J. Cent. South Univ. (2020) 27: 911-919

DOI: https://doi.org/10.1007/s11771-020-4340-z

Gradation equation of coarse-grained soil and its applicability

WU Er-lu(吴二鲁)^{1, 2}, ZHU Jun-gao(朱俊高)^{1, 2}, CHEN Ge(陈鸽)^{1, 2},BAO Meng-die(包孟碟)^{1, 2}, GUO Wan-li(郭万里)³

1. Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering,Hohai University, Nanjing 210098, China;

2. Jiangsu Research Center of Geotechnical Engineering Technology, Hohai University,Nanjing 210098, China;

3. Geotechnical Engineering Department, Nanjing Hydraulic Research Institute, Nanjing 210024, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Gradation equation is one way to describe the gradation of coarse-grained soil conveniently, exactly and quantitatively. With the gradation equation, the influence of gradation on the mechanical behaviors of coarse-grained soil can be expressed quantitatively. A new gradation equation with a parameter is proposed. The basic properties and applicability of the new equation are studied. The results show that the proposed equation has the applicability to express coarse-grained soil gradation (CSG), and the range of the parameter b is found to be 00.5, the gradation curve is sigmoidal, otherwise the gradation curve is hyperbolic. For well graded gradations, the parameter has the value of 0.13

Key words: gradation curve; gradation equation; coarse-grained soil; applicability

Cite this article as: WU Er-lu, ZHU Jun-gao, CHEN Ge, BAO Meng-die, GUO Wan-li. Gradation equation of coarse-grained soil and its applicability [J]. Journal of Central South University, 2020, 27(3): 911-919. DOI: https://doi.org/10.1007/s11771-020-4340-z.

1 Introduction

With the increasing height of earth-rockfill dam, the physical and mechanical properties of coarse-grained soil need to be accurately grasped [1-4]. It is generally accepted that the gradation has great influence on the engineering properties of soil and behaviors of soil and structure interaction [5-11]. Some researches indicate that there are great differences in engineering properties among coarse grained soils with different gradations [12-17]. It is known that the coefficient of uniformity (C_u) and the coefficient of curvature (C_c) are proposed to describe coarse-grained soil gradation (CSG), and coarse-grained soil is regarded as well-graded when C_u>6 and C_c=1-3, otherwise it is poorly graded [18]. However, C_u and C_c cannot describe the gradation completely. For example, different gradations can have the same values of C_u and C_c. That is, there are two or more gradations corresponding to the same C_u and C_c.

Therefore, it is not sufficient to describe CSG with C_u and C_c so that one method to describe CSG more accurately needs to be found.

Now the fraction contents, frequency curve of particle group, and gradation curve have been widely used to describe CSG [19, 20]. However, it is difficult to use these ways to obtain the quantitative relationship between the engineering properties and the gradation of coarse-grained soil. For example, researchers can only analyze difference of each granular group content one by one when comparing the difference among gradations, which causes that researchers can only analyze the trend of the engineering properties of coarse-grained soil with the coarse-grained content changing [21-25]. Consequently, it is essential to find one method to describe CSG accurately and quantitatively so as to make it potential to establish the quantitative relationship between the engineering properties and CSG.

Gradation equation can describe CSG exactly and quantitatively. If CSG can be expressed by an equation, such as P=f(d) (d is the particle size, P is the percent passing through the sieve opening of d) with gradation parameters, a₁, a₂, _…, the quantitative relationship between the engineering properties and the gradation may be achieved by an equation. For example, the relationship between the maximum dry density (ρ_dmax) and the gradation can be expressed quantitatively by ρ_dmax=f(a₁, a₂, …).

So far, a few equations for the expression of soil gradation have been proposed. FULLER et al [26], TALBOT et al [27], ROSIN [28] and SWAMEE et al [29] all presented their own gradation equation. However, these equations all have limitations to express gradation. Therefore, it is essential to find a more appropriate gradation equation.

Accordingly, to describe CSG exactly and quantitatively, one new gradation equation with a parameter is proposed based on the fractal gradation equation. The ability to describe CSG of the new gradation equation is discussed, and its applicability is verified by several gradation curves of coarse- grained soils used in earth-rockfill dams. The relationship between gradation curve shape and gradation parameter is investigated. Additionally, the range of the gradation parameter is deduced when the gradation is well graded, and the commonly used range of the gradation parameter is summarized.

