Experimental Investigation on Small-Strain Stiffness of Marine Silty Sand

Abstract

The significance of small-strain stiffness (G_max) of saturated composite soils are still of great concern in practice, due to the complex influence of fines on soil fabric. This paper presents an experimental investigation conducted through comprehensive bender element tests on G_max of marine silty sand. Special attention is paid to the influence of initial effective confining pressure (${\sigma }_{\mathrm{c}0}^{\prime }$), global void ratio (e) and fines content (FC) on G_max of a marine silty sand. The results indicate that under otherwise similar conditions, G_max decreases with decreasing e or FC, but decreases with increasing FC. In addition, the reduction rate of G_max with e increasing is not sensitive to ${\sigma }_{\mathrm{c}0}^{\prime }$, but obviously sensitive to changes in FC. The equivalent skeleton void ratio (e*) is introduced as an alternative state index for silty sand with various FC, based on the concept of binary packing material. Remarkably, the Hardin model is modified with the new state index e*, allowing unified characterization of G_max values for silty sand with various FC, e, and ${\sigma }_{\mathrm{c}0}^{\prime }$. Independent test data for different silty sand published in the literature calibrate the applicability of this proposed model.

1. Introduction

The small-strain stiffness G_max of marine deposits plays a fundamental role in liquefaction potential assessment, site seismic response analyses, and the design of marine structures (e.g., pipeline, immersed tunnel, caisson foundation) subjected to storm or earthquake loading [1,2,3,4]. Generally, G_max is defined as the stiffness of soil at small-strain level of 10⁻⁶, where the soil properties are considered to exhibit pure elasticity. Hardin and his co-authors [5,6,7] conducted comprehensive studies on G_max of clean, uniform, quartz sands through well-controlled resonant column tests, and these investigations indicated that the global void ratio e and initial effective confining pressure ${\sigma }_{\mathrm{c}0}^{\prime }$ are considered to be the most important ones among the various factors that may influence G_max. Similar results were also presented by Seed et al. (1986) [8], Youn et al. (2008) [9], Yang and Gu (2013) [10], and Payan et al. (2016) [1].

While a large number of attempts have been carried out to characterize G_max for clean sands, systematic studies on silty sand with different fines content (FC) are relatively few, despite the fact that naturally deposited sands are not clean, but contain a certain amount of fine particles [11,12,13,14]. A systematic study was first implemented by Iwasaki and Tatsuoka (1977) [11] to study the G_max influence factors of Iruma silty sand. Their results showed that G_max decreased with increasing FC, and at given e and ${\sigma }_{\mathrm{c}0}^{\prime }$, G_max exhibited a decreasing trend as uniformity coefficient C_u increasing. The state parameter of skeleton void ratio e_sk was introduced by Wichtmann et al. (2015) [12] to uniquely characterize G_max of silty sand. However, as discussed by Rahman et al. (2008) [13] and Yang and Liu (2016) [14], the application of e_sk might contribute to underestimation of G_max at high FC. Goudarzy et al. (2017) [15] developed a new G_max prediction method based on the binary packing state. A series of bender element tests has been conducted on Ottawa sand with FC = 5%–20% by Salgado et al. (2000) [16], the test results revealed that G_max decreases dramatically with the increasing of FC at a constant relative density and ${\sigma }_{\mathrm{c}0}^{\prime }$. Salgado et al. (2000) [16] introduced a state parameter ψ to estimate G_max in the framework of critical state soil mechanics by taking account of the stress dependence. However, compared with the Goudarzy et al. (2017) method [15], the introduction of state parameter ψ requires determination of the critical state line, thus complicating the application of this method [16].

