Characterizing the Effect of Water Content on Small-Strain Shear Modulus of Qiantang Silt

Due to the impact of natural and artificial influence, such as waves, tides, and artificial dewatering, the small-strain shear modulus of soils may vary with the water content of soil, causing deformation of excavations and other earth structures. The present study used a resonant column device to investigate the effects of water content, void ratio, and confining pressure on the small-strain shear modulus of a silt extracted from an excavation site near Qiantang River in Hangzhou, China. The test results revealed that the effects of the three factors are not coupled and can be characterized by three individual equations. In particular, the small-strain shear modulus decreases with increasing water content under otherwise similar conditions, which can be characterized by a power function. The classical Hardin’s equation is modified to consider the effect of water content by introducing an additional function of water content.


Introduction
Hangzhou, a city of China, is rapidly developing the alluvial region near the Qiantang River. However, the river tide, initiated from the sea, occurs periodically and has a strong impact along the alluvial regions. Waves and tides affect the underground water conditions [1], e.g., groundwater table, and thus affect the mechanical properties of soils due to change of water content or effective stress. Thus, earth structures, such as excavations and tunnels, constructed in areas under such impacts may encounter potential problems regarding serviceability or stability. In addition, the artificial drawn down of groundwater level in the excavation may also affect the groundwater level nearby. These natural and artificial changes of groundwater may cause a substantial change of the in-situ water content and thus a change of small-strain shear modulus of the soil [2,3], which is an important design parameter for geotechnical applications involving dynamic and cyclic loadings, such as seismic ground motion characteristic analysis and soil-structure interaction analysis [4,5]. For these reasons, it would be practically interesting to investigate and characterize the dynamic properties of the soils under such conditions.
The small-strain shear modulus (G 0 ) is an important soil property and an essential parameter in analyzing the deformation and stability of excavations. Extensive investigations have revealed that G 0 can be affected by confining pressure, void ratio (e), overconsolidation ratio, soil gradation, particle properties, and so on [6][7][8][9][10][11]. Among these factors, confining pressure and void ratio have been widely investigated [12][13][14][15][16][17][18], and their effects can be well-characterized by Hardin's equation, as follows: where A is a fitting parameter to reflect effects of soil properties and other factors; F(e) represents a void ratio function; P is the effective confining pressure; P a is a reference pressure equaling atmospheric pressure; n is a fitting parameter. Many previous investigations focused on G 0 of fully saturated soil and dry soil (particularly dry sand) [19,20]. Several other researchers investigated the G 0 of partially saturated soils (degree of saturation > 90%) and found G 0 of partially saturated soils is not affected by the degree of saturation [21]. However, the degree of saturation and water content of soils may change in projects associated with the change of groundwater level. Some researchers have found that the soil exhibits different dynamic and strength properties under different water contents. For example, Song et al. [22] found that the modulus measured under a given dynamic strain level decreases with increasing water content, while the decreasing rate increases with water content; Zhang et al. [23] investigated the modulus of soil under different water contents and found the modulus decreases with increasing water content as an exponential function. Both of the two studies, however, used a cyclic triaxial apparatus to measure the modulus under a strain range of 10 −4~1 0 −1 , which exceeds the range of small strain (10 −6~1 0 −5 ). In addition, small-strain apparatus (e.g., resonant column and bender element apparatus) can be modified to control and to measure the degree of saturation and matrix suction in the soil and thus provide more insightful investigations of the small-strain properties of unsaturated soils [24,25]. However, controlling and measuring matrix suction requires a relatively long time and specialized expertise. For routine practice, characterization of small-strain stiffness of soils with different water content under traditional framework should be a much simpler and easier way. It should also be noted that the impact of water content on G 0 may vary with soil properties [24] and thus it is important to establish a database for different types of working conditions and different regions.
In this study, G 0 of Qiantang River silt was measured by resonant column tests, with a range of water content, confining pressure, and void ratios. Particular attention was focused on the effects of water content on G 0 under different void ratios and confining pressures. Based on the laboratory data and analysis, a modified version of Hardin's equation was proposed to extend its applicability to soils with various water contents.

