# Influence of Particle Size Distribution on the Critical State of Rockfill

Authors: Gui Yang; Yang Jiang; Sanjay Nimbalkar; Yifei Sun; Nenghui Li
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

In order to study the effect of particle size distribution on the critical state of rockfill, a series of large-scale triaxial tests on rockfill with different maximum particle sizes were performed. It was observed that the intercept and gradient of the critical state line in thee−p′ plane decreased as the grading broadened with the increase in particle size while the gradient of the critical state line in the p′−q plane increased as the particle size increased. A power law function is found to appropriately describe the relationship between the critical state parameters and maximum particle size of rockfill.


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## Body

## 1. Introduction

The construction of earthfill and rockfill dams has given new impetus to investigation of the physical and mechanical properties of rockfill material [1]. In most cases, triaxial testing on the prototype rockfill using conventional laboratory equipment is unattainable as the sizes of aggregates used in the field are usually too large. This in turn emphasizes the need to develop appropriate scaling laws. However, the strength of rockfill is greatly influenced by the size distribution of the crushed stone aggregates. Results of triaxial tests on rockfill reported by Marachi et al. [2] revealed that the friction angle increased with the decrease in maximum particle size (dM). On the contrary, different trends were observed from the results of direct shear tests on glass beads [3] and triaxial tests on ballast [4]. Another important governing factor for the design of the hydraulic dam is the critical state behaviour of constituting rockfill aggregates [5, 6]. In these studies, the critical state line proposed by Li and Wang [7] was used to evaluate the effect of particle size distribution on the critical state of granular aggregates. Through discrete element modelling of granular aggregates, the critical state stress ratio (Mcs) was found to be insensitive to any change in the coefficient of uniformity (Cu), but the critical state parameters (ecs0, λs) in the e−p′/paξ plane decreased with the increase in Cu [8, 9]. However, the effect of dM on the critical state of granular soil is still largely unknown. The parallel gradation and combination methods [10] are widely acknowledged in testing and designing of rockfill, inevitably instigating the use of reduced dM.The purpose of this study is to investigate the influence of particle size distribution (PSD) adopting the combination method on the critical state of rockfill. Therefore, a series of monotonic drained tests using a large-scale triaxial apparatus were performed on graded rockfill materials with varyingdM. Based on laboratory findings, a general relationship between the critical state parameters and dM is proposed.


## 2. Laboratory Testing

### 2.1. Test Material

The rockfill material used in this study was collected from the Xiaolangdi Dam in Jiyuan, Henan Province of China. Aggregates were derived from the parent sandstone rock. Visual inspection revealed that aggregates larger than 5 mm in size had a rounded/subrounded shape. The finer fraction (size less than 5 mm) contained sand-gravel particles. Based on the values of coefficient of uniformity and coefficient of curvature (Cu = 90, Cc = 1.7), rockfill was classified as well graded [11]. The prototype grading (dM = 250 mm) and the corresponding four scaled down PSDs of Xiaolangdi rockfill via the method of combination are shown in Figure 1. The dM of each grading are 40 mm (T4), 60 mm (T3), 80 mm (T2), and 120 mm (T1), respectively. In tests, the different grading specimens are compacted with the same relative density 0.90. The sample diameter (Φ) and density (ρ) are also listed in Figure 1.Figure 1
Prototype and initial gradings of Xiaolangdi rockfill.


### 2.2. Test Procedure

A large-scale triaxial apparatus designed and built in Nanjing Hydraulic Research Institute was used. The maximum cell pressure is 4.0 MPa, maximum axial loading force is 5000 kN, and maximum axial displacement is 250 mm. The apparatus can accommodate samples of different diameters (Φ), such as Φ = 500, 300, and 200 mm. Particles selected from each size range were carefully washed and dried under natural sunlight. Subsequently, aggregates were weighed separately and mixed together before splitting into either sixteen equal portions (to prepare the test specimen with 500 mm in diameter and 1100 mm in height) or ten equal portions (to form the test specimen with a diameter of 300 mm and a height of 700 mm). Each portion was then compacted inside a split cylindrical mould using a vibrator (as shown in Figure2). The motor power is 1.2 kW (Φ = 300 mm) and 1.8 kW (Φ = 500 mm), respectively. Each portion was compacted for 60 s (Φ = 300 mm) and 90 s (Φ = 500 mm), respectively. According to the study’s results [12], the influence of different sample diameter sizes can be ignored. During vibratory compaction, a static pressure of 14 kPa and a frequency of 20 Hz were applied to achieve desired initial density simulating in-field conditions. The test specimen was then placed inside a test chamber. Before commencement of the monotonic shear test, the sample was saturated by allowing water to pass through the base of the triaxial cell under a back pressure of 10 kPa until Skempton’s B-value exceeded 0.95. The samples were then isotropically consolidated at effective confining pressures (σ3′) of 0.2–2.2 MPa before monotonic loading. Triaxial tests were then conducted under the monotonic drained condition with a constant axial displacement rate of 2.7 mm/min until the axial strain was accumulated up to 15%.Figure 2
Vibration equipment.


