γ) = 2.149 \cdot γ^{- 0.2343} + 4.337 \cdot e^{- 66.56 \cdot γ}

(10b)

C_{2} = \frac{P (γ)}{C_{1}}

(10c)

P (γ) = e^{0.405} \cdot {(γ - γ_{tv})}^{0.3291}

(10d)

C_{3} = 1.2

(10e)

where γ and γ_tv are in percent. The expressions for C₁, C₂, and C₃ given by Equations 9 and 10 can be used in conjunction with the simplified and non-simplified forms of the expanded Byrne model (Equations 8a and 8b, respectively). Note that when used in conjunction with the simplified form, γ in Equations 10a to 10d is γ_eff, and when used in conjunction with the non-simplified γ is γ_i. In the following section, both forms of the expanded Byrne model are used to analyze a field case history from the 2007, moment magnitude (M_w) 6.6 Niigata-ken Chuetsu-oki, Japan, earthquake.

Case history analysis

Background

The main shock of the M_w6.6 Niigata-ken Chuetsu-oki Japan earthquake occurred on 16 July 2007. The event affected an ∼100 km-wide area along the coastal regions of southwestern Niigata prefecture and triggered ground failures as far as the Unouma Hills, located in central Niigata approximately 50 km from the shore (Kayen et al., 2009). Of specific interest to this study is the seismic compression that occurred during this event at the Kashiwazaki-Kariwa Nuclear Power Plant (KKNPP) site (Yee, 2011; Yee et al., 2011). What makes this case history of particular value is that the motions at the site were recorded by a free-field downhole array (Service Hall Array (SHA)), and the magnitude of the seismic compression was accurately determined from the settlement of soil around a vertical pipe housing one of the array seismographs. The geometric mean values of the peak accelerations at bedrock and the ground surface were ∼0.55g and ∼0.4g, respectively, indicating NL site response of the soil column. The seismic compression at the site was ∼10–20 cm.

Yee et al. (2011) performed a detailed site investigation and determined that the profile at the strong motion array consists of ∼70 m of medium-dense sands overlying clayey bedrock and that the ground water table (gwt) is at a depth of ∼45 m. Suspension logging and standard penetration tests (SPTs) with energy measurements were performed at the site, with the former providing small-strain shear and compression wave velocities (i.e. V_s and V_p, respectively). In addition, laboratory tests were performed on disturbed and undisturbed samples to classify the soil, to determine index properties and shear strength of the soil, and to develop modulus reduction and damping (MRD) curves. The geologic log and instrument locations for the SHA site are shown in Figure 3. Also, shown in this figure are the results of SPT and suspension logging geophysical testing and some of their interpretations.

Site response analysis

One-dimensional equivalent linear (EQL) site response analyses were performed for the site using the software Strata (Kottke and Rathje, 2009) following the modeling details in Yee et al. (2011, 2013). The unprocessed ground motions recorded by the SHA array were obtained from the Tokyo Electric Power Company (TEPCO) and were processed following the procedures detailed in Boore (2005) and Boore and Bommer (2005). This involved adding zero pads at the beginning and end of each record equal to 1.5·n′/f_c·dt, where n′ = 4 (the high-pass filter order), f_c is the filter corner frequency, and dt is the sampling interval. An acausal high-pass filter was applied at the filter corner frequencies which were picked manually by comparing the signal with noise in frequency domain and visualizing the displacement. The same corner frequencies were used for all three components of motion recorded by a strong motion station during a given event. The horizontal motions were oriented in the EW and NS directions, and those corresponding to a depth of 99.4 m were specified as “with-in” input motions in the EQL analyses. The motions are shown in Figure 4.

