# Effect of Mean Grain Size on the Small-Strain Dynamic Properties of Calcareous Sand

Authors: Liwei Wen
Journal: Advances in Civil Engineering
 (2022)

---

## Abstract

Calcareous sand was selected as the prior material for island reclamation in many coastal regions. The mechanical properties of the granular materials are greatly affected by their grain size distribution conditions. The shear modulus and damping ratio are two important parameters for earthquake ground response analysis and liquefaction evaluation. A series of resonant column tests had been performed on calcareous sands with varying median grain diameter and uniform coefficient. The dependence of the shear modulus and damping ratio of the calcareous sand on grain size has been confirmed in this examination. The test results reveal that the shear modulus decreases with a rise in shear strain for calcareous sand samples at a given confining pressure and relative density. The maximum shear modulus tends to increase with confining pressure and relative density. On the maximum shear modulus and void ratio plane, the trend lines of the measured results shift toward up and right position with a rise in grain diameter. The measured results indicate that the influence of uniform coefficient on the maximum shear modulus is neglectable. A revised empirical equation based on the Hardin model had been proposed considering the influence of grain diameter to estimate the maximum shear modulus of calcareous sand. The predicted values show satisfactory agreement with the measured results. The results manifest that the effect of grading condition on small-strain dynamic properties of calcareous sands cannot be neglected for the evaluation of seismic safety for reclamation engineering sites.


---

## Body

## 1. Introduction

Calcareous sands are biogenic medium and widely distributed in the coastal region all around the world. They are originated from marine shells and organisms. The major composition of such marine medium is calcium carbonate. The calcareous sands possess an intrapore structure, angular shape, and a rough surface. The calcareous sands are liable to be damaged at normal working stress [1–4]. Thus, calcareous sediments exhibit different mechanical response compared to the terrigenous sand. Recently, the calcareous sand is selected as the prior material for reclamation construction. The demand for fundamental understanding of the mechanical properties of calcareous sand is highly urgent.Some costal island reclamation work in South China Sea had been constructed in past years [5–8]. Besides, the occurrence of earthquakes is frequently reported in that area. Therefore, well understanding of the dynamic characteristic of the sand deposition site is significantly important for the earthquake of reef reclamation site. Moreover, the small-level dynamic properties are parameters affecting the ground motion characteristics. Those indexes can also be adopted to estimate the settlement of reclamation ground subjected to earthquake.The shear stiffness of the sand at the shear strain level less than 10–5 is generally defined as Go. Past studies verified that the shear modulus was the function of void ratio and confining stress. Additionally, the small-level dynamic properties of sand are also susceptible to many other factors including gradation, distribution, and particle characteristics. The bender element and resonant column tests are the two tradition laboratory approaches to measure the small-level dynamic index of sands. The resonant column tests had been extensively carried out to measure the small-level dynamic properties of sands [9–12]. Phamet al. [13] found that the maximum shear modulus of the calcareous was larger than that of silica sand under the same test condition. Goudarzy et al. [14] noticed that the maximum shear modulus significantly reduced with increasing fines content and adopted the equivalent skeleton void ratio [15] to revise the Hardin model [16] to include the effect of fines particles on fabric and force chains of sand assembly. Jafarian and Javdanian [17] tested the dynamic properties of siliceous-carbonate sand and understood that the maximum shear modulus of the samples subjected to anisotropic condition was larger than the isotropically consolidated samples. The effect of uniform coefficient and mean grain size on the dynamic properties of sand at the small-strain level is mixed. Anastasiadis et al. [18] reported that the mean grain size affected the normalized shear modulus of the sand. However, some researchers held the opposite opinions [19, 20]. Therefore, the investigation on the effect of grain diameter on the small-level dynamic properties of the calcareous sand is still insufficient. Further research should be conducted to complement the existing literature.This study carries out a series of resonant column tests to examine the influences of median diameter and uniform coefficient on the shear modulus and damping ratio of the calcareous sand at different confining pressures and densities. The dependence of the shear modulus of calcareous sand on the void ratio and confining pressure plane is also studied. The test results indicate that the effect of gradation condition on the small-strain dynamic properties of calcareous sand should be seriously considered when the earthquake ground response analysis and liquefaction evaluation of reclamation site are conducted. The well-proved Hardin model [16, 21] had been revised to consider the effect of gradation condition to estimate the maximum shear modulus of calcareous sand under different test conditions. The predicted values agree well with measured results for calcareous sand with varying gradation conditions.