Many natural silty sands contain a significant amount of fines. This is particularly true for marine deposits, which in most cases behave as composite soils. Therefore, study is needed on whether the G_max prediction method established for clean sand is also applicable to that of marine silty sand. The main purpose of this study is to explore how FC, initial effective confining pressure (${\sigma }_{\mathrm{c}0}^{\prime }$), and global void ratio (e) affect the G_max of marine silty sand and whether the G_max of silty sand can be predicted within the established framework based on clean sand. In addition, the influence of parameters in the Hardin model for G_max prediction was discussed in a traditional way. In particular, the binary packing state concept [17,18,19] is implemented to establish the modified Hardin model for evaluation of G_max of marine silty sand. For this purpose, a series of bender element tests were conducted on marine silty sand with FC = 0%~30%.

2. Materials and Methods

2.1. Testing Apparatus

The measurement of shear wave velocity (V_s) or the associated G_max was performed using a pair of piezoceramic bender elements (BE) installed in the cell chamber of a dynamic hollow/solid cylinder apparatus (HCA) [20], as shown in Figure 1. For each of the BE tests, a set of sinusoid signals from 1 to 40 kHz, rather than a single signal, was used as the excitation, and the received signals corresponding to these excitation frequencies were examined in whole to better identify the travel time of the shear wave, then, G_max can be calculated as following [16].

$$G_{\max }=\rho V_{\mathrm{s}}^2$$

(1)

2.2. Tested Materials

Nantong marine sand was used as clean sand and Nantong marine silt with sub-angular particles was used as pure fines to investigate the effects of FC on the G_max of silty sand. Figure 2 shows the grain size distributions and scanning electron microscopy image of clean sand and pure fines, and the material properties are given in

Table 1. Index properties of clean sand and pure fines.

	Clean Sand	Pure Fines
Material	Nantong sand	Nantong silt
$d_{50}$/mm	0.114	0.040
$d_{10}$/mm	0.080	0.016
$C_{\mathrm{u}}$	1.672	2.931
$G$	2.672	2.719
$e_{\max }$	1.262	1.481
$e_{\min }$	0.662	0.764

. Although the ASTM D4253 [21] and D4254 [22] test methods for the determination of minimum and maximum void ratios (e_min and e_max) are applicable to silty sand with FC < 15%, these methods were also used for silty sands with FC ≥ 15% in order to provide consistent measurements [23]. The clean sand was mixed with non-plastic Nantong silt (pure fines) corresponding to various FC from 0% to 30% by mass. The e_min and e_max of the silty sand are shown in

Table 2. Physical index of Nantong marine silty sand with different FC.

	FC of Silty Sand (%)
	0	10	20	30
emax	1.290	1.232	1.221	1.212
emin	0.731	0.587	0.431	0.364
G	2.669	2.680	2.690	2.701
d50	0.113	0.104	0.097	0.091
Cc	0.796	0.829	1.453	1.752
Cu	1.646	1.681	2.826	3.201

.

2.3. Specimen Preparation, Saturation and Consolidation

The bender element tests were conducted on specimens with 100 × 200 mm (diameter × height), and all specimens of the tested silty sands were prepared by the moist tamping method; considering this method can ensure a very wide range of e for the specimens and contribute to preventing segregation and enhancing uniformity [24], all specimens of the tested silty sands were prepared by the moist tamping method using an under-compaction procedure. All samples were tested under saturated rather than other conditions, as the former is more practical [14]; in order to saturate the specimen fully, carbon dioxide flushing from bottom to top of the specimen was applied firstly; then, de-aired water flushing followed immediately [19]; finally, back pressure saturation at the back pressure of 400 kPa was used to guarantee Skempton’s B-value greater than 0.95 [25]. After saturation, all the specimens were isotropically consolidated.

2.4. Testing Program and Process

For the bender element tests, the 10 kHz excitation signal was found to consistently yield a clear arrival of the shear wave for both clean sand and silty sand with various FC, which is consistent with the test results of Yang and Liu (2016) [14]. Figure 3 presents a set of typical received signals captured from the bender element in different silty sand specimens. The first arrival time method was introduced to determine the shear wave travel time in this study [26,27,28], and the zero after first bump point corresponds to Point C marked in Figure 3, suggested by Yoo et al. (2018) [29] and Lee and Santamarina (2005) [30], was selected as the shear wave arrival time.