2 Gradation equation for CSG

2.1 Shortages of existing gradation equations

In order to show shortages of the existing gradation equations, existing gradation equations need to be introduced.

FULLER et al [26] proposed a parabolic gradation equation

                         (1)

where d_max is the maximum particle size.

TALBOT et al [27] put forward a gradation equation based on the fractal theory, which can be described as

                    (2)

where D is called fractal dimension.

ROSIN [28] raised a gradation equation to express soil gradation

             (3)

where γ and λ are fitting parameters.

One gradation equation was proposed for natural sediment by SWAMEE et al [29], which can be written as

                    (4)

where d_*, a and b are three fitting parameters.

CSGs used in earth-rockfill dams are all continuous and have a maximum particle size (d_max) [20, 30, 31]. Gradation equation should satisfy the condition that when d=d_max, the value of P should be 100%. Additionally, ZHU et al [32] summarized two shapes of CSG curves in the P-logd coordinate by investigating many CSGs. The two shapes of CSG curves are sigmoidal and hyperbolic, as seen in Figure 1.

However, Eqs. (1) and (2) can describe hyperbolic gradation curves, and the two equations lack the ability to express sigmoidal curves. Comparing with Eqs. (1) and (2), there is no maximum particle size in Eqs. (3) and (4), which makes the two equations inconvenient to use.

Figure 1 Two types of CSG curves

2.2 New gradation equation

As discussed in the previous section, it is essential to develop a new gradation equation.A reasonable gradation equation should have the ability to describe sigmoidal curves and hyperbolic curves. Additionally, the gradation equation should satisfy that when d=d_max, P is equal to 100%.

The fractal gradation equation Eq. [2] is the most commonly used gradation equation among Eqs. (1)-(4), and satisfies the condition that P is 100% when d=d_max. However, the fractal gradation equation can only reflect the hyperbolic curve, as can be seen in Figure 2. That is, the fractal gradation equation lacks the ability to describe the sigmoidal curve. If this deficiency can be solved, there will be no limitations when Eq. (2) is used to express the gradation. Thus, improvements are made to the ability of Eq. (2) to express the sigmoidal curve, and the improved gradation equation can reflect the two typical gradation curves.

Figure 2 Gradation curves expressed by Eq. (2)

In accordance with the previous analysis, the improved gradation equation is put forward as

              (5)

where β is called gradation parameter.

Therefore, CSG can be expressed by β and d_max. When Eq. (5) is used to describe the gradation curve, the gradation parameter β needs to be determined by the gradation curve. There is no doubt that it is effective to get the parameter value of a known function through the optimization method. If the gradation curve is known, d_max can be obtained from the curve. Then several software can be used to determine the value of β, such as MATLAB. Noting that as many data points (d, P) as possible should be selected from the gradation curve to carry out the optimization.

3 Applicability of equation to coarse grained soil

To show the applicability of Eq. (5), d_max is set to 60 mm, and β is set to different values. Then these gradation curves are plotted in Figure 3. As shown, with the variation of the parameter β, the shapes of the curves which the equation describes can be different, which shows that the new gradation equation proposed in this paper can describe two typical gradation curves.

In order to show the applicability and advantage of Eq. (5) over Eq. (2), taking the two typical gradation curves shown in Figure 1 as examples, they are fitted by Eqs. (2) and (5), respectively (Figure 4).

As shown in Figure 4, for hyperbolic gradation curve, the fitted curve of Eq. (2) is well matched with it, however, there is a big difference between the fitted curve of Eq. (2) and sigmoidal gradation curve. By contrast, whatever gradation curve, the fitted curve of Eq. (5) is in good agreement with the gradation curve. Thus, it can be seen that the gradation equation proposed in this paper has a more wide applicability than Eq. (2).

To further validate the applicability of Eq. (5) for CSG, several CSG data from domestic and foreign earth-rockfill dams are arranged [20, 33-37], and these CSG curves are fitted by Eq. (5), as seen in Figure 5.

Figure 3 Gradation curves expressed by Eq. (5):

Figure 4 Comparison of applicability between Eq. (5) and Eq. (2)

As presented in Figure 5, excellent agreement is found between fitted curves and gradation data. That is, Eq. (5) has the applicability to reflect CSGs used in earth-rockfill dam engineering, indicating that the proposed gradation equation is practicable to quantitatively describe the gradation distribution of coarse-grained soils.