## 2. Materials and Methods

The calcareous sand was collected from the Nansha island, South China Sea. The grain size distribution range for the calcareous sand of the Nansha island region is quite wide, and these biogenetic sediments contain large and small grains by arbitrary proportion. To investigate the influence of gradation property on the small-strain dynamic characteristics of calcareous sand, the grain size distribution curves of calcareous sand had been artificially adjusted. Figure1 shows the grain size distribution curves of all samples. Three median diameters (d50) of 0.75 mm, 1.5 mm, and 2.5 mm had been selected. For each grain diameter, the uniform coefficient Cu had been prepared as 1.2, 1.8, and 2.4. The larger the uniform coefficient Cu is, the wider the grain size distribution curve becomes. The grading curve basically covers the representative grain size distribution range in the reclamation reef island. It is noticed that less fines particle with a grain diameter smaller than 0.075 mm exist in tested materials [22–24]. Table 1 shows the representative physical parameters of the calcareous sand with varying median diameters and uniform coefficients. It is noted that the maximum and minimum void ratios of calcareous sand are affected by the gradation condition. For the same median diameter d50, both the maximum and minimum void ratios decrease as coefficient uniform Cu increases. The maximum and minimum void ratios also increase as the median diameter increases. It is because more void space is created for the skeleton structure formed by large grains. The specific gravity Gs is less dependent on the median diameter and uniform coefficient.Figure 1
Grain size distribution curves of calcareous sand with varying gradation conditions.Table 1
The physical parameters of calcareous sand with different gradation conditions.
Sample no.emaxemind50 (mm)CuGs11.2640.9020.751.22.7621.2790.8540.751.82.7631.2140.7740.752.42.7641.2730.9681.51.22.7651.1390.8961.51.82.7661.0720.8351.52.42.7671.3901.0842.51.22.7681.2951.0062.51.82.7691.2330.9702.52.42.76Figure2 shows the scanning electron microscopy of calcareous sand grains. The image shows that the particle shape of the calcareous sand is irregular and complex. The pores are distributed on the grain surface, and some pores are connected into the internal portion. The grain surfaces seem quite rough. The complex grain shape and microstructure of calcareous sand make its dynamic properties quite different from terrigenous sands.Figure 2
Scanning electron microscopy of calcareous sand.The calcareous sand samples were prepared using the wetting tamping method [25–31]. The calcareous sand was mixed with an initial water content of 10%. The cylindrical sample is 50 mm in diameter and 100 mm in height. The moist calcareous sand was gradually poured into mold by five layers. The samples were prepared at different densities by specific tamping times. Three target relative densities of samples as 30% (loose state), 60% (medium-dense state), and 80% (dense state) were prepared. The saturation process for the calcareous sand sample is difficult due to its porositic nature. Initially, the calcareous sand is kept in vacuum for de-air. CO2 is injected into the calcareous sand sample to replace the air, and this process lasts for 45 minutes. The saturation process for calcareous sand sample is accomplished by gradually elevating the back pressure. The Skempton B value for all samples is over 0.95 before testing. The isotropic consolidation pressures are set as 50 kPa, 100 kPa, 200 kPa, and 300 kPa before testing.The GDS resonant column testing apparatus had been adopted to study the small-strain dynamic response of calcareous sand in this investigation. The resonant column testing apparatus is classified into the Stokoe type. The sample is excited in sinusoidal vibrations. The confining and back pressure range are 1.0 MPa, and the operating frequency range is between 5 and 30 Hz. The cylindrical sample is installed in triaxial cell, and the cell base is embraced by water in an inner tube.