In order to investigate the influences of FC, e, and ${\sigma }_{\mathrm{c}0}^{\prime }$ on G_max of silty sand, FC = 0, 10, 20, and 30% were considered, and three samples were prepared at different e for silty sand at a fixed FC. The G_max were measured subjected to ${\sigma }_{\mathrm{c}0}^{\prime }$ at 100, 200, 250, 300, and 400 kPa in five stages,

Table 3. Schemes of bender element tests for Nantong marine silty sand.

ID	FC/%	Dr/%	e	ρ (g/cm3)	b Value	e*	${\sigma }_{\mathrm{c}0}^{\prime }/\mathbf{kPa}$
S1	0	35	1.076	1.286	0	1.286	100 200 250 300 400
S2	0	50	0.973	1.352	0	1.352
S3	0	60	0.890	1.412	0	1.412
S4	10	35	1.009	1.334	0.321	1.155
S5	10	50	0.934	1.386	0.321	1.075
S6	10	60	0.883	1.424	0.321	0.953
S7	20	35	0.936	1.348	0.454	1.189
S8	20	50	0.947	1.382	0.454	1.077
S9	20	60	0.824	1.475	0.454	0.998
S10	30	35	0.948	1.386	0.555	1.248
S11	30	50	0.865	1.448	0.555	1.152
S12	30	60	0.792	1.506	0.555	1.042

details the test conditions.

3. Results and Discussion

3.1. Factors Influencing Maximum Shear Modulus

Figure 4 present the comprehensive view of the measured G_max values of silty sand with different FC, e, and ${\sigma }_{\mathrm{c}0}^{\prime }$. A remarkable finding from the figure is that FC, e, or ${\sigma }_{\mathrm{c}0}^{\prime }$ all has a significant impact on G_max, the increase of e will significantly reduce G_max for silty sand at different FC and ${\sigma }_{\mathrm{c}0}^{\prime }$. Furthermore, in each plot, the five trend lines describe the effect of e on G_max, and the range of trend lines revealed the influence of varying ${\sigma }_{\mathrm{c}0}^{\prime }$. Under otherwise similar conditions, G_max decreases with increasing e or FC, but increases with increasing FC. The existing explanation that: as e increases, the dense state changes from compact to loose, which reduces the amount of force chain between particles, contribute to a attenuation in the stiffness of silty sand; while at a fixed e, the amount of sand grains composed of soil skeleton is constant as FC increases, a certain amount of grains participate in the composition of soil skeleton, and the grain contact area increases, eventually leading to an increase in G_max. In addition, the relationship between G_max and e is insensitive to ${\sigma }_{\mathrm{c}0}^{\prime }$, but obviously sensitive to FC. According to Yang and Liu (2016) [14], there is a linear function relationship between G_max with e for Toyoura silty sand, and the void ratio dependence appears to be similar to silty sand with different FC. Incorporating the test results in the study, an obvious soil-specific relationship between G_max and e can be found, and a more comprehensive study needs to be conducted for addressing this concern.

For silty sand at a specific FC, given that G_max is dependent on both e and ${\sigma }_{\mathrm{c}0}^{\prime }$, e must be taken into account when quantifying the impact of ${\sigma }_{\mathrm{c}0}^{\prime }$. Therefore, a void ratio function F(e) was introduced to characterize the influence of e on G_max:

$$F\left(e\right)=\frac{{\left(c-e\right)}^2}{1+e}$$

(2)

where c is a soil-specific fitting parameter dependent on the particle shape—2.97 for angular particles and 2.17 for rounded particles [5,15]. Considering that the particles of marine silty sand are angular (Figure 2b), c = 2.97 was used. An empirical relation for G_max prediction, incorporating material, particle shape, e and ${\sigma }_{\mathrm{c}0}^{\prime }$, was proposed originally by Hardin and Black (1966) [5], then a more general form was developed based on the research of Iwasaki and Tatsuoka (1977) [11], Seed et al. (1986) [8], Youn et al. (2008) [9], Yang and Gu (2013) [10], Wichtmann et al. (2015) [12], and Payan et al. (2016) [1]:

$$G_{\max }=A\frac{{\left(c-e\right)}^2}{1+e}{\left(\frac{{\sigma }_{\mathrm{c}0}^{\prime }}{P_{\mathrm{a}}}\right)}^n$$