## 2.1. Test Material

The rockfill material used in this study was collected from the Xiaolangdi Dam in Jiyuan, Henan Province of China. Aggregates were derived from the parent sandstone rock. Visual inspection revealed that aggregates larger than 5 mm in size had a rounded/subrounded shape. The finer fraction (size less than 5 mm) contained sand-gravel particles. Based on the values of coefficient of uniformity and coefficient of curvature (Cu = 90, Cc = 1.7), rockfill was classified as well graded [11]. The prototype grading (dM = 250 mm) and the corresponding four scaled down PSDs of Xiaolangdi rockfill via the method of combination are shown in Figure 1. The dM of each grading are 40 mm (T4), 60 mm (T3), 80 mm (T2), and 120 mm (T1), respectively. In tests, the different grading specimens are compacted with the same relative density 0.90. The sample diameter (Φ) and density (ρ) are also listed in Figure 1.Figure 1
Prototype and initial gradings of Xiaolangdi rockfill.


## 2.2. Test Procedure

A large-scale triaxial apparatus designed and built in Nanjing Hydraulic Research Institute was used. The maximum cell pressure is 4.0 MPa, maximum axial loading force is 5000 kN, and maximum axial displacement is 250 mm. The apparatus can accommodate samples of different diameters (Φ), such as Φ = 500, 300, and 200 mm. Particles selected from each size range were carefully washed and dried under natural sunlight. Subsequently, aggregates were weighed separately and mixed together before splitting into either sixteen equal portions (to prepare the test specimen with 500 mm in diameter and 1100 mm in height) or ten equal portions (to form the test specimen with a diameter of 300 mm and a height of 700 mm). Each portion was then compacted inside a split cylindrical mould using a vibrator (as shown in Figure2). The motor power is 1.2 kW (Φ = 300 mm) and 1.8 kW (Φ = 500 mm), respectively. Each portion was compacted for 60 s (Φ = 300 mm) and 90 s (Φ = 500 mm), respectively. According to the study’s results [12], the influence of different sample diameter sizes can be ignored. During vibratory compaction, a static pressure of 14 kPa and a frequency of 20 Hz were applied to achieve desired initial density simulating in-field conditions. The test specimen was then placed inside a test chamber. Before commencement of the monotonic shear test, the sample was saturated by allowing water to pass through the base of the triaxial cell under a back pressure of 10 kPa until Skempton’s B-value exceeded 0.95. The samples were then isotropically consolidated at effective confining pressures (σ3′) of 0.2–2.2 MPa before monotonic loading. Triaxial tests were then conducted under the monotonic drained condition with a constant axial displacement rate of 2.7 mm/min until the axial strain was accumulated up to 15%.Figure 2
Vibration equipment.


## 3. Result Analysis

### 3.1. Stress-Strain Behaviour

The stress-strain behaviour of Xiaolangdi rockfill tested at varying maximum particle sizes (dM) of 40, 60, 80, and 120 mm are shown in Figures 3–6, respectively. Different values of initial confining pressure (σ3′) such as 0.2, 0.4, 0.8, 1.6, and 2.2 MPa were chosen. Specimen tested at low pressure (σ3′ = 0.2 MPa, dM = 120 mm) exhibited mild strain-softening behaviour and dilatancy. On the contrary, prominent strain-hardening behaviour and contraction [13, 14] were also observed in the specimen tested under higher σ3′ (>0.2 MPa). The volumetric strain decreased with the increase in dM while the axial strain increased with decrease in dM at the same stress level. Moreover, with the increase of dM, the peak shear strength of rockfill (σf) was improved (for example, σf = 3.10, 3.14, 3.27, and 3.44 MPa corresponding to σ3′ = 0.8 MPa). This finding is contrary to that reported by Marachi et al. [2] where an improved strength was associated with the reduced dM of the (angular) rockfill aggregates. This was probably due to the different aggregate shapes used in this study (rounded/subrounded and flaky) which had influenced the frictional interlock and breakage of aggregates. More discussion on these aspects is given by Varadarajan et al. [15] and Dai et al. [3].Figure 3
Stress-strain behaviors of rockfill withdM = 40 mm.
(a)
(b)Figure 4
Stress-strain behaviors of rockfill withdM = 60 mm.
(a)
(b)Figure 5
Stress-strain-volume behaviors of rockfill withdM = 80 mm.
(a)
(b)Figure 6
Stress-strain-volume behaviors of rockfill withdM = 120 mm.
(a)
(b)


