Influence of Particle Size Distribution on the Critical State of Rockfill

Hindawi Advances in Civil Engineering Volume 2019, Article ID 8963971, 7 pages https://doi.org/10.1155/2019/8963971

Research Article Influence of Particle Size Distribution on the Critical State of Rockfill

Gui Yang ,1 Yang Jiang,1 Sanjay Nimbalkar,2 Yifei Sun,1 and Nenghui Li3

1Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China 2School of Civil and Environmental Engineering, Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW 2007, Australia 3Nanjing Hydraulic Research Institute, Nanjing 210024, China

Correspondence should be addressed to Gui Yang; [email protected]

Received 24 June 2019; Revised 20 August 2019; Accepted 17 September 2019; Published 20 October 2019

Academic Editor: Chun-Qing Li

Copyright © 2019 Gui Yang et al. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 the e − 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.

1. Introduction aggregates, the critical state stress ratio (Mcs) was found to be insensitive to any change in the coefficient of uniformity ,e construction of earthfill and rockfill dams has given new (Cu), but the critical state parameters (ecs0, λs) in the e − ξ impetus to investigation of the physical and mechanical (p′/pa) plane decreased with the increase in Cu [8, 9]. properties of rockfill material [1]. In most cases, triaxial However, the effect of dM on the critical state of granular soil testing on the prototype rockfill using conventional labo- is still largely unknown. ,e parallel gradation and com- ratory equipment is unattainable as the sizes of aggregates bination methods [10] are widely acknowledged in testing used in the field are usually too large. ,is in turn em- and designing of rockfill, inevitably instigating the use of phasizes the need to develop appropriate scaling laws. reduced dM. However, the strength of rockfill is greatly influenced by the ,e purpose of this study is to investigate the influence of size distribution of the crushed stone aggregates. Results of particle size distribution (PSD) adopting the combination triaxial tests on rockfill reported by Marachi et al. [2] method on the critical state of rockfill. ,erefore, a series of revealed that the friction angle increased with the decrease in monotonic drained tests using a large-scale triaxial appa- maximum particle size (dM). On the contrary, different ratus were performed on graded rockfill materials with trends were observed from the results of direct shear tests on varying dM. Based on laboratory findings, a general re- glass beads [3] and triaxial tests on ballast [4]. Another lationship between the critical state parameters and dM is important governing factor for the design of the hydraulic proposed. dam is the critical state behaviour of constituting rockfill aggregates [5, 6]. In these studies, the critical state line 2. Laboratory Testing proposed by Li and Wang [7] was used to evaluate the effect of particle size distribution on the critical state of granular 2.1. Test Material. ,e rockfill material used in this study aggregates. ,rough discrete element modelling of granular was collected from the Xiaolangdi Dam in Jiyuan, Henan 2 Advances in Civil Engineering