(3)

where A = material constant depends on soil type; P_a = atmospheric pressure (≈100 kPa); n = stress exponent, the values of n typically distribute between 0·35 and 0·6 for silty sand. Iwasaki and Tatsuoka (1977) [11] and Yang and Liu (2016) [14] present a common phenomenon that the stress exponent n is a soil-specific constant.

In order to explore the distribution of A and n values, the G_max values of silty sand are plotted as function of ${\sigma }_{\mathrm{c}0}^{\prime }$/P_a and F(e) in Figure 5. Under otherwise identical conditions, G_max increases with increasing in normalized effective confining stress ${\sigma }_{\mathrm{c}0}^{\prime }$/P_a and void ratio function F(e). In addition, R-square of the Hardin model are all greater than 0.9, which means that the Hardin model can characterize the influence of e and ${\sigma }_{\mathrm{c}0}^{\prime }$ on G_max of silty sand at a specific FC well. However, for a specific silty sand, the exponent n is insensitive to FC and e, which is consistent with the results demonstrated by Iwasaki and Tatsuoka (1977) [11] and Yang and Liu (2016) [14]. The exponent n, reflecting the incremental rate of G_max due to the enhancement of ${\sigma }_{\mathrm{c}0}^{\prime }$, is highly dependent on the types of silty sand and present as a soil-specific constant. Using the generalized nonlinear regression model for the test data of marine silty sand tested in this study and six silty sands compiled from the literature, the soil-specific constant n is closely related to the synthesizing material parameter $C_{\mathrm{u}}^{\mathrm{s}}\cdot C_{\mathrm{u}}^{\mathrm{f}}$ of sandy soils (as shown in Figure 6). It is seen that n increases with the increase of $C_{\mathrm{u}}^{\mathrm{s}}\cdot C_{\mathrm{u}}^{\mathrm{f}}$, indicating a logarithmic function relation. The soil-specific constant n can be determined empirically by the following equation:

$$n=0.086\ln \left(C_{\mathrm{u}}^{\mathrm{s}}\cdot C_{\mathrm{u}}^{\mathrm{f}}\right)+0.302,R^2=0.98$$

(4)

It is worth noting that the addition of FC will obviously alter the material-specific fitting parameter A, which describe the increment ratio of G_max/F(e) caused by the increasing of (${\sigma }_{\mathrm{c}0}^{\prime }$/P_a)ⁿ (Figure 7), and a fairly good exponential relationship can be given as following:

$$A\left(FC\right)=A_0\times \exp \left(m\cdot FC\right)$$

(5)

where, the value of A₀ represents the parameter A for clean sand (FC = 0%) in the Hardin model, m is the fitting parameter and the value of m is −1.52 for Nantong silty sand. It is worth noting that care should be exercised when the Hardin model is directly used for predicting the G_max of silty sand, considering the sensitivity of the A to FC. Therefore, a modified Hardin model needs to be explored for unified charactering G_max of silty sand with different FC.

3.2. Modified Hardin Model Based on Binary Packing Model

The binary packing state concept [17,31] is adopted herein to interpret the behavior of granular soil. For the binary packing system, the FC_th has been introduced to distinguish the difference of “coarse-dominated behavior” from “fines-dominated behavior” for silty sand with various FC [32,33]. The FC_th can be determined empirically by semi-experience formula [13]:

$$F{C}_{\mathrm{th}}=0.40\times \left(\frac{1}{1+\exp \left(0.5-0.13\cdot \chi \right)}+\frac{1}{\chi }\right)$$

(6)

where $\chi =d_{10}^{\mathrm{s}}/d_{50}^{\mathrm{f}}$ is the particle size disparity ratio, $d_{10}^{\mathrm{s}}$ is the grain size at 10% finer for clean sand, $d_{50}^{\mathrm{f}}$ is the grain size at 50% finer for pure fines.

