STRESS-STRAIN RELATIONSHIPS AND NONLINEAR MOHR STRENGTH CRITERIA OF FROZEN SANDY CLAY

Transcription

1 SOILS AND FOUNDATIONS Vol. 50, No. 1, 45 53, Feb. 010 Japanese Geotechnical Society STRESS-STRAIN RELATIONSHIPS AND NONLINEAR MOHR STRENGTH CRITERIA OF FROZEN SANDY CLAY LAI YUANMING i),gao ZHIHUA i),zhang SHUJUAN i) and CHANG XIAOXIAO i) ABSTRACT A series of triaxial compressive tests were performed on frozen sandy clay at -4 and-69c under conˆning pressures from 0 to 18 MPa. The experimental results indicate that the stress-strain curves show strain softening and hardening phenomena when the conˆning pressures are below and above 3.0 MPa, respectively. Since the generally hyperbolic model can not describe the strain hardening behavior very well and the Duncan-Chang model can not ideally describe the strain softening behavior of the frozen sandy clay, an improved Duncan-Chang model is proposed. This model can describe not only the strain softening behavior but also the strain hardening behavior of the frozen sandy clay, and the calculated results are rather coincident with the corresponding experimental data. In addition, it is also suitable for frozen silty clay with a high precision. Due to pressure melting, the shear strength of the frozen sandy clay changes nonlinearly with increasing conˆning pressures. In order to solve the problem that the linear Mohr-Coulomb criteria can not exactly re ect the shear strength of the frozen sandy clay, a nonlinear Mohr criteria of the frozen sandy clay is presented. The calculated results illustrate that it has higher precision and can describe the shear strength of frozen sandy soils more accurately than the linear Mohr-Coulomb criteria does. Key words: frozen clay, improved Duncan-Chang model, nonlinear Mohr strength criteria, stress-strain relationship (IGC: D6) INTRODUCTION There are million km of permafrost, which accounts for about 4z of the world's land area (French, 1996). Thus, the strength and deformation of frozen soil often must be considered in the designing of railways, roadways, oil pipelines, airports and buildings. Bragg and Andersland (1981) investigated the strain rate, temperature, and sample size ešects with tests on the compression and tensile properties of frozen sand, and determined the tensile strength using the splitting cylinder experiments. Tsytovich et al. (1981) provided the relationship between the elastic modulus and temperature according to experiments on frozen coarse-grained soil. Aas (1981) performed tensile, bending and shearing tests on frozen clay in Oslo, and found that there was a critical shearing stress for clay at a given temperature. Chen et al. (1998) studied the ešects of conˆning pressure on the shear strength of clay by the artiˆcially frozen method. Wu et al. (1994) researched the in uence of loading rates on the strength of frozen sandy soil. Shen (1995a) studied the in uence of temperature and loading rate on the results of the axial splitting test, the possibility of using axial splitting method to indirectly determine the tensile strength of frozen soil, and then measured the tensile strength of frozen loess (Shen et al., 1995b). Ma et al. (00) discussed the ešect of the unloading stress path on the strength of frozen soils. Arenson and Springman (005a) conducted triaxial tests on ice-rich frozen soils from the Alps in Switzerland, and found that the minimum strain rate of creep increases exponentially with increasing temperature and applied deviator stress. They also provided mathematical descriptions for the behavior of ice-rich frozen soils at temperatures close to 09C (Arenson and Springman, 005b). According to uniaxial compressive tests, Zhu et al. (199) proposed a uniaxial constitutive equation for frozen soils. Lai et al. (008) investigated strength distributions of warm frozen clay and proposed a stochastic damage constitutive model according to the uniaxial test results of warm frozen clay. Wang et al. (004) researched the stress-strain characteristic of frozen loess subjected to K 0 consolidation under unloading conditions, and showed the relationships between the initial conˆning pressure and temperature, and the initial tangent modulus and ultimate deviator stress. In our investigations, it was found that, when stresses are adjusted by increasing