1.Introduction

Currently,many super-high rockfill dams are under construction or being planned in southwest China,an area with high earthquake intensity.The stress level of a dam increases with the dam height,and the volumetric contraction behavior during earthquakes is more significant for high rockfill dams.Existing earthquake disaster data show that seismic subsidence,cracks,landslides,and other types of earthquakerelated damage to rockfill dams are closely related to the earthquake-induced permanent deformation(Wang et al.,2000).It is therefore necessary to develop an effective model to evaluate the deformation of rockfill materials in high dams(Xu et al.,2012;Zou et al.,2013;Xiao and Liu,2017).

Through the use of empirical permanent deformation models of rockfill materials(Shen and Xu,1996;Zou et al.,2008),a simplified method of calculating the earthquakeinduced deformation of a dam was developed based on an equivalent static nodal force theory(Serff et al.,1976).In the theory,the equivalent static nodal force is determined by means of the cyclic shear strain of the dam and the cyclic shear strain-permanent strain relationship.The cyclic shear strain of the dam during an earthquake can be obtained by dynamic analysis,and the cyclic shear strain-permanent strain relationship can be obtained by cyclic triaxial tests(Chi et al.,1998).At present,there are mainly two kinds of empirical permanent deformation models that have significant differences.In the first kind,only the permanent shear strain is considered(Kong and Han,1994;Jia and Kong,2004),while both the permanent volumetric strain and permanent shear strain are considered in the second kind(Shen and Xu,1996;Zou et al.,2008;Ling et al.,2010;Zhu and Zhou,2010;Wang et al.,2013).Therefore,the second kind of models is more reasonable.

Experimental results have shown that the use of a semi-log linear relationship between the cyclic number and the permanent volumetric strain leads to underestimation of the volumetric deformation of rockfill materials with a large cyclic number,and the error increases with the confining pressure(Shen and Xu,1996;Zou et al.,2008).The earthquakeinduced deformation of a high rockfill dam during a strong earthquake may be underestimated by the existing permanent deformation models.

In this study,a series of cyclic triaxial tests were performed on two kinds of rockfill materials to study the permanent deformation.A simple hyperbolic permanent deformation model was developed,and the effect of the mean effective stress on the permanent volumetric strain as well as the influence of the consolidation stress ratio on the permanent shear strain were incorporated.The proposed model was validated with the experimental results of Zou et al.(2008).

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2.Test apparatus and programs

由于初始缺陷或环境的作用,作为结构主要构件的梁——钢筋混凝土梁或木梁等在服役期会不可避免地产生性能退化,并出现各类裂纹.裂纹尖端的应力集中及疲劳往往导致裂纹扩展,使梁刚度和承载能力降低,最终发生破坏,对生命和财产造成严重威胁,因此裂纹梁的加固与修复成为相关研究领域的关注课题.

Fig.1.Particle size distributions.

Each specimen was divided into six equal portions.The dry weight of each portion was prepared separately to ensure a uniform particle distribution.Each portion was compacted to 100 mm in height with three layers and placed inside a 3-mmthick rubber cylinder that was supported with a split cylindrical mold.The specimen was saturated by running deoxygenated water from the bottom with an upper drainage system until the pore pressure coefficient was greater than 0.95.

All test specimens were first consolidated under the confining pressure σ3 of 400,1200,and 2000 kPa,respectively.A constant cyclic axial stress σd with amplitudes of both 0.3σ3 and 0.9σ3 was applied,and the consolidation stress ratio Κc reached the values of 1.5 and 2.0,respectively,in the test.The total cyclic number was 30 with a frequency of 0.1 Hz.The permanent deformation of the rockfill materials was investigated under different test conditions.

3.Permanent deformation

At present,it is generally assumed that the permanent deformation is related to the cyclic number N at different earthquake intensities(Yin,2007).In some commonly used models of permanent deformation(Shen and Xu,1996;Zou et al.,2008;Ling et al.,2010;Wang et al.,2013;Zhu and Zhou,2010),it is assumed that there is a linear relationship between the permanent volumetric strain εvr and N+1 in the semi-log coordinates.Comparisons between experimental results and the commonly used semi-log linear curve are shown in Fig.2(a)(Zou et al.,2008),where R2 is the coefficient of determination.

When the semi-log linear relationship is used,εvr is overestimated with a small cyclic number,but underestimated with a large cyclic number.The error increases with the confining pressure and cyclic stress amplitude,and the volumetric contraction deformation of the dam during a strong earthquake is underestimated(Zou et al.,2008).It is clear that εvr does not have a linear relationship with N+1 in the semi-log coordinates.