Province of China. Aggregates were derived from the parent 100 sandstone rock. Visual inspection revealed that aggregates larger than 5 mm in size had a rounded/subrounded shape. 80 ,e finer fraction (size less than 5 mm) contained sand- gravel particles. Based on the values of coefficient of uni- 60 formity and coefficient of curvature (Cu � 90, Cc � 1.7), rockfill was classified as well graded [11]. ,e prototype 40 grading (dM � 250 mm) and the corresponding four scaled down PSDs of Xiaolangdi rockfill via the method of com- Percentage passing (%) 20 bination are shown in Figure 1. ,e dM of each grading are 40 mm (T4), 60 mm (T3), 80 mm (T2), and 120 mm (T1), 0 respectively. In tests, the different grading specimens are 0.1 1 10 100 compacted with the same relative density 0.90. ,e sample Particle size (mm) diameter (Φ) and density (ρ) are also listed in Figure 1. Prototype ρ = 2230 kg/m3 T-1, Ф = 500 mm, ρ = 2190 kg/m3 2.2. Test Procedure. A large-scale triaxial apparatus T-2, Ф = 500 mm, ρ = 2150 kg/m3 designed and built in Nanjing Hydraulic Research Institute T-3, Ф = 300 mm, ρ = 2100 kg/m3 was used. ,e maximum cell pressure is 4.0 MPa, maxi- T-4, Ф = 300 mm, ρ = 2060 kg/m3 mum axial loading force is 5000 kN, and maximum axial Figure displacement is 250 mm. ,e apparatus can accommodate 1: Prototype and initial gradings of Xiaolangdi rockfill. 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. Sub- sequently, 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 Figure 2). ,e 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. ,e test specimen was then placed inside a test chamber. Before com- Figure 2: Vibration equipment. mencement 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 dM � 120 mm) exhibited mild strain-softening behaviour Skempton’s B-value exceeded 0.95. ,e samples were then and dilatancy. On the contrary, prominent strain-hardening isotropically consolidated at effective confining pressures behaviour and contraction [13, 14] were also observed in the ′ (σ3) of 0.2–2.2 MPa before monotonic loading. Triaxial specimen tested under higher σ3′ (>0.2 MPa). ,e volumetric tests were then conducted under the monotonic drained strain decreased with the increase in dM while the axial strain condition with a constant axial displacement rate of increased with decrease in dM at the same stress level. 2.7 mm/min until the axial strain was accumulated up Moreover, with the increase of dM, the peak shear strength of to 15%. rockfill (σf ) was improved (for example, σf � 3.10, 3.14, 3.27, and 3.44 MPa corresponding to σ3′ � 0.8 MPa). ,is finding is 3. Result Analysis contrary to that reported by Marachi et al. [2] where an improved strength was associated with the reduced dM of the 3.1. Stress-Strain Behaviour. ,e stress-strain behaviour of (angular) rockfill aggregates. ,is was probably due to the Xiaolangdi rockfill tested at varying maximum particle sizes different aggregate shapes used in this study (rounded/ (dM) of 40, 60, 80, and 120 mm are shown in Figures 3–6, subrounded and flaky) which had influenced the frictional respectively. Different values of initial confining interlock and breakage of aggregates. More discussion on pressure (σ3′) such as 0.2, 0.4, 0.8, 1.6, and 2.2 MPa were these aspects is given by Varadarajan et al. [15] and Dai chosen. Specimen tested at low pressure (σ3′ � 0.2 MPa, et al. [3]. Advances in Civil Engineering 3

5 10

(%) 4 8 v 6 3 4 2 2 1 Volume strain ε

Deviator stress q (MPa) 0 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 ε ε Axial strain 1 (%) Axial strain 1 (%) σ′ σ′ σ′ σ′ 3 = 0.2 MPa 3 = 1.6 MPa 3 = 0.2 MPa 3 = 1.6 MPa σ′ σ′ σ′ σ′ 3 = 0.4 MPa 3 = 2.2 MPa 3 = 0.4 MPa 3 = 2.2 MPa σ′ σ′ 3 = 0.8 MPa 3 = 0.8 MPa (a) (b)

Figure 3: Stress-strain behaviors of rockll with d 40 mm. M

5 10 (%) 8 v 4 6 3 4 2 2 1 Volume strain ε

Figure 4: Stress-strain behaviors of rockll with d 60 mm. M

5 12

10 (%) 4 v 8 3 6 2 4 2 1 Volume strain ε

Figure 5: Stress-strain-volume behaviors of rockll with d 80 mm. M

3.2. Critical State Strength. Critical state of rockll is the of each test; therefore, the critical state of each rockll core concept in elastoplastic constitutive modelling [16]. specimen could be obtained by measuring the nal state of As evident from the laboratory data, the deviator stress shearing (Figures 3–6). e critical state line is plotted in and volumetric strain became almost stabilized at the end the p q plane as shown in Figure 7. It could be − ′ 4 Advances in Civil Engineering

12 3

10 (%) v 8 2 6 4 1 2 Volume strain ε

Figure 6: Stress-strain-volume behaviors of rockll with d 120 mm. M

σ′ 8 σ′ 8 3 = 2.2 MPa T-4 3 = 2.2 MPa T-3

σ′ σ′ 6 3 = 1.6 MPa 6 3 = 1.6 MPa

Simulation equation 4 σ′ Simulation equation 4 σ′ = 0.8 MPa q = M p′ 3 = 0.8 MPa q M p′ 3 cs = cs M M cs = 1.61 σ′ cs = 1.59 σ′ R2 = 0.99 3 = 0.4 MPa R2 3 = 0.4 MPa Deviator stress q (MPa)

Deviator stress q (MPa) = 0.99 2 2 σ′ = 0.2 MPa σ′ 3 3 = 0.2 MPa

0 0 012345 012345 p′ Effective mean stress p′ (MPa) Effective mean stress (MPa)