## 3. Test Results

Figure3 shows the shear modulus G-shear strain γ curve of calcareous sand with a median diameter of 0.75 mm and a uniform coefficient Cu = 1.2 at three densities of 30%, 60%, and 80%. The varying range of shear modulus measured in this study is between 5∗10−5 and 10−4. The test results display that the shear modulus G tends to decline with a rise in the shear strain regardless of the initial relative density. The attenuating tendency of the shear modulus is steady at a small-level shear strain and intensifies as the shear strain increases. The confining stress-dependence of the shear modulus for calcareous sand is also clearly seen. At a specific confining pressure, the shear modulus-shear strain curves move upward with a rise in relative density. The varying range for the shear modulus of calcareous sand at loose, medium-dense, and dense states is between 40 MPa and 210 MPa.Figure 3
Shear modulusG and shear strain γ for calcareous sand with d50 = 0.75 mm and Cu = 1.2 at different confining pressures. (a) Loose state. (b) Medium dense state. (c) Dense state.
(a)(b)(c)Figures4 and 5 present the plot of shear modulus G against the shear strain γ of calcareous sand with median diameters of 1.5 mm and 2.5 mm at varying confining pressures and densities. The shear modulus-shear strain curves expressed in Figures 4 and 5 exhibit quite similar varying tendency to the calcareous sand with a small median grain diameter shown in Figure 3. A rise in the confining pressure largely elevates the position of the G-γ curve. A rise in deposition state markedly enhances the shear modulus under the same test condition. It demonstrates that the densification of the calcareous sediment is an effective approach to increase the resilience of the reclamation ground when the earthquake occurs.Figure 4
Shear modulusG and shear strain γ for calcareous sand with d50 = 1.5 mm and Cu = 1.2 at different confining pressures. (a) Loose state. (b) Medium dense state. (c) Dense state.
(a)(b)(c)Figure 5
Shear modulusG and shear strain γ for calcareous sand with d50 = 2.5 mm and Cu = 1.2 at different confining pressures. (a) Loose state. (b) Medium dense state. (c) Dense state.
(a)(b)(c)Figures6(a) and 6(b) show the varying tendency of maximum shear modulus Go with the level of confining stress for calcareous sand samples with median diameters of 0.75 mm and 2.5 mm, respectively. The maximum shear modulus Go is determined as the intersection point of vertical axis and the fitting curve to measured data on shear modulus and shear strain γ plane. The dependence of maximum shear modulus Go on the stress and density levels is distinctive. It is seen that the maximum shear modulus Go linearly increases with a rise in confine pressure. An increasing confining pressure makes the calcareous sand grains closely contacted during the isotropic consolidation process. A rise in relative density enhances the maximum shear modulus for calcareous sand sample with small and large grains. Confining stress and deposition state are two influential parameters determining the maximum shear modulus for sandy material. It indicates that a denser calcareous ground owns a stronger modulus to resist small-level vibration loading. The results are consistent with past investigations on silica and calcareous sands [9, 32].Figure 6
Maximum shear modulusGo and confining pressure for calcareous sands with Cu = 1.2 at varying densities. (a) d50 = 0.75 mm and (b) d50 = 2.5 mm.