The V_s profile used in the analyses is shown in Figure 5, and the total unit weights (γ_t) of the soil are listed in Table 1. The Menq (2003) MRD curves were used to model the sandy soil above the gwt, with Yee et al.’s (2013) strength-adjustment applied and a minimum damping of 5% used. To account for the influence of effective confining stress, the reference strain (γ_r) used in the Menq (2003) modulus reduction curves (i.e. curves of (G/G_max)_γeff vs γ_eff) was adjusted using:

γ_{r} = γ_{r, 1} \cdot {(\frac{{σ'}_{o}}{P_{a}})}^{n}

(11)

where $σ'_{o}$ is the mean effective confining stress; P_a has the same units as $σ'_{o}$ ; γ_r_,1 is the reference strain for $σ'_{o} = 1 atm$ ; and n is an empirical soil-specific factor. Per Yee et al. (2013), γ_r_,1 = 0.0904 and n = 0.4345, based on the MRD test data for sandy soils above the gwt from the site. No samples from below the gwt from the site were tested, and it was assumed that the γ_r_,1 and n values proposed by Menq (2003) applied for sandy soils below the gwt: γ_r_,1 = 0.0684 and n = 0.4345. The Darendeli (2001) MRD curves were used for the relatively plastic soils and rock materials below 70 m.

Table 1.

Assumed soil types and unit weights used in analysis (Motamed et al., 2016)

Depth range (m)	Soil type	Total unit weight, γ_t (kN/m³)
0–4	Sand	16
4–45	Sand	17.75
45–70	Sand	20.8
70–99.4	Clay	20.8

To validate the EQL model, computed and recorded motions were compared at depths of 2.4 and 50.8 m. As shown in Figure 6, the peak ground accelerations (PGAs) for the recorded and computed motions are in good agreement, as are the response spectra. In addition, the validity of the one-dimensional profile response assumption was assessed using the criteria detailed in Toa and Rathje (2019) and shown to be valid. Accordingly, the EQL model was used to compute the shear strain time histories at the center of each of the model layers above 45 m (i.e. above the gwt). As discussed next, these time histories were used to compute the ε_v in each layer and the overall settlement at the site due to seismic compression.

Seismic compression

Yee et al. (2011) performed a series of drained cyclic simple shear tests on samples from the KKNPP site and developed soil-specific calibration parameters for the Duku et al. (2008) simplified model (Equation 2) for Dr ≈ 35% and 60%. These calibration parameters are listed in Table 2. Using these, the calibration parameters for the expanded Byrne model were computed:

C_{1} = K_{σ, ε} \cdot 1.28 \cdot e^{- 0.019 \cdot Dr}

(12a)

C_{2} = \frac{0.7864}{C_{1}}

(12b)

C_{3} = 1.2

(12c)

where Dr is in percent. Figure 7 shows a comparison of the ε_v_,15 versus γ_eff and C_N versus n_eqγ for the Duku et al. (2008) and expanded Byrne models using the KKNPP soil-specific calibration parameters. As may be observed from these plots, the model predictions are in very good agreement.

Table 2.

KKNPP soil-specific calibration parameters for Duku et al. (2008) model

	Dr ≈ 35%	Dr ≈ 60%
a _1 atm	2.15	1.33
B	1.2	1.2
R	0.31	0.31
γ_tv	0.03%	0.03%

KKNPP: Kashiwazaki-Kariwa Nuclear Power Plant.

Dr for the soil was estimated using the relationship:

Dr = 100 \cdot \sqrt{\frac{N_{1, 60}}{C_{d}}}

(13)

where Dr is in percent, N_1,60 is the corrected SPT blow count, and C_d is a soil-specific parameter. Per Skempton (1986), C_d was assumed to be 55 (natural deposit of fine sand).

The non-simplified expanded Byrne model (Equation 8b) calibrated using Equation 12 was used in conjunction with the shear strain time histories computed at the center of each of the Strata model layers above the gwt. The total settlement at the ground surface (S_T) at the site was then computed from the resulting ε_v values for each layer:

S_{T} = \sum_{j} ε_{v_{j}} \cdot Δ z_{j}

(14)

where ε_vj is the volumetric strain in the jth layer and Δz_j is the thickness of the jth layer. The seismic compression computed using the EW and NS motions was ∼3.5 and ∼1.3 cm, respectively, resulting in a total settlement of ∼4.8 cm, assuming the direct addition of the settlements due to each of the horizontal components of motion (e.g. Pyke et al., 1975).