Tested Soils
The tested soil was extracted from an excavation site near Qiantang River, Hangzhou, China. It displays a silky touch and disintegrates in water. According to the particle size distribution curve presented in Figure 1, it is seen that 80.16% of the particles are within the range of 0.005 to 0.075 mm (silt-size particles according to Chinese standard), giving a mean particle size of 0.035 mm. The plastic limit (PL) is 17.3, and the liquid limit (LL) is 27.9 using the Chinese standard method, resulting in a plastic index (PI) of 10.6. The specific gravity of the silt is 2.69. Figure 2 presents the SEM images with 2000 times of magnification, showing that the particle is of irregular shape but more or less uniform size. Some very small clayey particles adhere to the surface of the larger particles. The other physical properties are listed in Table 1.

Tested Procedures
The moist-tamping method [19,26] was adopted to prepare the cylindrical specimens with a 50 mm diameter and a 100 mm height for the resonant column tests. The preparation procedures are given as follows: (1) gravel-size clusters in the air-dried soil were removed by passing a 2 mm sieve; (2) a certain amount of water was mixed into the soil based on the target water content to form a uniform wet mixture; (3) a designed amount of wet soil was compacted in the mold to the desired height (20 mm for each layer); (4) the surfaces of the first four layers were scratched to provide interlocking between each layer and procedure; (3) was repeated until the last layer was compacted; (5) the sample was retrieved from the mold to measure the sample weight and dimension, which will be used to calculate the water content and void ratio after sample preparation with using the ovendried sample mass after the resonant test. The targeted water content ranges from 7% to

Tested Procedures
The moist-tamping method [19,26] was adopted to prepare the cylindrical specimens with a 50 mm diameter and a 100 mm height for the resonant column tests. The preparation procedures are given as follows: (1) gravel-size clusters in the air-dried soil were removed by passing a 2 mm sieve; (2) a certain amount of water was mixed into the soil based on the target water content to form a uniform wet mixture; (3) a designed amount of wet soil was compacted in the mold to the desired height (20 mm for each layer); (4) the surfaces of the first four layers were scratched to provide interlocking between each layer and procedure; (3) was repeated until the last layer was compacted; (5) the sample was retrieved from the mold to measure the sample weight and dimension, which will be used to calculate the water content and void ratio after sample preparation with using the ovendried sample mass after the resonant test. The targeted water content ranges from 7% to

Tested Procedures
The moist-tamping method [19,26] was adopted to prepare the cylindrical specimens with a 50 mm diameter and a 100 mm height for the resonant column tests. The preparation procedures are given as follows: (1) gravel-size clusters in the air-dried soil were removed by passing a 2 mm sieve; (2) a certain amount of water was mixed into the soil based on the target water content to form a uniform wet mixture; (3) a designed amount of wet soil was compacted in the mold to the desired height (20 mm for each layer); (4) the surfaces of the first four layers were scratched to provide interlocking between each layer and procedure; (3) was repeated until the last layer was compacted; (5) the sample was retrieved from the mold to measure the sample weight and dimension, which will be used to calculate the water content and void ratio after sample preparation with using the oven-dried sample mass after the resonant test. The targeted water content ranges from 7% to 18%, and the void ratio varies from 0.67 to 0.8. The tested degree of saturation covers a wide range from about 20% to about 70%. The actual water contents and void ratios after sample preparation deviate slightly from the targeted values. Table 2 summarizes the detailed testing conditions of the program. The prepared sample was placed in the pressure chamber, with the resonant components mounted on the top of it. This study aims to simulate an unsaturated soil condition due to artificial drawdown of groundwater, in which the pores are connected to the air. Thus, back pressure saturation is skipped, and the pores of the sample are connected to the atmosphere. Then, a series of confining pressures were applied (50, 100, 200, 300, and 400 kPa) to simulate different total overburden pressures. Although the in-situ stress condition is anisotropic rather than isotropic, the results under isotropic stress state can be converted to an anisotropic state by multiplying a certain function regarding stress anisotropy in most cases. The effects of other factors (e.g., void ratio, water content, etc.) are not affected by the stress anisotropy. The resonant column test was conducted after each pressure level was reached and the measured axial deformation ceased. The measured axial deformation was used to estimate the volumetric deformation and thus void ratio by assuming the volumetric strain is three times as the axial strain.