(a)(b)Figures7(a) and 7(b) show the maximum shear modulus Go plotted against the median diameter d50 for the calcareous sand. For calcareous sand with a uniform coefficient Cu = 1.2, the maximum shear modulus Go is insensitive to the grain size at both loose and dense states. However, for calcareous sand with a relatively larger uniform coefficient, the maximum shear modulus Go initially increases and reverses to reduce with diameter varying from 0.8 to 2.4. For a well-graded calcareous sand with a specific median diameter, the total grain number within a specimen and the major “force chain” subjected to torsion are larger than the other two cases. It is supposed that the mechanism about the influences of grain size and gradation condition on the small-level dynamic shear modulus is quite complicated. Further investigation should be carried out to reveal the relevant micromechanism.Figure 7
Maximum shear modulusGo and median diameter d50. (a) Cu = 1.2, Dr = 30%; (b) Cu = 1.2, Dr = 80%; (c) Cu = 2.4, Dr = 30%; (d) Cu = 2.4. Dr = 80%.
(a)(b)(c)(d)To directly understand the effect of uniform coefficient Cu on the maximum shear modulus, the test data under different test conditions are compared on the maximum shear modulusGo-void ratio e plane. Figure 8 shows the relationship between the maximum shear modulus and void ratio for calcareous sand with varying uniform coefficient. The maximum shear modulus Go tends to decrease with a rise in the void ratio. The calcareous sands subjected to dynamic loading loses its deformation resistance ability when it is at a loose state. It is noted that the effect of uniform coefficient on the Go-e curve is neglectable. In Figure 8, all test results could be simulated using a curve on the maximum shear modulus and the void ratio plane when the median diameter and confining pressure are known. The test results acquired in this work are different to previous investigations on silica sand [11, 33]. The difference is probably originated from the grain morphology between the calcareous sand and silica sand.Figure 8
Shear modulus and void ratio for calcareous sand with varying uniform coefficient Cu. (a)d50 = 0.75 mm, 100 kPa; (b) d50 = 0.75 mm, 200 kPa; (c) d50 = 1.5 mm, 100 kPa; (d) d50 = 1.5 mm, 200 kPa.
(a)(b)(c)(d)Figure9 expresses the maximum shear modulus Go plotted against the void ratio e for calcareous sand with uniform coefficient Cu = 1.2 and different median diameters. The vertical axis for Figures 9(a)–9(d) is different because the applied confining pressure level is different. With similarity to Figure 8, the maximum shear modulus displays reducing tendency as the sample becomes loose. Under the same test condition, the effect of the grain diameter on the maximum shear modulus Go is remarkable on the Go-e plane. It is clearly seen that a decrease in grain diameter d makes the measured result points move downward. The difference in the grain diameter leads to the different fabric and coordination number among the grain particles. The complex contact condition in samples with small grain diameter leads to the reduction in maximum shear modulus for calcareous sand [34–38].Figure 9
Maximum shear modulus and void ratio for calcareous sand with Cu = 1.2. (a) 50 kPa, (b) 100 kPa, (c) 200 kPa, and (d) 300 kPa.
(a)(b)(c)(d)Figures10 and 11 show the relationship between the maximum shear modulus Go and void ratio e for calcareous sands at different confining pressures and two uniform coefficients. It indicates that the maximum shear modulus reduces as samples become loose. Besides, a rise in the grain diameter increases the maximum shear modulus at the same void ratio although only two measured points are acquired for a given median diameter. It is noted that the dependence of maximum shear modulus Go on the grain size is applicable for samples with different Cu. Therefore, the effect of median diameter on the dynamic index of calcareous sand should be reasonably treated when the estimation equation of the maximum shear modulus is proposed.Figure 10
Maximum shear modulus and void ratio for calcareous sand with Cu = 1.8. (a) 50 kPa, (b) 100 kPa, (c) 200 kPa, and (d) 300 kPa.
(a)(b)(c)(d)Figure 11
The maximum shear modulus and void ratio for calcareous sand with Cu = 2.4. (a) 50 kPa, (b) 100 kPa, (c) 200 kPa, and (d) 300 kPa.
(a)(b)(c)(d)