4 Range of gradation parameter

There is no doubt that concrete gradation distribution of coarse-grained soil can be determined by the gradation parameter, and the engineering properties of coarse-grained soil depends largely on the gradation. Therefore, an investigation on the range of β is essential.

Above all, according to (the factor of Eq. [5]), the value of β can be determined and it is in the range of β≥0. Additionally, β cannot be 0 or 1. If β is 0 or 1, P will be 100% according to Eq. (5), so that Eq. (5) cannot describe the relationship between P and d. Notably, for any CSG, the value of P increases with increasing d. Hence, the derivative with respect to d must be greater than 0, which can be expressed as

              (6)

where the denominator is positive.

As seen from Eq. (6), when 1-β>0, this inequality will be satisfied. As a result, the value of β is in the range of β<1.

Given that gentle and steep curves are the two extreme cases of gradation curve, normal gradation curve ought to be located between these two curves. Gentle curve, steep curve, and common curve are shown in Figure 6. ZHU et al [32] has proposed the conditions to avoid the gentle and steep gradation curves. The conditions are that the value of lgd_max-lgd₁₀ is in the range of 0.125 to 7 (d₁₀ is the particle size when P is 10%). That is, the steep gradation curve can be avoided with lgd_max- lgd₁₀>0.125, and the gentle gradation curve can be avoided with lgd_max-lgd₁₀<7.

According to Eq. (5), lgd_max-lgd₁₀ can be written as

          (7)

Therefore, 0.125max-lgd₁₀<7 can be expressed as the inequalities of β.

               (8)

Figure 5 Applicability of Eq. (5) to CSGs used in engineering

Figure 6 Three gradation curves

By solving inequality (8), β is in the range of (0.021, 1).

In fact, there are no gentle and steep gradation curves used in earth-rockfill dam. In other words, the range of 0.021<β<1 is too wide for coarse-grained soil used in laboratory experiment and earth-rockfill dam engineering. Therefore, in order to further obtain the practical range of β for CSG, the common range of β will be given based on C_u and C_c.

According to Eq. (5), d₆₀, d₃₀ and d₁₀ can be expressed as

                  (9)

Based on the definition of C_u and C_c [8], C_u and C_c can be written as

   (10)

For CSG, the condition of gradation regarded as well graded is C_u>6 and C_c =1-3. According to Eq. (10), C_u>6 and C_c=1-3 can be expressed as

           (11)

Because the value of β is in the range of 0-1, and 3(1-0.1β)(1-0.6β)- 2(1-0.3β)²=1-0.9β>0, then it can be obtained that and Therefore, C_u>6 and C_c>1 are always valid.

As mentioned, inequality (11) will be valid when C_c<3. In order to gain the range of β with C_c<3, the curve expressed by the function of C_c is shown in Figure 7. As seen, when β>0.13, inequality (11) is valid and the gradation expressed by Eq. (5) is well graded. In other words, the practical range of β is (0.13, 1).

Accordingly, in order to investigate the commonly used range of β of CSG used in earth-rockfill dam engineering, gradation parameter values of CSGs (Figure 5) are shown in Figure 8. As indicated in Figure 8, the value of β for coarse-grained soil used in earth-rock dam is concentrated in a common area of 0.18-0.97. Notably, the determinant coefficients are all larger than 0.97, indicating that Eq. (5) has the ability to express CSG.

Figure 7 Curve expressed by function of C_c

Figure 8 Gradation parameters of CSGs used in earth-rockfill dams

As discussed previously, the commonly used value of β is in the range of 0.18 to 0.97 for CSG used in laboratory test and earth-rockfill dam. The confirmation of this range is of great significance for gradation design and investigation on the engineering properties of coarse-grained soil used in earth-rockfill dam.

5 Relationship between curve shape and parameter

There are two different CSG curves in P-lgd coordinate, e.g., hyperbolic and sigmoidal. Obviously, with an inflection point on the gradation curve, the curve is hyperbolic, if not, the gradation curve is sigmoidal.

There is no doubt that the second derivative equals zero at the inflection point of the gradation curve. Since the CSG curve is shown in P-lgd coordinate, Eq. (5) can be rewritten as

              (12)

As seen from Eq. (12), it is hard to obtain the derivative of P to lgd. Comparing with this, the first derivative of lgd to P can be easily obtained, which can be expressed as

               (13)

Then, according to Eq. (13), the second derivative of lgd to P can be developed as

           (14)

As shown in Eq. (14), when P=1/(2β), the second derivative equals zero. That is, the point where P=1/(2β) is the inflection point.