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The measured value of permanent shear strain γr is larger than the predicted value obtained with the semi-log linear relationship with a small cyclic number,while the measured value is smaller than the predicted value with a large cyclic number.Similar conclusions can be drawn from experimental results in Zou et al.(2009),Ling et al.(2010),and Zhu and Zhou(2010).

In this study,the relationships of the permanent volumetric and shear strains vs.the cyclic number were represented by hyperbolic curves,as shown in Fig.2(b).It can be concluded that the hyperbolic law can represent the relationships well,and the correlation coefficients of the fitted hyperbolic curves are larger than those of the semi-log linear relationships.

In Eq.(5),d3 and d4 are the intercept at γd=0.01%and the slope of the linear relationship between lg［A sr/（K c-1）1.5］and lgγd,respectively.The model parameters are shown in Table 1.

Fig.2.Permanent volumetric and shear strains vs.cyclic number for σ3=600 kPa,Κc=2.0,and σd=1.0σ3.

where A vr is the ultimate permanent volumetric strain,A sr is the ultimate permanent shear strain,εvr1 is the permanent volumetric strain at the first cycle,and γr1 is the permanent shear strain at the first cycle.

A vr and B vr can be determined according to the linear relationship between 1/εvr and 1/N,and A sr and B sr can be determined according to the linear relationship between 1/γr and 1/N.Fig.3 and Fig.4 show comparisons between experimental results and the fitted hyperbolic curves,and the model can properly reflect the variations of the permanent volumetric and shear strains with the cyclic number under different conditions.

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3.1.Ultimate permanent strains Avr and Asr

Fig.3.Relationships between εvr and N under different test conditions.

Fig.4.Relationships between γr and N under different test conditions.

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In this study,the mean effective stress was introduced to reflect the variation of permanent volumetric deformation.The ultimate permanent volumetric strain A vr is expressed as a function of the mean effective stress and cyclic shear strain as follows:

In Eq.(3),d1 and d2 are the intercept at γd=0.01%and the slope of the linear relationship between lg［A vr/（p/p a）0.5］and lgγd,respectively.It should be mentioned that there is an advantages to choosing the intercept at γd=0.01%:the plastic behavior of rockfill materials is negligible when γd is smaller than 0.01%(Kong et al.,2001),and the increases of the intercept d1 and slope d2 are consistent with the increase of the permanent volumetric strain,which helps in determining the characteristics of materials intuitively.When d1 is considered the intercept at γd=1%(Shen and Xu,1996;Zou et al.,2008;Ling et al.,2010;Wang et al.,2013),the increase of the permanent volumetric strain does not agree with the increases of d1 and d2 simultaneously,because γd is in the range of 0.02%-0.2%during a strong earthquake(Qian and Yin,1996).

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As shown in Fig.3,the permanent volumetric strain increases with the confining pressure and cyclic axial stress.However,the influence of confining pressure on the permanent volumetric strain was not considered in previous studies(Shen and Xu,1996;Zou et al.,2008;Ling et al.,2010;Wang et al.,2013),resulting in the overestimation of the permanent volumetric contraction deformation near the dam crest.Moreover,there is no significant permanent volumetric deformation occurring at the dam crest,but dilatancy may occur there(Liu and Chi,2013).

As shown in Fig.4,the permanent shear strain increases with the consolidation stress ratio Κc and cyclic axial stress σd(or cyclic shear strain γd).The stress level is introduced to reflect the permanent shear strain in many existing models(Shen and Xu,1996;Zou et al.,2008;Ling et al.,2010;Wang et al.,2013).However,it is not convenient for application because of the necessity of the shear strength parameter in the calculation of the stress level.The consolidation stress ratio Κc is used herein instead of the stress level.A sr is expressed as a function of the consolidation stress ratio and cyclic shear strain as follows:

where d3 and d4 are the experimental parameters,and n is the power exponent.A sr is equal to 0 at Κc=1,and the permanent shear strain is neglected in the isotropic stress state(Shen and Xu,1996).

The correlation analysis of the power exponent n was carried out according to the experimental results for limestone and basalt.When n is close to 1.5,the correlation coefficient reaches its peak value,as shown in Fig.5.Therefore,a value of 1.5 was used for n in this study.Eq.(4)can be simplified as follows:

As illustrated in Fig.6,the linear relationship between lg［A vr/（p/p a）0.5］and lgγd shows agreement with the test results of the two kinds of rockfill materials,indicating that Eq.(3)can capture the experimental responses well.Likewise,a linear relationship between lg［A sr/（K c-1）1.5］and lgγd can be obtained,as shown in Fig.7,and the experimental results can also be reflected well by Eq.(5).