The simplified expanded Byrne model (Equation 8a) calibrated using Equation 12 was used in conjunction with the γ_eff values computed using Equation 1. Equation 1 was solved iteratively using the shear modulus reduction curves used in the Strata analyses and a_max = 0.4g (i.e. geometric mean of the recorded peak accelerations at ground surface). The r_d relationship proposed by Idriss (1999) was used. Although Lasley et al. (2016a) shows that the Idriss (1999) relationship generally predicts too rigid of profile response for liquefaction triggering analyses, the profiles for seismic compression analyses tend to be stiffer than sites evaluated for liquefaction due to deeper gwt (or higher effective confining stresses). As a result, it is recommended that the Idriss (1999)r_d relationship be used to compute γ_eff in seismic compression analyses.

Figure 8 shows the plot of the γ_eff computed using Equation 1 and computed from the shear strain time histories from the EQL analyses. For the latter values, γ_eff was computed as 0.65 times the geometric mean of the peak shear strains in each layer resulting from the Strata analyses using the EW and NS motions. As may be observed from Figure 8, γ_eff values computed using Equation 1 have a similar trend with depth as those from the Strata analyses, but are slightly larger in magnitude for most depths.

The relationship proposed by Lee and Green (2017) was used to compute n_eqγ:

\ln (n_{eq γ}) = \exp (b_{1} \cdot z) + b_{2} \cdot {R_{rup}}^{b_{3}} + b_{4} \cdot M_{w} + b_{5}

(15)

where z is depth below the ground surface in m; R_rup is the closest distance to the fault rupture plane (km); and b₁–b₅ are regression coefficients. Yee et al. (2011) give R_rup = 16 km and values of b₁–b₅ are listed in Table 3 for shallow crustal events in stable continental and active tectonic regimes (e.g. Central-Eastern US (CEUS) and Western US (WUS), respectively). This relationship is preferred over others because it was specifically developed for computing the n_eqγ for seismic compression analyses. Its use is in contrast to the common practice of using n_eqτ relationships developed for liquefaction triggering analyses in seismic compression analyses, which fails to recognize the potential differences between n_eqγ and n_eqτ (see details in Green and Terri, 2005). Figure 9 shows a plot of the computed n_eqγ versus depth.

Table 3.

Regression coefficients and standard deviations of inter-event, intra-event, and total error (Lee and Green, 2017)

CEUS
b ₁	b ₂	b ₃	b ₄	b ₅	τ_ln	σ_ln	σ_ln _{_total}
−0.020	0.80	0.22	0.19	−1.30	0.26	0.47	0.54
WUS
b ₁	b ₂	b ₃	b ₄	b ₅	τ_ln	σ_ln	σ_{ln_total}
−0.0099	0.67	0.21	0.28	−1.79	0.24	0.41	0.48

CEUS: Central-Eastern US; WUS: Western US.

The resulting seismic compression using the expanded simplified Byrne model is ∼3.7 cm. However, this value only reflects the seismic compression resulting from the profile being subjected to the geometric mean of the two horizontal components of motion (i.e. both the a_max and n_eqγ were based on geometric means of the horizontal components of motion). Accounting for both horizontal components of shaking, the above value for seismic compression is multiplied by a factor of 2 (e.g. Pyke et al., 1975), resulting in the predicted settlement due to seismic compression using the expanded simplified Byrne model being ∼7.5 cm.