the cross-sectional area during the tests (Bardet, 1997), the stress-strain behavior of frozen sandy clay has a strain softening phenomenon for low conˆning pressures and a strain hardening phenomenon for higher conˆning pressures, respectively. The shear strength increases as the conˆning pressure increases when the con- i) State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, China (ymlai@lzb.ac.cn). The manuscript for this paper was received for review on September 9, 008; approved on October, 009. Written discussions on this paper should be submitted before September 1, 010 to the Japanese Geotechnical Society, 4-38-, Sengoku, Bunkyo-ku, Tokyo , Japan. Upon request the closing date may be extended one month. 45

2 46 YUANMING ET AL. ˆning pressure is below a certain critical value. However when the conˆning pressure is beyond that critical value, the shear strength decreases with further increase in con- ˆning pressure. Based on rock triaxial data, Baker (004) presented a procedure for estimating parameters from Mohr form of Hoek and Brown empirical failure criteria. Jiang et al. (003) investigated the ešect of strength envelope nonlinearity on slope stability computations. The studies mentioned above represent a great advance in the study on strength envelope nonlinearity. However, due to the particular properties of frozen soil, the present nonlinear Mohr formulae, proposed by Baker and Jiang et al. for rocks, are not suitable. Therefore, in order to better describe the mechanical properties for frozen sandy clay, an improved Duncan-Chang model and nonlinear Mohr strength criteria of frozen sandy clay are presented in this paper. When the strength and deformation of frozen soil engineering is analyzed, the stress-strain relationship and strength criterion of frozen soil need to be ascertained. For the engineering requirement of Qinghai-Tibet railway, we spent over two years to obtain the data and present the model in this paper. Moreover, with the wide use of the artiˆcial ground freezing technology in constructions, engineering activities in areas with deep alluvium, such as in mining, underground power stations, railways, tunnels, etc., are gradually able to be carried out to completion. In such areas, the artiˆcial freezing method is usually used to increase the soil strength. When the deep frozen soil is excavated and high lateral pressure is released, high conˆning pressures are usually encountered. For example, the stresses at the frozen wall of coal, mining 800 m under ground in Shandong province in China, can reach 17.6 MPa. Sometimes, even in excavations without very large depth, stress concentration may also lead to very high stress level. The purpose of this paper is to present the strength criteria and stress-strain relationship of frozen sandy clay for a large range of stress states, with particular attention to the high stress level. Fig. 1. Grain size distribution curve of sandy clay Fig.. Detailed ˆgure of the pressure chamber TEST CONDITIONS AND METHOD The material used in tests was remoulded Qinghai- Tibetan Sandy clay. The sandy clay is mainly quartz and its main ingredients were SiO,Al O 3,Fe O 3,MgO,K O and TiO in sequence according to their mass contents. The grain size distribution curve of this soil is shown in Fig. 1. The density of the soil particles was.65 g/cm 3. The process of preparing specimens is described as follows. Firstly, distilled water was added to air-dried soil to make an initial water content of about 13z by weight. Then the soil was put in a cylindrical rigid mold and compacted to the desired dry density, and they were quickly frozen from top to bottom in a freezing cabinet. After freezing, the specimens were taken out from the molds andmachinedto15.0mminlengthand61.8mmindiameterinacoldroom(wuetal.,1994;laietal.,008). At last, the average water content and dry density of the tested specimens were 1.9z (with a saturated content of Fig. 3. MTS-810 triaxial testing machine about.0z) and kg/m 3, respectively. The tests were conducted as follows. The specimen at the required temperature was ˆrst placed into the triaxial pressure cell for a duration of about 3 4 h, which allowed the specimen to reach temperature equilibrium (the tem-