Fig.5.Relationship between correlation coefficient and n.

where d1 and d2 are the experimental parameters,γd is the cyclic shear strain,p is the mean effective stress,and p a is the atmospheric pressure.As shown in Eq.(3),although the dilatancy cannot be reflected,the parameter A vr would be very small at low mean effective stress,indicating an insignificant volumetric contraction deformation near the dam crest.

The permanent volumetric and shear strains varying with the cyclic number can be expressed as follows:

The large-scale static and dynamic triaxial apparatus developed by the Earthquake Engineering Research Institute at the Dalian University of Technology was used in this study.The specimens were 300 mm in diameter(D)and 600 mm in height(H).The test materials included two different kinds of rockfill materials:limestone and basalt.The particle size distributions(Fig.1)and densities of the two kinds of rockfill materials were the same.The particle size distributions of the materials were determined from the original rockfill material using a mixed method according to the Specification of Soil Test(SL237-1999).The density of the test specimens was 2.21 g/cm3.

3.2.Parameters Bvr and Bsr

Fig.8 shows the distributions of the experimental parameters B vr and B sr of the two kinds of rockfill materials varying with the cycle shear strain under different consolidation stress ratios.It is clear that the ranges of B vr and B sr are narrow under different experimental conditions.The mean values were calculated,as shown in Table 1.

Fig.6.Relationships between A vr/（p/p a）0.5 and γd for different kinds of rockfill materials.

Fig.7.Relationships between A sr/（K c-1）1.5and γd for different kinds of rockfill materials.

Table 1 Permanent deformation model parameters.

Test material d1(10-4) d2 d3(10-4) d4 Mean B vr Mean B sr Limestone 2.40 0.77 10.14 0.81 6.54 5.01 Basalt 3.45 0.83 13.25 0.94 6.10 4.87

Taking limestone as an example,the influence of B vr on the permanent volumetric strain was analyzed.The lower limit,mean value,and upper bound of B vr,with the values of 4.90,6.54,and 7.00,respectively,were substituted into Eq.(1)to calculate the permanent volumetric strain at N=20 and N=30.The deviations of the permanent volumetric strains obtained with the upper bound and lower limit of B vr from that obtained with its mean value were-1.88%and 6.04%,respectively,at N=20,and the deviations were-1.37%and 4.38%,respectively,at N=30.Similar analysis was conducted to analyze the influence of B sr on the permanent shear strain.It can be seen that the variations of B vr and B sr have slight influences on the permanent deformation of limestone.The same conclusions can be obtained for basalt.Therefore,B vr and B sr can be assumed to be constants for rockfill materials.

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4.Model validation

The experimental results regarding rockfill materials from Zou et al.(2008)were also used to verify the proposed model in this study.Fig.9 and Fig.10 show comparisons between experimental results and the fitted hyperbolic curves,and demonstrate that the model can properly reflect the relationships between the permanent strains and the cyclic number under different conditions.

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Fig.11(a)shows the relationship between A vr/（p/p a）0.5 and γd.It can be concluded that the permanent volumetric strain characteristics of rockfill materials can be represented well by Eq.(3).Fig.11(b)shows the relationship between A sr/（K c-1）1.5 and γd.The permanent shear strain characteristics of rockfill materials can be represented well by Eq.(5),and the equation is reasonable when the value of n is equal to 1.5.The ranges of B vr and B sr were 13-18 and 4-7,respectively.They showed insignificant differences with the changes of the cyclic shear strain,confining pressure,and consolidation stress ratio,and the mean values of B vr and B sr were 15 and 5,respectively.Model parameters from Zou et al.(2008)are different from those in this study,due to the differences in densities and particle size distributions.

Fig.8.Variations of B vr and B sr.

Fig.9.Relationships between εvr and N for experimental results of Zou et al.(2008).

Fig.10.Relationships between γr and N for experimental results of Zou et al.(2008).

Fig.11.Fitted results of test data.

5.Conclusions

The following conclusions can be drawn:

(1)Hyperbolic relationships can represent the permanent volumetric and shear strains varying with the cyclic number well and avoid underestimating the volumetric contraction deformation of a dam during strong earthquakes.

(2)The effect of mean effective stress was incorporated in the expression of permanent volumetric strain,which can better reflect the permanent deformation characteristics of rockfill materials under low confining pressure and avoid overestimating the volumetric contraction deformation near the dam crest.

(3)The consolidation stress ratio was introduced into the expression of permanent shear strain so that the shear strength parameters are no longer required for calculating the stress level in most existing models.

In the future,more studies need to be conducted to analyze the range of the model parameters for different rockfill materials,including reinforced soil(Zou et al.,2009)and gravels improved by polyurethane foam adhesive(Xiao et al.,2018).

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