Discussion

The predicted settlement using the non-simplified, expanded Byrne model is about half to of the lower end of the range of observed settlements (∼4.8 vs 10–20 cm), while the predicted settlement using the simplified variant is about three-quarters of the lower end of the observed range (∼7.5 vs 10–20 cm). Given that the site was very well characterized, the site response model was validated and the motions used in modeling were those that were recorded at the site, and the seismic compression model was calibrated using soil from the site; potential reasons for the under-predictions are explored. These include the estimated Dr and γ_tv of the soil, orientation of the ground motions used in the analyses, influence of settlement of the of the soil below the ground water table, and how multidirectional shaking is being accounted for. However, additional issues that may contribute to the under-predictions that are not explored extensively herein are the influence of soil fabric on the seismic compression response behavior of the soil and the use of NL site response analyses, in lieu of EQL analyses, to compute the shear strains at depth in the soil profile in implementing the non-simplified variant of the expanded Byrne model.

As mentioned previously, Equation 13 was used to estimate Dr from the SPT blow count assuming C_d = 55 (e.g. Skempton, 1986). This resulted in Dr values of ∼60% for most depths. This value is about double that determined from samples (triple-barrel pitcher samples and frozen samples) which were ∼30%–40%, as shown in Figure 3. To assess the influence of the assumed value of C_d on the predicted magnitude of seismic compression, C_d = 40 (lower bound value for clean sands reported in literature) and 145 (value that results in Dr ≈ 35%) were used to analyze the case history. This resulted in the magnitude of seismic compression ranging from 3.9 to 7.8 cm for the non-simplified analyses and 5.6 to 12.2 cm of for the simplified analyses. As expected, the smaller value of C_d results in a reduction of the predicted settlement due to seismic compression while the larger value results in an increase. However, the predicted magnitude of seismic compression for non-simplified analysis for C_d = 145 (i.e. Dr ≈ 35%) is still only about three-quarters of the lower end of the observed range (∼7.8 vs 10–20 cm).

Note that although the simplified variant is predicting more accurate settlements than the non-simplified procedure in this instance, this should not be interpreted as the simplified procedure being a superior or more accurate approach. It is doubtful that the simplified variant will always predict more accurate settlements (or even larger settlements) than the non-simplified variant. Rather, the non-simplified procedure should be viewed as providing more accurate estimates of the predicted seismic compression if the required inputs and model assumptions used in the analyses are appropriate. In this instance, the larger predictions yielded by the simplified variant relates to the n_eqγ estimated using the Lee and Green (2017) relationship, because the γ_eff values for the simplified procedure are approximately equal to those from the site response analyses (Figure 8).

Based on laboratory tests, Yee et al. (2011) measured values of γ_tv for the soil at the KKNPP site to range from 0.03% to 0.044% (γ_tv = 0.03% was used in the analyses presented above), which is considerably higher than the value recommended by Dobry et al. (1982) of 0.01% for clean sands. To assess the influence of γ_tv on the magnitude of the predicted settlements due to seismic compression, the case history was re-analyzed assuming γ_tv = 0.01%, 0.02%, and 0.04%. The resulting predictions ranged from 4.5 to 2.9 cm, 5.5 to 4.5 cm, and 9.0 to 7.3 cm for γ_tv = 0.01%–0.04% and for C_d = 40, 55, and 145, respectively, with lower values of γ_tv and higher values of C_d resulting in larger values of predicted settlements using the non-simplified form of the expanded Byrne model. However, the combination that yields the maximum settlement (i.e. γ_tv = 0.01% and C_d = 145) still yields a prediction that is less than the lower end of the observed range (∼9 vs 10–20 cm).