3 NONLINEAR MOHR'S CRITERION 47 perature was measured with three thermistors laid at dišerent positions close to the specimen, shown in Fig. ). The conˆning pressure was then increased over a period of about 30 s to the desired value, and maintained for a period of about h for the temperature and pressure of the whole apparatus to equilibrate. The compression test was then conducted at a constant conˆning pressure. The axial deformation rate was 1.5 mm/min -1, equivalent to an axial strain rate of /s -1.All tests were conducted using the modiˆed MTS-810 triaxial testing machine shown in Fig. 3. The test temperatures were -49C and-69c, respectively. The both temperatures were chosen because the soil tested was taken from the railway line in Qinghai-Tibetan Plateau, at which the typical ground temperature is about -4 to-69c atthe depth of soil sampling. The test pattern was the axis-symmetry, the specimen was free to expand at the specimenplaten interface. The conˆning pressures were MPa. The tests were controlled by MTS test application software. TEST RESULTS AND ANALYSES Aseriesoftriaxialtestsat-4.0 and -6.09C on frozen sandy clay under conˆning pressures s 3 from 0 to 18.0 MPa were carried out, the corresponding test results are showninfig.4.fromthisˆgure,itcanbeseenthatthe Fig. 4. Stress-strain curves of frozen sandy soil at -4 and-69c under various conˆning pressures stress-strain curves show strain softening and hardening phenomena when the conˆning pressures are below and above 3.0 MPa, respectively (Lai et al., 008, Zhang et al., 007). Because the plastic-volume strain becomes larger with loading when the conˆning stress is small, the overall volumetric strain gradually increases. In addition, the strain softening phenomena of this soil is obvious due to the low yielding strength of sandy clay. With the increase of the conˆning stress the compressive strain is predominant, and because the yielding or peak strength becomes larger, the stress-strain curves appear to be the strain hardening phenomena. Generally, the hyperbolic model (Shen, 005) is used to describe the strain softening phenomenon on soft soil, s 1 -s 3 = ed(a+ced) (a+be d) (1) Where (s 1 -s 3) is the deviator stress, e d =e 1 -e 3, e 1 and e 3 are axial and radial strains, respectively. a, b and c are ˆtting parameters. When the conˆning pressures s 3 are larger than 3.0 MPa, the stress-strain curves of -6.09C frozen sandy clay shows a strain hardening phenomena, and is generally described by the Duncan-Chang model (Duncan et al., 1970), e1 s 1 -s 3 = () d+ee 1 Where d and e are ˆtting constants. Using ˆtting data, it is found that the generally hyperbolic model (Eq. (1)) could not describe the stress-strain softening behavior of frozen sandy clay when the conˆning pressures are below 3.0 MPa, and the Duncan-Chang model (Eq. ()) could not ideally re ect the situation when the conˆning pressures are above 3.0 MPa. Hence, an improved Duncan-Chang model is presented here, and used to describe the stress-strain relationships of frozen sandy clay at -6.09C when the conˆning pressures s 3 are in the range of 0 to 18.0 MPa. That is, s 1 -s 3 = (3) m+ne 1 +le 1 Where m=1/e 0, n=1/(s 1 -s 3) m -/(e 1mE 0), l=1/ (E 0e 1m), E 0 is the initial elastic modulus and determined by lim e1ª0 (d(s 1 -s 3)/de 1). The physical meanings of limiting stress, (s 1 -s 3) m, and limiting strain, e 1m, are illustrated in Fig. 5. m, n and l can be determined according to the formulae mentioned above and Fig. 5. They can also be determined by the data regression method when a series data of stress and strain are obtained. The parameter m, n and l canalsobedeterminedbytheleast square method. i.e., V=S K i=1 «m+ne1i +le 1i- e1i $ Where e 1i is the value of e 1 at the i-th measured point and =(s 1 -s 3) i, i.e., the value of (s 1 -s 3)atthei-th measured point. K is the total number of measured data. e 1