As FC increases, fines may come in between the contact of sand grains and participate in the force chain. Thus, the effect of fines on the force transfer mechanism is considered by introducing an alternative equivalent skeleton void ratio e* [31,34], as defined by Equation (7).

$$e*=\frac{e+\left(1-b\right)\cdot FC}{1-\left(1-b\right)\cdot FC}$$

(7)

The physical meaning of b is the fraction of fines that participate in the force chain between soil grains and 0 ≤ b ≤ 1. Equation (7) is based on coarse-dominated behavior soil fabric, this meaning b requires FC < FC_th. Rahman and his co-authors developed a semi-empirical relation to predict the parameter b [13,15,35]:

$$b=\left\{1-\exp \left(-\mu \frac{{\left(FC/F{C}_{\mathrm{th}}\right)}^{n_{\mathrm{b}}}}{k}\right)\right\}{\left(r\times \frac{FC}{F{C}_{\mathrm{th}}}\right)}^r$$

(8)

where r = 1/χ, and k = 1 − r^0.25, μ and n_b are the fitting parameters which depend on the specific soil type. The experimental results, presented by Lashkari (2014), suggested that a μ of 0.30 and n_b of 1.0 satisfy a large dataset and were later verified with new datasets. Goudarzy et al. (2016) acknowledged that these parameters might vary for different types of soil, the μ and n_b value were optimized in Equation (8) to obtain the maximum value of R². It has been well recognized that e*, instead of e, can well capture various aspects of the mechanical behavior of silty sand [36]. Notably, the binary packing state parameter has been introduced to uniquely quantify the critical state line, steady state line, and liquefaction resistance, etc. of the silty sand with different FC. Hence, an effort has been made to investigate whether e* determined by Rahman’s approach can better characterize G_max by replacing e with e* in Equation (3):

$$F\left(e*\right)={\left(c-e*\right)}^2/\left(1+e*\right)$$

(9)

Figure 8 show the relationship between G_max, F(e*), and normalized effective confining stress (${\sigma }_{\mathrm{c}0}^{\prime }$/P_a)ⁿ of silty sands. Despite the variation in FC, e, or ${\sigma }_{\mathrm{c}0}^{\prime }$ of the specimens, all of the test data points are located in a narrow surface, which means that e* appears to adequately capture the effects of FC, e, and particle gradations when FC < FC_th. Therefore, the modified Hardin model based on the binary packing state parameter can be established:

$$G_{\max }=A^{*}\frac{{\left(c-e*\right)}^2}{\left(1+e*\right)}{\left(\frac{{\sigma }_{\mathrm{c}0}^{\prime }}{P_{\mathrm{a}}}\right)}^n$$

(10)

A* = 59.3 MPa and R-square = 0.938 for Nantong silty sand, n was determined using Equation (4).

To validate the accuracy of the modified Hardin model, the comparison between the predicted G_max in Equation (10) and the measured G_max are presented in Figure 9. Almost all of the data pairs are close to the bisecting line, with the errors within 10%, indicating that the measured and predicted G_max values are basically consistent. Considering the complexity of the effect of fines and man-made errors, such an error is acceptable. Therefore, the modified Hardin model can be used to predict the G_max of silty sand when FC < FC_th in a simple yet reliable way.

Given the complexity of material properties, further work to validate the applicability of the modified Hardin model evaluation G_max by using experimental data is worthwhile. The similar G_max testing series were carried out on four types of silty sand by Goudarzy et al. (2016) [15], Salgado et al. (2000) [16], Chien and Oh (1998) [37], and Thevanayagam and Liang (2001) [38].

Table 4. Physical index properties and fitting parameters of silty sands considered in this study.