All of the predictions presented above were based on horizontal ground motions oriented in the NS and EW directions (i.e. the directions of the motions recorded by the strong motion station). To assess the influence of the motion orientation of the magnitude of the predicted settlements due to seismic compression, the motions were rotated in 5° increments and the case history re-analyzed. For γ_tv = 0.03% and C_d = 145, the predicted settlements ranged from 7.3 to 10.3 cm (Figure 10), with the maximum predicted settlement occurring for motions rotated between 60° and 65° clockwise. This relatively large variation in predicted settlements due to motion orientation highlights the importance of both the absolute amplitude and sequencing of pulses in the load history on predicted magnitude of seismic compression (i.e. the significance of the load-dependent, interaction macro-level fatigue behavior exhibited by the seismic compression phenomenon), which, again, is commonly ignored in computation of n_eqγ (e.g. Green and Terri, 2005; Lee and Green, 2017).

As commonly defined, seismic compression is a phenomenon that occurs in unsaturated or partially saturated sandy soils (e.g. Stewart et al., 2004a). However, the observed surface settlement at the KKNPP likely reflects the seismic compression that occurred in the unsaturated or partially saturated sandy soil above the gwt and settlement that occurred in the saturated sandy soil below the gwt (the soil below the gwt is assumed to be saturated based on V_p measurements of ∼1500 m/s shown in Figure 3). Pyke (2019) provides some guidance on how to compute the settlement of the saturated sandy soil subjected to earthquake shaking:

While not checked experimentally, it was assumed in the studies reported by Martin et al. (1975) and Seed et al. (1978) that the latent settlement generated in a fully saturated sand was equal to the actual settlement of a dry sand up to the point of initial liquefaction, and this assumption appeared to yield good results. Once initial liquefaction (or 100 percent excess pore pressure ratio) is reached, larger latent settlements will be generated.

In essence, co-seismic and post-seismic settlement are separate phenomena. Co-seismic settlement (i.e. seismic compression) results from the rearrangement of soil particles during dynamic loading and is assumed to dominate in unsaturated and saturated soil where the excess pore water pressures are limited. The post-seismic settlement (i.e. consolidation) is caused by the dissipation of excess pore water pressures and is assumed to dominate in saturated soil when the excess pore water pressures are significant (Thum et al., 2021).

Yee et al. (2011) evaluated liquefaction triggering for the KKNPP site for the event of interest and showed that the factor of safety was greater than 1. More importantly, none of the ground motion recordings from the SHA array exhibited characteristics of the ground softening due to higher levels excess pore pressure generation (e.g. Upadhyaya et al., 2019; Wotherspoon et al., 2015). Accordingly, following the guidance in Pyke (2019), the settlement of the sandy soil both above and below the gwt at the KKNPP site was evaluated using the using the non-simplified form of the expanded Byrne model, with the resulting predicted settlements ranging from 10.4 to 12.7 cm for γ_tv = 0.03% and C_d = 145 (Figure 10). For this case, the entire range of predicted settlements is within the observed range, although still toward the lower end of the observed range (i.e. 10.4–12.7 vs 10–20 cm).

The predictions presented above were based on analyses that only considered horizontal shaking and assumed that the settlements predicted for each horizontal component of motion were additive. This approach was based on a detailed, but somewhat limited, laboratory study performed by Pyke et al. (1975). Alternatively, Lasley and Green (2012) (also see Nie et al., 2017) proposed the values tabulated in Table 4 to relate seismic compression in soil subjected to geometric mean motions to that resulting from the soil being subjected to two horizontal components of motions simultaneously. The values listed in Table 4 are based on a series of numerical analyses with soil elements subjected to multidirectional motions, wherein the soil response was modeled using a reduced-order bounding surface hypoplasticity model (Li et al., 1992). Using Table 4 and assuming Dr ranges from 35% to 60% (Figure 3 and Equation 13), a factor of ∼1.7 should be applied to the geometric mean of the settlement computed from the two horizontal components of motion.

Table 4.