Resonant Column Tests
This study adopted the Stokoe-type resonant column system manufactured by GDS Instruments (GDS-RCA), as shown in Figure 3. The electric-magnetic driver generates a series of excitations with different frequencies. The sample may respond differently to these excitations, and such responses will be recorded. Figure 4 illustrates three frequencyresponse curves of samples with the same confining pressure (200 kPa) and similar void ratios (0.683~0.698) but three different water contents. Each response curve exhibits a distinct peak, corresponding to the resonant frequency (f n ). G 0 is calculated using the following equations based on the theory of elasticity.
where I and I 0 are the mass polar moments of inertia of the specimen and the driving system, respectively; f n is the resonant frequency (Hz); L is the sample length (m); V s is the shear wave velocity of the sample (m/s); and ρ is the sample density (kg/m 3 ). where I and I0 are the mass polar moments of inertia of the specimen and the driving system, respectively; fn is the resonant frequency (Hz); L is the sample length (m); Vs is the shear wave velocity of the sample (m/s); and ρ is the sample density (kg/m 3 ).    where I and I0 are the mass polar moments of inertia of the specimen and the driving system, respectively; fn is the resonant frequency (Hz); L is the sample length (m); Vs is the shear wave velocity of the sample (m/s); and ρ is the sample density (kg/m 3 ).