It is known that the value of β is in the range of 0 to 1. When 0.5<β<1, the value of 1/(2β) is in the range of 0 to 1. That is, the gradation curve can show the inflection point at P=1/(2β). Therefore, the gradation curve is sigmoidal, as indicated as curve A (P=1/(2β)=52.6%) in Figure 9. Otherwise, the value of β is in the range of 0 to 0.5, and the inflection point is at the point where P>1. That is, the gradation curve cannot show the inflection point. Hence, the gradation curve shape is hyperbolic, which has been shown as curve B (P=1/(2β)= 125%) in Figure 9. It is apparent from Figure 3 that when β>0.5, the gradation curve shape is sigmoidal, and the degree of sigmoidal bending is becoming less and less evident with the decrease of β.

Figure 9 Inflection point of gradation curve

Based on the discussion previously, the value of the parameter β determines the gradation curve type. If the value of β is in the range of 0.5 to 1, the gradation curve is sigmoidal, otherwise the gradation curve is hyperbolic.

6 Potential application of equation

This proposed gradation equation brings about the possibility to quantitatively describe the relationship between CSG and the physical and mechanical behaviors, instead of the qualitative description. For example, when studying the compaction property of coarse-grained soil, if the quantitative relationship between the maximum dry density and the gradation can be built, its corresponding optimal gradation can be easily obtained by obtaining the extremum of the quantitative relationship. Furthermore, with the gradation equation, the quantitative relationship between natural gradation parameter and gradation parameter after size reduction can be built to further study the scale effect. Notably, the new gradation equation can be used to design specific CSGs used for laboratory tests.

7 Conclusions

One new gradation equation is proposed that can describe continuous gradation of coarse-grained soil. The properties of the new equation are investigated and its applicability is verified through gradation data from earth-rockfill dams worldwide. The main conclusions are as follows.

1) The proposed gradation equation has the ability to describe sigmoidal and hyperbolic gradation curves of coarse-grained soils, and has the applicability to express CSGs used in earth-rockfill dam.

2) The value of β is in the range of 0 to 1. When β>0.13, the gradation expressed by Eq. (2) is well graded. The commonly used value of β is in the range of 0.18 to 0.97 for CSG used in earth-rockfill dam, and the determination of this range is of value to guide the design for CSG of earth-rockfill dam.

3) The gradation curve shape depends on the value of β. When 0.5<β<1, the gradation curve is sigmoidal, when 0<β<0.5, the gradation curve is hyperbolic.

4) The gradation equation presented in this paper has the ability to describe CSG, which is more practical than the fractal gradation equation. It provides a basis for quantitative study of effect of gradation on the engineering properties of coarse-grained soil.

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(Edited by FANG Jing-hua)

中文导读

粗粒料的级配方程及其适用性研究

摘要：级配方程可以准确和定量地表示粗粒料的级配，级配方程使得定量研究粗粒料级配与其物理力学性质之间的关系成为可能。提出了一个适用于描述粗粒料连续级配的级配方程，并对该级配方程的基本性质和适用性进行了研究。结果表明：提出的级配方程对粗粒料的级配具有良好的适用性。方程参数β的范围为(0, 1)，参数的大小决定了级配曲线的类型。当β大于0.5时，方程表示的级配曲线为反S形，否则为双曲线形。当β为0.13～1时，方程表示的级配为良好级配。对国内外土石坝用粗粒料的级配曲线进行了探讨，其参数值为0.18～0.97。该范围的确定对土石坝用粗粒料级配的设计具有指导价值。

关键词：级配曲线；级配方程；粗粒料；适用性

Foundation item: Project(2018YFC1508505) supported by the National Key Research and Development Program of China; Project(U1865104) supported by Yalong River Joint Fund of Natural Science Foundation of China-Yalong River Basin Hydropower Development Co., Ltd., China; Project(51479052) supported by National Natural Science of China; Project(2019T120443) supported by China Postdoctoral Science Foundation

Received date: 2019-09-05; Accepted date: 2019-12-09

Corresponding author: ZHU Jun-gao, PhD, Professor; Tel: +86-13851754908; E-mail: zhujungao@hhu.edu.cn; ORCID: 0000-0002- 4154-5168