Correction factor, C_2D, for two-dimensional shaking (Lasley and Green, 2012; also see Nie et al., 2017)

Dr (%) (N_1,60)	Moment magnitude, M_w
	5–6	6–7	7–8
45 (9)	1.5	1.6	1.7
60 (17)	1.9	1.8	1.8
80 (30)	2	1.9	1.8
100 (46)	2	2.1	2.1

As shown in Figure 10, using the factors in Table 4 will tend to reduce the magnitude of the settlements predicted in comparison to those predicted by directly adding the settlements predicted for each of the two horizontal components of motion separately. This is counter to bringing the predictions made herein into better accord with field observations. However, up to this point, the influence of vertical motions on seismic compression has not been considered. Again, in a detailed, but somewhat limited, laboratory study, Pyke et al. (1975) examined vertical motions having PGAs ranging from 0.15g to 0.3g and acting in combination with horizontal motions. They found that the vertical motions can increase the seismic compression by 20% to 50%, relative to the seismic compression resulting from horizontal motions alone. This finding is significant, but interestingly Pyke et al. (1975) is the only study that the authors could find that examined the influence of vertical motions on seismic compression. And, this aspect of the Pyke et al. (1975) study has largely not been accounted for by researchers and practitioners, to include those that were involved in the Pyke et al. (1975) study (e.g. Pyke, 2019; Tokimatsu and Seed, 1987). Nevertheless, Yee et al. (2011) accounted for vertical accelerations in predicting the magnitude of seismic compression at the KKNPP SHA site using an effective peak vertical acceleration of 0.4g for the event, which increases the predicted seismic compression by ∼50% per Pyke et al. (1975). Assuming C_2D = 1.7 (Table 4), γ_tv = 0.03, C_d = 145, and accounting for the influence of the vertical component of motion per Pyke et al. (1975), the predicted settlement using the non-simplified form of the expanded Byrne model ranges from 11.6 to 15.2 cm, as shown in Figure 10. For completeness, the predicted settlement using the simplified form of the expanded Byrne model ranges from 20.5 to 26.8 cm, assuming C_2D = 1.7, γ_tv = 0.03, C_d = 145, and accounting for the influence of the vertical component of motion per Pyke et al. (1975). This range of values is larger than the post-earthquake field observations.

The last set of predicted settlements using the non-simplified form of the expanded Byrne model are in very good accord with the post-earthquake field observations, as shown in Figure 10. However, additional studies are needed to fully investigate how to account multidirectional shaking on the seismic compression response of soil, to include vertical motions.

In addition to the input variables and modeling assumptions discussed above, two others merit mention, soil fabric and method of site response analysis. As mentioned previously, Duku et al. (2008) examined the influence of soil fabric on seismic compression, among other factors, and found that it did not significantly influence the seismic compression response of clean sands. As a result, the expanded Byrne model proposed herein does not account for soil fabric. However, the recent study by Bhaumik et al. (2019) somewhat contradicts the findings of Duku et al. (2008), concluding that soil fabric does influence the seismic compression response of loose- and medium-dense sands, but not dense sands. Further study is needed to resolve these seemingly contradictory findings. Nevertheless, the expanded Byrne model can be readily modified to account for the influence of soil fabric on the seismic compression response of soil if deemed appropriate.

In the analyses presented herein, the non-simplified variant of the expanded Byrne model was implemented using EQL site response analyses. Alternatively, NL site response analyses could be used (e.g. Yee and Stewart, 2018; Yee et al., 2011). However, while the variability in site response is small for EQL analyses, regardless of what software is used (e.g. Lasley et al., 2014), this is not the case for NL analyses. This issue is highlighted in Yee and Stewart (2018) wherein two versions of DEEPSOIL, v4.0 and v6.1.5, were used to analyze the KKNPP site and the resulting peak shear strains as a function of depth varied significantly for one components of motion. Undoubtedly, the variation in the compute shear strains will only increase if different software and different constitutive models are used in the NL analyses. Ultimately, the accuracy of an NL analysis is tied to the inherent constitutive model used, and more advanced constitutive models typically require a more-detailed calibration. As a result, the use of NL analyses to increase the accuracy of the predicted magnitude of seismic compression will likely increase the uncertainty in the predicted values, significantly so.