Effects of Void Ratio and Confining Pressure
G 0 of soil is state-dependent, namely, it is a function of both void ratio and confining pressure, but the relationship between G 0 and the two factors is soil-specific, i.e., the relationship, particularly the fitting parameters of Equation (1), can be affected by soil properties and characteristics. For this reason, it is necessary to evaluate the effects of the two factors for the tested soil, especially when different water contents are considered. Selected data of the G 0 -e relationship are presented in Figure 5, showing that G 0 decreases with increasing void ratio under all tested confining pressures and water contents (w c ). This observation is the same as that for fully saturated soils. There have been two commonly used void ratio functions (Equations (3) and (4)) [27,28] to characterize the G 0 -e relationship.
where a and x are fitting parameters. The fitting curves using Equation (3) with a = 2.17 are compared with the test data in Figure 5 for different confining pressures and water contents, showing good agreement. Hardin and Black [29] suggested that a = 2.97 for angular particles and a = 2.17 for rounded particles. However, some other researchers suggested that a = 2.17 is suitable for most soil types [30]. In addition, x = 1.3 in Equation (4) can produce good predictions.
where a and x are fitting parameters. The fitting curves using Equation (3) with a = 2.17 are compared with the test data in Figure 5 for different confining pressures and water contents, showing good agreement. Hardin and Black [29] suggested that a = 2.97 for angular particles and a = 2.17 for rounded particles. However, some other researchers suggested that a = 2.17 is suitable for most soil types [30]. In addition, x = 1.3 in Equation (4) can produce good predictions. The data in Figure 5 also show that higher confining pressure leads to higher G0 at a given void ratio for a given water content. In order to clearly illustrate the effect of confining pressure, G0 is plotted against confining pressure in Figure 6, showing that G0 increases non-linearly with increasing confining pressure for a given combination of void ratio and water content. By increasing the confining pressure from 50 to 400 kPa, G0 generally increases by a factor of about 2. In addition, Figure 6 also reflects the effects of void ratio that echoes the observation in Figure 5. To better characterize the effects of confining pressure, a void ratio-normalized parameter, G0/F(e), is calculated and plotted against the confining pressure in Figure 7, where F(e) adopts either Equation (3) or Equation (4), with a = 2.17 or x = 1.3, respectively. Clearly, higher confining pressure leads to higher G0/F(e), as shown in Figure 7. A power function can be used to characterize the non-linear effect of confining pressure on G0/F(e), for each water content. It should be noted that different void ratio functions have nearly no effect on the exponents. The data in Figure 5 also show that higher confining pressure leads to higher G 0 at a given void ratio for a given water content. In order to clearly illustrate the effect of confining pressure, G 0 is plotted against confining pressure in Figure 6, showing that G 0 increases non-linearly with increasing confining pressure for a given combination of void ratio and water content. By increasing the confining pressure from 50 to 400 kPa, G 0 generally increases by a factor of about 2. In addition, Figure 6 also reflects the effects of void ratio that echoes the observation in Figure 5. To better characterize the effects of confining pressure, a void ratio-normalized parameter, G 0 /F(e), is calculated and plotted against the confining pressure in Figure 7, where F(e) adopts either Equation (3) or Equation (4), with a = 2.17 or x = 1.3, respectively. Clearly, higher confining pressure leads to higher G 0 /F(e), as shown in Figure 7. A power function can be used to characterize the non-linear effect of confining pressure on G 0 /F(e), for each water content. It should be noted that different void ratio functions have nearly no effect on the exponents.  Based on the above analysis, it is clear that the effects of void ratio and confining pressure can be characterized separately by a void ratio function (either Equation (3) or Equation (4)) and a power function of confining pressure, respectively. Moreover, the parameters of the void ratio function are not affected by confining pressure, and the power index of confining pressure is not affected by the void ratio. In other words, classical Hardin's equation can be used to characterize the effects of confining pressure and void ratio under a given water content. Based on the above analysis, it is clear that the effects of void ratio and confining pressure can be characterized separately by a void ratio function (either Equation (3) or Equation (4)) and a power function of confining pressure, respectively. Moreover, the parameters of the void ratio function are not affected by confining pressure, and the power index of confining pressure is not affected by the void ratio. In other words, classical Hardin's equation can be used to characterize the effects of confining pressure and void ratio under a given water content.

Effects of Water Content
In Figure 8, G0/F(e) decreases non-linearly with increasing water content for each confining pressure, regardless of the selection of the type of void ratio function. By using Equation (1) with either void ratio function (Equation (3) with a = 2.17 or Equation (4) with x = 1.3), the parameters A and n can be best fitted by using the G0/F(e) data for each water content. The fitted parameters are summarized in Table 3.

Effects of Water Content
In Figure 8, G 0 /F(e) decreases non-linearly with increasing water content for each confining pressure, regardless of the selection of the type of void ratio function. By using Equation (1) with either void ratio function (Equation (3) with a = 2.17 or Equation (4) with x = 1.3), the parameters A and n can be best fitted by using the G 0 /F(e) data for each water content. The fitted parameters are summarized in Table 3.  The effects of water content on parameters A and n are shown in Figure 9. Increasing water content leads to a decrease of parameter A, and the following equation can be used to characterize the relationship between parameter A and the water content.
where the fitting parameter A * is dependent on the selection of void ratio function; wc is in percentage; and m is a fitting parameter with a value of 0.263. Parameter n increases slightly with increasing water content (n1 = 0.388~0.438, n2 = 0.390~0.442). In addition, it seems that A and n have a unique relationship for a given type of void ratio function as shown in Figure 10.  The effects of water content on parameters A and n are shown in Figure 9. Increasing water content leads to a decrease of parameter A, and the following equation can be used to characterize the relationship between parameter A and the water content.
where the fitting parameter A * is dependent on the selection of void ratio function; w c is in percentage; and m is a fitting parameter with a value of 0.263. Parameter n increases slightly with increasing water content (n 1 = 0.388~0.438, n 2 = 0.390~0.442). In addition, it seems that A and n have a unique relationship for a given type of void ratio function as shown in Figure 10. The effects of water content on parameters A and n are shown in Figure 9. Increasing water content leads to a decrease of parameter A, and the following equation can be used to characterize the relationship between parameter A and the water content.
where the fitting parameter A * is dependent on the selection of void ratio function; wc is in percentage; and m is a fitting parameter with a value of 0.263. Parameter n increases slightly with increasing water content (n1 = 0.388~0.438, n2 = 0.390~0.442). In addition, it seems that A and n have a unique relationship for a given type of void ratio function as shown in Figure 10.
(a) (b) Figure 9. The curve of relations between fitting parameters and water contents (a) parameter A and (b) parameter n.