Data from	Material	Index Properties			In Equation (8)		In Equation (10)
		erange(s)	Cu(s)	χ	μ	nb	A*	R2
This study	Nantong sand + Nantong silt	0.60	1.67	2.0	0.32	0.94	62.1	0.932
Goudarzy et al. (2017)	Hostun sand + Quartz powder	0.35	2.01	63.3	0.33	1.05	30.3	0.943
Salgado et al. (2000)	Ottawa sand + Sil-co-Sil	0.30	1.48	11.8	0.34	0.92	44.7	0.895
Chien and Oh (1998)	Yunling sand + Yunling silt	0.55	1.69	2.17	0.27	1.08	64.9	0.883
Thevanayagam and Liang (2001)	Foundary sand + Sil-co-Sil	0.19	1.69	17.1	0.29	0.89	43.2	0.902

Note: e_range(s)—void ratio range of clean sand (=e_max − e_min); C_u(s)—uniformity coefficient of clean sand; μ and n_b—fitting parameters in Equation (8); A*—fitting parameter in Equation (10); R²—coefficient of determination for Equation (10).

presents the physical index properties and fitting parameters of Nantong marine silty sand tested in this study and four silty sands using compiled data from the literature. Best fitting values of μ and n_b in Equation (8) are 0.27~0.34 and 0.89~1.08, and the R-square value of the modified Hardin model for experimental data compiled from the literature are all over 0.9, which means the modified Hardin model can characterize G_max for different types of silty sands well. It should be noted that A* for different types of silty sand presents an obvious soil-specific diversification. In addition, as shown in Figure 10, a power function relationship between A* and the synthesizing material property parameters ln(e_range(s)∙C_u(s)∙χ) was established:

$$A*=54.6\times {\left[\ln \left(e_{\mathrm{range}\left(\mathrm{s}\right)}·C_{\mathrm{u}}{}_{\left(\mathrm{s}\right)}·\chi \right)\right]}^{-0.43}$$

(11)

Thus, the modified G_max prediction method based on the binary packing model can be established by combining Equations (4), (10), and (11), only considering basic indices of the clean sand and pure fines. It is worth noting that the application of the binary packing model should not be limited to the evaluation of G_max. Existing test results show that e* presents a unified correlation with static liquefaction characteristics [39], drained and undrained triaxial compression behaviors [40], critical strength [41], liquefaction strength [42], etc., of silty sand, and the proposed procedure in this paper provides a significant improvement in the evaluation of the above mechanical properties in geotechnical engineering practice.

4. Conclusions

In order to investigate how e, FC and ${\sigma }_{\mathrm{c}0}^{\prime }$ alter the G_max of marine silty sand, comprehensive bender element tests were performed under isotropic consolidation, and a modified procedure based on the Hardin model was established to predict the G_max. The main obtained results are summarized as follows.

(1)

Under otherwise similar conditions, G_max decreases with decreasing e or FC, but decreases with increasing FC. In addition, the reduction rate of G_max with e increasing is not sensitive to ${\sigma }_{\mathrm{c}0}^{\prime }$, but obviously sensitive to changes in FC.

(2)

For a specific FC, the traditional Hardin model can well characterize the influence of e and ${\sigma }_{\mathrm{c}0}^{\prime }$ on the G_max of silty sands. The stress exponent n does not appear to be sensitive to changes in FC and e, but sensitive to changes in the types of silty sand. In addition, the soil-specific constant n increases with increasing $C_{\mathrm{u}}^{\mathrm{s}}\cdot C_{\mathrm{u}}^{\mathrm{f}}$ and shows a logarithmic function. However, the material-specific fitting parameter A in the Hardin model is sensitive to FC. The traditional Hardin model cannot incorporate the influence of FC on G_max of marine silty sand.

(3)

e*, instead of e, can be an appropriate proxy to characterize the G_max of marine silty sand with various FC. The modified Hardin model, established in the framework of the binary packing model, allowing unified characterization of Gmax values for silty sands, only considering basic indices of the clean sand and pure fines. The predicted errors are within 10% for the Nantong marine silty sand tested. Independent test data in the literature validate the applicability of this modified model.