### 3.2. Critical State Strength

Critical state of rockfill is the core concept in elastoplastic constitutive modelling [16]. As evident from the laboratory data, the deviator stress and volumetric strain became almost stabilized at the end of each test; therefore, the critical state of each rockfill specimen could be obtained by measuring the final state of shearing (Figures 3–6). The critical state line is plotted in the p′−q plane as shown in Figure 7. It could be concluded that the critical state stress ratio (Mcs=q/p′) is constant for a given PSD. However, unlike past published studies which reported independency of Mcs with Cu [8, 9], Mcs was found to increase as dM increased in this study (Figure 8). This dependency of Mcs with dM could be described by adopting the power law relationship given below:(1)Mcs=a1dMdnb1,where dn is equal to 60 mm (nominal particle diameter chosen arbitrarily for the purpose of normalization) and a1 and b1 are material constants, equal to 1.61 and 0.0332, respectively.Figure 7
(a) Critical state line in thep′−q plane for rockfill with dM = 40 mm. (b) Critical state line in the p′−q plane for rockfill with dM = 60 mm. (c) Critical state line in the p′−q plane for rockfill with dM = 80 mm. (d) Critical state line in the p′−q plane for rockfill with dM = 120 mm.
(a)
(b)
(c)
(d)Figure 8
Variation of critical state stress ratio with particle size.This is similar to the test results of Honkanadavar and Sharma [14]. Mcs is calculated according to internal friction angle φ (related to the critical state value Mcs=6sinφ/3−sinφ).The critical state line of granular aggregates in thee−lnp′ plane may shift or rotate with increasing loading stress due to particle degradation [17, 18]. Therefore, the nonlinear critical state line proposed by [7] is used:(2)ecs=ecs0−λsp′paξ,where ecs denotes the critical state void ratio; ecs0 and λs are dimensionless material constants; and pa = 101 kPa is the atmospheric pressure used for the purpose of normalization. The parameter ξ = 0.7 is used in this study which shows good resemblance with laboratory data. Figure 9 shows the critical state lines along with the critical state stress points of Xiaolangdi rockfill with differing dM. A linear correlation between ecs and p′/paξ could be observed. With the increase of dM, ecs was found to decrease. The influence of dM on the critical state line (equation (2)) is indicated by the variation of the gradients and intercepts of samples with different gradings. As evident from Figure 10, both ecs0 and λs showed decrease with the increase in dM, which can be described by the following power law relationships:(3)ecs0=a2dMdnb2,λs=a3dMdnb3,where a2, a3, b2, and b3 are material constants, equal to 0.274, −0.18, 0.313, and −0.47, respectively. This simple empirical relationship could be introduced into an elastoplastic model [19, 20] developed within the framework of critical-state soil mechanics to simulate the stress-strain behaviour of rockfill more appropriately.Figure 9
(a) Critical state line in thee−p′/pa0.7 plane for rockfill with dM = 40 mm. (b) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 60 mm. (c) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 80 mm. (d) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 120 mm.
(a)
(b)
(c)
(d)Figure 10
(a) Variation ofecs0 with particle size. (b) Variation of λs with particle size.
(a)
(b)


## 3.1. Stress-Strain Behaviour

The stress-strain behaviour of Xiaolangdi rockfill tested at varying maximum particle sizes (dM) of 40, 60, 80, and 120 mm are shown in Figures 3–6, respectively. Different values of initial confining pressure (σ3′) such as 0.2, 0.4, 0.8, 1.6, and 2.2 MPa were chosen. Specimen tested at low pressure (σ3′ = 0.2 MPa, dM = 120 mm) exhibited mild strain-softening behaviour and dilatancy. On the contrary, prominent strain-hardening behaviour and contraction [13, 14] were also observed in the specimen tested under higher σ3′ (>0.2 MPa). The volumetric strain decreased with the increase in dM while the axial strain increased with decrease in dM at the same stress level. Moreover, with the increase of dM, the peak shear strength of rockfill (σf) was improved (for example, σf = 3.10, 3.14, 3.27, and 3.44 MPa corresponding to σ3′ = 0.8 MPa). This finding is contrary to that reported by Marachi et al. [2] where an improved strength was associated with the reduced dM of the (angular) rockfill aggregates. This was probably due to the different aggregate shapes used in this study (rounded/subrounded and flaky) which had influenced the frictional interlock and breakage of aggregates. More discussion on these aspects is given by Varadarajan et al. [15] and Dai et al. [3].Figure 3
Stress-strain behaviors of rockfill withdM = 40 mm.
(a)
(b)Figure 4
Stress-strain behaviors of rockfill withdM = 60 mm.
(a)
(b)Figure 5
Stress-strain-volume behaviors of rockfill withdM = 80 mm.
(a)
(b)Figure 6
Stress-strain-volume behaviors of rockfill withdM = 120 mm.
(a)
(b)