Critical state point Critical state point (a) (b)

σ′ = 2.2 MPa 8 σ′ 8 3 T-2 3 = 2.2 MPa T-1

σ′ σ′ = 1.6 MPa 6 3 = 1.6 MPa 6 3

4 σ′ 4 σ′ Simulation equation 3 = 0.8 MPa Simulation equation 3 = 0.8 MPa q M p′ q M p′ = cs = cs M M σ′ cs = 1.65 σ′ = 1.62 = 0.4 MPa 2 Deviator stress q (MPa) = 0.4 MPa cs 3 R

3 Deviator stress q (MPa) = 0.98 2 R2 = 0.99 2 σ′ 3 = 0.2 MPa

0 0 012345 012345 Effective mean stress p′ (MPa) Effective mean stress p′ (MPa)

Critical state point Critical state point (c) (d)

Figure 7: (a) Critical state line in the p q plane for rockll with d 40 mm. (b) Critical state line in the p q plane for rockll with d 60 mm. − M − M (c) Critical state line in the p q plane for rockll with dM 80 mm. (d) Critical state line in the p q plane for rockll with dM 120 mm. − ′ − ′ ′ ′ b concluded that the critical state stress ratio (Mcs q/p ) is 1 dM constant for a given PSD. However, unlike past published Mcs a1 , dn ( ) studies which reported independency of Mcs with′ Cu [8, 9], Mcs was found to increase as dM increased in this where dn is equal to 60 mm (nominal particle diameter1 study (Figure 8). is dependency of Mcs with dM could be chosen arbitrarily for the purpose of normalization) and a1 described by adopting the power law relationship given and b1 are material constants, equal to 1.61 and 0.0332, below: respectively. Advances in Civil Engineering 5

Honkanadavar, [14] 1.8 RD = 87%

cs Honkanadavar, [14] RD = 75% 1.7 0.0332 Mcs = 1.61(dM/dn) R2 = 0.97

1.6

Ф500 mm

Critical state stress ratio M 1.5 Ф300 mm

1.4 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1

Normalised maximum particle size dM/dn Figure 8: Variation of critical state stress ratio with particle size.

0.31 0.29 T-4 T-3 0.30 0.28 Simulation equation 0.29 e e λ p p 0.7 0.27 σ′=0.2 MPa Simulation equation σ′ = 0.4 MPa cs = cs0 – s ( / a) 3 e e λ p p 0.7 3 e λ –2 cs = cs0 – s ( / a) 0.28 = 0.293; = 0.37 × 10 –2 σ′ = 0.2 MPa cs0 s 0.26 σ′ e = 0.275; λ = 0.33 × 10 3 3 = 0.4 MPa cs0 s 0.27 σ′ = 0.8 MPa 3 0.25 σ′ = 0.8 MPa 0.26 3 Void ratio e σ′ Void ratio e 0.24 σ′ 0.25 3 = 1.6 MPa 3 = 1.6 MPa 0.23 σ′ 0.24 3 = 2.2 MPa σ′ 0.23 3 = 2.2 MPa 0.22 0.22 0.21 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 p p 0.7 p p 0.7 Normalised mean effective stress ( / a) Normalised mean effective stress ( / a) Critical state point Critical state point (a) (b) 0.25 0.26 T-2 T-1 0.24 σ′ = 0.4 MPa 0.25 σ′ 3 Simulation equation Simulation equation 3 = 0.2 MPa e e λ p p 0.7 e e λ p p 0.7 cs = cs0 – s( / a) cs = cs0 – s( / a) e λ –2 0.23 σ′ –2 = 0.261; = 0.27 × 10 3 = 0.4 MPa e = 0.241; λ = 0.22 × 10 0.24 σ′ = 0.8 MPa cs0 s cs0 s 3 σ′ 3 = 0.8 MPa σ′ 0.22

= 1.6 MPa Void ratio e Void ratio e 0.23 3 σ′ 3 = 1.6 MPa σ′ 3 = 2.2 MPa 0.21 σ′ 0.22 3 = 2.2 MPa

0.21 0.20 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 p p 0.7 p p 0.7 Normalised mean effective stress ( / a) Normalised mean effective stress ( / a) Critical state point Critical state point (c) (d)