Conclusion

Together, the simplified and non-simplified forms of the expanded Byrne model provide a versatile approach for evaluating seismic compression that is scalable based on available data and the importance of the project. Both forms of the model use the same calibration parameters, which have been developed herein for clean sands and non-plastic-to-moderately-plastic (PI ≤ 10) silty sands/sandy silts using the extensive laboratory data performed by researchers at the University of California at Los Angeles. The non-simplified form is relatively easy to implement and thus overcomes the complexity issues with implementing other non-simplified models (e.g. Lasley et al., 2016b).

Both the simplified and non-simplified expanded Byrne models were used to evaluate seismic compression at the KKNPP SHA site during the main shock of the 2007, M_w6.6 Niigata-ken Chuetsu-oki, Japan, earthquake. The non-simplified model was used in conjunction with shear strain time histories computed at varying depths in the profile using EQL site response analyses. Using “best estimate” input parameter estimations and common modeling assumptions, the simplified variant predicted more accurate settlements than the non-simplified procedure for this case history, although both under-predicted the post-earthquake field observed settlement. However, this should not be interpreted as the simplified procedure being a superior or more accurate approach. Rather, the non-simplified procedure should be viewed as providing more accurate predictions of seismic compression if the required inputs and model assumptions used in the analyses are appropriate. Toward this end, the influence of some of the input parameters and modeling assumptions on the computed settlement was explored. Although additional studies are needed to further validate these, some of the findings made herein are as follows:

Estimation of Dr and γ_tv, and ground motion orientation, individually, have moderate-to-significant influences on the computed magnitude of seismic compression, but in combination, they can have significant influence.

The seismic compression models can seemingly be used to predict the settlement in fully saturated sand when the excess pore water pressures are limited, as has been hypothesized by others.

Accounting for multidirectional shaking has a significant influence on the computed magnitude of seismic compression.

Expanding on the last finding listed above, in the author’s view, the greatest uncertainty in seismic compression predictions relates to the influence of vertical motions. This is because few studies have examined this issue (i.e. the author is only aware of the study by Pyke et al., 1975), and the resulting adjustments increase the predicted seismic compression by 20%–50%. This aspect of the Pyke et al. (1975) study has largely been ignored by researchers and practitioners, to include those that were involved in the Pyke et al. (1975) study (e.g. Pyke, 2019; Tokimatsu and Seed, 1987). Accordingly, seismic compression evaluations would benefit from a more-detailed analysis of the influence of vertical motions on seismic compression, as well as the development of calibration parameters for additional types of soils/states/fabric.

Supplemental Material

Jiang_et_al_-_mod_byrne_seis_compress_model_-_Appendix_A – Supplemental material for Expanded Byrne model for evaluating seismic compression

Supplemental material, Jiang_et_al_-_mod_byrne_seis_compress_model_-_Appendix_A for Expanded Byrne model for evaluating seismic compression by Yusheng Jiang, Russell A Green and Oliver-Denzil Taylor in Earthquake Spectra

Footnotes

Acknowledgements

The authors greatly appreciate the help from Professors Jonathan Stewart and Ramin Motamed regarding the Kashiwazaki-Kariwa Nuclear Power Plant (KKNPP) case history and from Mr Mahdi Bahrampouri in processing the KKNPP ground motions. The ground motion data used in this study belong to Tokyo Electric Power Company, and the distribution license of the data belongs to Japan Association for Earthquake Engineering (JAEE). Finally, any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of ERDC, NSF, or JAEE or those that have provided help on this study.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was based on work supported by the US Army Engineer Research and Development Center (ERDC) Grant W912HZ-13-C-0035 and US National Science Foundation (NSF) Grants CMMI-1825189 and CMMI-1937984. The authors gratefully acknowledge this funding.

Supplemental material

Supplemental material for this article is available online.

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