## 3.2. Critical State Strength

Critical state of rockfill is the core concept in elastoplastic constitutive modelling [16]. As evident from the laboratory data, the deviator stress and volumetric strain became almost stabilized at the end of each test; therefore, the critical state of each rockfill specimen could be obtained by measuring the final state of shearing (Figures 3–6). The critical state line is plotted in the p′−q plane as shown in Figure 7. It could be concluded that the critical state stress ratio (Mcs=q/p′) is constant for a given PSD. However, unlike past published studies which reported independency of Mcs with Cu [8, 9], Mcs was found to increase as dM increased in this study (Figure 8). This dependency of Mcs with dM could be described by adopting the power law relationship given below:(1)Mcs=a1dMdnb1,where dn is equal to 60 mm (nominal particle diameter chosen arbitrarily for the purpose of normalization) and a1 and b1 are material constants, equal to 1.61 and 0.0332, respectively.Figure 7
(a) Critical state line in thep′−q plane for rockfill with dM = 40 mm. (b) Critical state line in the p′−q plane for rockfill with dM = 60 mm. (c) Critical state line in the p′−q plane for rockfill with dM = 80 mm. (d) Critical state line in the p′−q plane for rockfill with dM = 120 mm.
(a)
(b)
(c)
(d)Figure 8
Variation of critical state stress ratio with particle size.This is similar to the test results of Honkanadavar and Sharma [14]. Mcs is calculated according to internal friction angle φ (related to the critical state value Mcs=6sinφ/3−sinφ).The critical state line of granular aggregates in thee−lnp′ plane may shift or rotate with increasing loading stress due to particle degradation [17, 18]. Therefore, the nonlinear critical state line proposed by [7] is used:(2)ecs=ecs0−λsp′paξ,where ecs denotes the critical state void ratio; ecs0 and λs are dimensionless material constants; and pa = 101 kPa is the atmospheric pressure used for the purpose of normalization. The parameter ξ = 0.7 is used in this study which shows good resemblance with laboratory data. Figure 9 shows the critical state lines along with the critical state stress points of Xiaolangdi rockfill with differing dM. A linear correlation between ecs and p′/paξ could be observed. With the increase of dM, ecs was found to decrease. The influence of dM on the critical state line (equation (2)) is indicated by the variation of the gradients and intercepts of samples with different gradings. As evident from Figure 10, both ecs0 and λs showed decrease with the increase in dM, which can be described by the following power law relationships:(3)ecs0=a2dMdnb2,λs=a3dMdnb3,where a2, a3, b2, and b3 are material constants, equal to 0.274, −0.18, 0.313, and −0.47, respectively. This simple empirical relationship could be introduced into an elastoplastic model [19, 20] developed within the framework of critical-state soil mechanics to simulate the stress-strain behaviour of rockfill more appropriately.Figure 9
(a) Critical state line in thee−p′/pa0.7 plane for rockfill with dM = 40 mm. (b) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 60 mm. (c) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 80 mm. (d) Critical state line in the e−p′/pa0.7 plane for rockfill with dM = 120 mm.
(a)
(b)
(c)
(d)Figure 10
(a) Variation ofecs0 with particle size. (b) Variation of λs with particle size.
(a)
(b)


## 4. Conclusions

Critical state strength of granular aggregates, including rockfill, is an important aspect for constitutive modelling. However, due to the limitation of conventional laboratory equipment, the critical state behaviour of field rockfill is usually obtained by scaling down aggregate sizes. This in turn can lead to inaccuracies in prediction of material performance in “in situ” conditions. In this study, the influence of particle size distribution on the critical state strength of rockfill was investigated by adopting the combination method. A series of drained large-scale triaxial tests were performed on rockfill materials procured from the Xiaolangdi dam. Stress-strain analysis was performed on the test specimen with four different maximum particle sizes (i.e.,dM = 40, 60, 80, and 120 mm) and five different initial confining pressures (i.e., σ3′ = 0.2, 0.4, 0.8, 1.6, 2.2 MPa). The major findings of this study are summarized below:(a)
The volumetric strain decreased with the increase indM while the axial strain increased with a decrease in dM, at the same stress level. With the increase of dM, the peak shear strength of rockfill was observed to improve.(b)
The critical state lines in bothp′−q and e−p′ planes were influenced by the particle sizes. The critical state stress ratio (Mcs) increased with an increase in dM while the gradient (λs) and intercept (ecs) decreased as dM increased.(c)
Based on the laboratory data presented in this study, a power law relationship was proposed for describing the evolution of the critical state parameters withdM. This simple yet accurate empirical relationship could be introduced into a critical-state constitutive model to simulate the stress-strain behaviour of rockfill under a range of particle sizes. This empirical relationship is based on the limited test data and it needs further validation for a wider range of particle sizes.



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*Source: 8963971-2019-10-20.xml*