0.7 0.7 Figure 9: (a) Critical state line in the e p /pa plane for rockll with dM 40 mm. (b) Critical state line in the e p /pa plane for −( ) 0.7 −( ) rockll with dM 60 mm. (c) Critical state line in the e p /pa plane for rockll with dM 80 mm. (d) Critical state line in the e 0.7 −( ) − p /pa plane for rockll with dM 120 mm.′ ′ ( ) ′ ′ is is similar to the test results of Honkanadavar and e critical state line of granular aggregates in Sharma [14]. M is calculated according to internal friction the e ln p plane may shift or rotate with increasing loading cs − angle φ (related to the critical state value M 6 stress due to particle degradation [17, 18]. erefore, the cs sin φ/ 3 sin φ ). nonlinear critical′ state line proposed by [7] is used: ( − ) 6 Advances in Civil Engineering

0.4 0.4 Simulation equation s cs0 –0.47 Simulation equation λs = 0.313 (dM/dn) –0.18 ecs0 = 0.274 (dM/d ) 0.3 0.3 n

0.2 0.2

Ф300 mm Ф300 mm 0.1 Critical state void ratio constant λ Critical state void ratio constant e Ф500 mm Ф500 mm 0.1 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 Normalised maximum particle size / Normalised maximum particle size d /d dM dn M n (a) (b)

Figure 10: (a) Variation of ecs0 with particle size. (b) Variation of λs with particle size.

p ξ performance in “in situ” conditions. In this study, the in- e e , cs cs0 λs ( ) uence of particle size distribution on the critical state − pa ′ strength of rockll was investigated by adopting the com- 2 bination method. A series of drained large-scale triaxial tests where ecs denotes the critical state void ratio; ecs0 and λs are dimensionless material constants; and p 101 kPa is the were performed on rockll materials procured from the a atmospheric pressure used for the purpose of normalization. Xiaolangdi dam. Stress-strain analysis was performed on the e parameter ξ 0.7 is used in this study which shows good test specimen with four dierent maximum particle sizes (i.e., d 40, 60, 80, and 120 mm) and ve dierent initial resemblance with laboratory data. Figure 9 shows the critical M conning pressures (i.e., σ 0.2, 0.4, 0.8, 1.6, 2.2 MPa). e state lines along with the critical state stress points of 3 Xiaolangdi rockll with diering d . A linear correlation major ndings of this study are summarized below: M ′ between e and p /p ξ could be observed. With the in- cs ( a) (a) e volumetric strain decreased with the increase crease of dM, ecs was found to decrease. e inuence of dM in dM while the axial strain increased with a de- ′ on the critical state line (equation (2)) is indicated by the crease in dM, at the same stress level. With the variation of the gradients and intercepts of samples with increase of dM, the peak shear strength of rockll dierent gradings. As evident from Figure 10, both ecs0 and was observed to improve. λs showed decrease with the increase in dM, which can be (b) e critical state lines in both p q and e p − − described by the following power law relationships: planes were inuenced by the particle sizes. e b2 d critical state stress ratio (Mcs) increased′ with an′ e a M , cs0 2 increase in dM while the gradient (λs) and intercept dn (ecs) decreased as dM increased. b ( ) 3 (c) Based on the laboratory data presented in this study, a dM λs a3 , power law relationship was proposed for describing dn 3 the evolution of the critical state parameters with dM. is simple yet accurate empirical relationship could where a2, a3, b2, and b3 are material constants, equal to 0.274, 0.18, 0.313, and 0.47, respectively. is simple empirical be introduced into a critical-state constitutive model − − relationship could be introduced into an elastoplastic model to simulate the stress-strain behaviour of rockll [19, 20] developed within the framework of critical-state soil under a range of particle sizes. is empirical re- mechanics to simulate the stress-strain behaviour of rockll lationship is based on the limited test data and it needs more appropriately. further validation for a wider range of particle sizes.

4. Conclusions Data Availability

Critical state strength of granular aggregates, including e data used to support the ndings of this study are rockll, is an important aspect for constitutive modelling. available from the corresponding author upon request. However, due to the limitation of conventional laboratory equipment, the critical state behaviour of eld rockll is Conflicts of Interest usually obtained by scaling down aggregate sizes. is in turn can lead to inaccuracies in prediction of material e authors declare that they have no conicts of interest. Advances in Civil Engineering 7

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