## 4. Discussion

Past studies revealed that the shear modulus of the sand was greatly dependent on the density and confining stress [39–43]. Hardin and Richart [21] proposed an empirical formula to estimate the shear modulus Go of sandy soil based on a database. The shear modulus Go could be expressed as follows:(1)Go=Ac−e21+eppan,where A, n, and c are the material constant. pa represents atmospheric pressure (100 kPa), and p indicates the consolidation stress. Additionally, parameter c is affected by the angularity of the grains and takes the value of 2.97 for angular sand. The item related with void ratio e is also expressed by other forms in some studies.Figure12 shows the Go/p/pa0.5−e curve of calcareous sand using three median diameters. The normalization form of vertical axis is adopted to eliminate the effect of confining stress. The points representing the test results with the same Cu at different confining pressures are closely distributed. The test results could be simulated using the Hardin model. However, the model parameters are greatly dependent on the median diameter d50. Therefore, it is necessary to consider the effect of median diameter when the maximum shear modulus is estimated. For calcareous sands with a given median diameter d50, a set of physical parameters including A and n could be obtained.Figure 12
Go/p/pan and void ratio for calcareous sand at different confining pressure. (a) Cu = 1.2, (b) Cu = 1.8, and (c) Cu = 2.4.
(a)(b)(c)Figure13 shows that model parameter A linearly increases with a median diameter of d50. Besides, parameter n has an exponent power relationship with median diameter d50. The goodness of fitting curves in Figure 13 is 0.99. These two parameters can be closely related and expressed using the median diameter d50. Previous studies indicated that parameter c was insensitive to particle characteristics. Thus, parameter c is regarded to be a constant in the prediction.Figure 13
The median diameter and parameters associated with experienced equations. (a) ParameterA and (b) parameter n.
(a)(b)Figure14 represents all the test results for calcareous with three grain diameters at different confining pressures and densities. The varying tendency of normalization value of Go with void ratio e could be described using the unique trend line. The trend line is determined using the revised Hardin model. The physical parameter d50 is directly introduced in the Hardin model to quantitatively describe the parameters A and n. Parameters A and n are determined using the best-fitting method to the measured results. The revised Hardin model considering the effect of grain diameter is capable of predicting the maximum shear modulus of calcareous with different gradation conditions at different densities and stresses.Figure 14
Go/p/panApa and void ratio for calcareous sands with three values of Cu.Figures15 and 16 show the predicted and measured maximum shear modulus of the calcareous sand with varying grading curves under different test conditions. The shear modulus estimated using the original Hardin model is displayed in Figure 15. It is noted that the points representing samples with a median diameter of 0.75 are above the test results. It is also seen that the points are out of the zone indicating the relative error value within 10%.Figure 15
Measured and predicted maximum shear modulusGo of the calcareous sand with different values of d50.Figure 16
Measured and predicted maximum shear modulusGo of the calcareous sand using the proposed estimated equation.Figure16 shows that the predicted values match well with measured results with a relative error value of less than 10%. The maximum shear modulus is predicted using the revised Hardin model. The estimation capacity is largely enhanced by considering the influence of the grain size. The calcareous sand ground owns a wide range of grain size distribution curves. The revised Hardin model is expected to provide the predicted maximum shear modulus to accurately reflect the influence of gradation characteristics of calcareous sand ground in engineering practice.


## 5. Conclusions

The grain median diameter is noticed to be an important parameter governing the monotonic and dynamic properties of the calcareous sand. A series of resonant column tests had been conducted to investigate the effect of gradation condition (median diameter and uniform coefficient) on the small-level dynamic property of the calcareous sand under different test conditions. The well-known Hardin model had been revised to include the influence of grain size on the small-level dynamic properties. The new findings of this work can be summarized as follows:(1)
The shear modulus of the calcareous sand tends to decrease with a rise in the shear strain at a given confining pressure and density. The confining stress-dependence and density-dependence of the shear modulus and damping ratio for calcareous sand had been confirmed in this investigation. The maximum shear modulusGo linearly increases with the level of confining pressure at each relative density. On the maximum shear modulus and void ratio plane, the influence of uniform coefficient Cu on the measured maximum shear modulus is minimal.(2)
On the shear modulus and void ratio plane, at the same confining pressure and uniform coefficient, a rise in the median diameter enhances the maximum shear modulus. The maximum shear modulus displays a decreasing tendency with increasing void ratio. The looser the calcareous sand sample is, the smaller the capacity of the resisting torsion loading becomes.(3)
The physical parameterA in the Hardin model increases with median diameter, while parameter n in the Hardin model exponentially decreases with the median diameter. The well-proved Hardin equation had been revised to consider the influence of median diameter. The proposed equation is capable of characterizing the small-level dynamic property of the calcareous sand with varying gradation conditions.



---

*Source: 9291890-2022-07-18.xml*