Revised Hardin's Equation
The previous analysis has revealed that water content mostly affects parameter A but has negligible effects on parameter n. It would be reasonable to assume the effects of water content and the confining pressure are not coupled, i.e., a constant value of n can be assumed for all tested water contents. A function of water content (Equation (6)) can be introduced to the original Hardin's equation.
where the value of the fitting parameter m is 2.36 for the tested material. The original Hardin's equation can be modified as follows.
where P is the total confining pressure for this study. Equation (7) takes into account the effects of water content and can be applied to engineering problems in which the smallstrain shear modulus changes with the drawdown of groundwater level. Table 4 summarizes the fitted parameters, assuming a constant n.

Revised Hardin's Equation
The previous analysis has revealed that water content mostly affects parameter A but has negligible effects on parameter n. It would be reasonable to assume the effects of water content and the confining pressure are not coupled, i.e., a constant value of n can be assumed for all tested water contents. A function of water content (Equation (6)) can be introduced to the original Hardin's equation.
where the value of the fitting parameter m is 2.36 for the tested material. The original Hardin's equation can be modified as follows.
where P is the total confining pressure for this study. Equation (7) takes into account the effects of water content and can be applied to engineering problems in which the small-strain shear modulus changes with the drawdown of groundwater level. Table 4 summarizes the fitted parameters, assuming a constant n.

Discussions
The present study has shown that G 0 of the Qiantang River silt decreases with increasing water content under a given void ratio and confining pressure. This G 0 -w c trend is related to the decrease of the matric suction due to the increase of w c . However, measuring the matrix suction may not be feasible for many routine projects, and thus, the proposed revision of Hardin's equation can be applied easily to characterize the effects of water content. It should be noted that the matrix suction can be affected by various factors, such as particle size distribution, pore characteristics, soil mineralogy, etc., [31,32]; the empirical parameters of the proposed equation (Equation (7)) may be dependent on these factors. Further studies are needed to investigate the combined effects of water content and soil properties on G 0 .
The proposed power function to characterize the effects of water content is only based on the tested range of water content. For the lower range and the higher range, more tests are needed, as the power function may not capture the trend. In addition, the shear modulus may also change due to dry-wet cycles, which simulate the drawdown and recovery of the groundwater table due to artificial and natural process. This is because the dry-wet cycles may alter the micro-structure (e.g., pore size distribution) and stress/strain history. This process needs to be taken into account in future studies.

Conclusions
The small-strain shear modulus, G 0 , of Qiantang River silt is measured using a resonant column test, with consideration of a range of different water contents, confining pressures, and void ratios. G 0 decreases with increasing void ratio for all tested confining pressures and water contents. The effect of void ratio can be characterized by a void ratio function for a given water content and confining pressure. G 0 increases with increasing confining pressure for all tested void ratios and water contents. The effects of confining pressure can be characterized by a power function for a given void ratio and water content. Since the effects of confining pressure and void ratio are not coupled for all the tested water contents, the classical Hardin's equation can be used to characterize the effects of the two factors. G 0 decreases with increasing water content, and the effect is not coupled with the effects of void ratio and confining pressure. An equation to characterize these effects is proposed by adding a water content function to the original version of Hardin's equation.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.