4 48 YUANMING ET AL. Fig. 5. Physical meanings of (s 1-s 3) m and e 1m Letting &m =0, =0, and &n &l =0, we can obtain the following equations: &m =mk+n Se 1i +l-s e1i =0 &n =m Se 1i +n+ls e 3 1i-S e 1i =0 &l =m +nse 3 1i+lSe 4 1i-S e3 1i =0 From the three equations above, the parameter m, n and l can also be determined by the following formulae. S e 1i Se 1i Se 1i m= 1 S e 1i Se 3 D S e3 1i Se 3 1i Se 4 Fig. 6. Comparisons between experimental data and modeling results of (a) general hyperbolic model and (b) improved Duncan-Chang model when s 3=0.3 MPa at -69C K n= 1 D K l= 1 Se 1i D Where D= Se 1i S e1i S e 1i S e3 1i Se 1i Se 3 1i K Se 1i Se 3 Se 4 S e1i S e 1i i S e3 1i q Se 1i Se 3 1i 3 Se1i. 4 By investigation, it is found that with the increasingly conˆning pressure, variations of m and n are not obvious, whereas variation of l are apparent. As such, according to the test results, m and n are taken as their average values of and 0.09 at -4.09C, and 0.01 and 0.05 at -6.09C, respectively; and l=1.586 (s 3/R t +

5 NONLINEAR MOHR'S CRITERION 49 Table 1. Maximal residual errors between the results calculated and the corresponding experimental data at dišerent strains at -69C when s 3=0.3 MPa Axial strain (z) À0.0 General hyperbolic model (MPa) Improved Duncan-Chang model (MPa) Table. Maximal residual errors between the results calculated and the corresponding experimental data at dišerent strains at -69C when s 3=10.0 MPa Axial strain (z) À0.0 Duncan-Chang model (MPa) Improved Duncan-Chang model (MPa) Fig. 7. Comparisons between experimental data and modeling results of (a) Duncan-Chang model and (b) improved Duncan-Chang model when s 3=10 MPa at -69C 0.36) , R t is the tensile strength of frozen sandy clay, 0.53 MPa at -4.09C and0.6mpaat-6.09c, respectively. To investigate the precision of the models mentioned above in describing the stress-strain relationship, the comparisons of the results calculated by Eqs. (1), () and (3) with the corresponding experimental data of sandy clay at -6.09C are shown in Figs. 6 and 7 when s 3 =0.3 MPa and s 3 =10.0 MPa, respectively. Maximal residual errors between the results calculated and the corresponding experimental data are also listed in Table 1 and at dišerent strains, respectively. From Fig. 6, it can be seen that the agreement of the calculated results using the improved Duncan-Chang model (Eq. (3)) with the corresponding experimental data is better than that of the calculated results using the generally hyperbolic model (Eq. (1)) with the corresponding experimental data. Table 1 also shows that the simulated precision of Eq. (3) is better than that of Eq. (1) when the strain softening behavior of frozen sandy clay is described by it. In particular, when the axial strain is in the range of 0 5.0z, the calculated results of Eq. (3) are much closer to the experimental values than those of Eq. (1). According to Fig. 7 and Table, it can be seen that the agreement of the calculated results of the improved Duncan-Chang model (Eq. (3)) with the corresponding experimental data is better than that of the calculated results of the Duncan-Chang model (Eq. ()) with the corresponding experimental data. Table also shows that maximal residual error between the calculated results of Eq. (3) and the corresponding experimental data is about half of the residual error between the calculated results of Eq. () and the corresponding experimental data in the axial strain of 0 0.0z. However, when the axial strain is above 0.0z, the absolute value of maximal residual error between the calculated results by the improved Duncan-Chang model and the corresponding experimental data is about one third of the results obtained from the Duncan-Chang model and the corresponding experimental data. Our results clearly show that the modeling precision of the improved Duncan-Chang model is better than that of the Duncan-Chang model when it is used to describe the strain hardening behavior of frozen sandy clay. From Figs. 6 and 7, it is known that the generally hyperbolic model can only describe the strain softening behavior of frozen sandy clay, and the Duncan-Chang model can only describe the strain hardening behavior. On the other hand, the improved Duncan-Chang model can describe not only the strain softening behavior of frozen sandy clay but also the strain hardening behaviour. In order to illustrate the modeling precision of the improved Duncan-Chang model in conˆning pressures of MPa, some results from the improved model and the corresponding experimental data are shown in Figs. 8(a) (f). From Figs. 8(a) (f), it can be seen that the calculated results approximate well with corresponding experimental data with increasingly conˆning pressure, so there is clearly very good agreement. NONLINEAR MOHR STRENGTH CRITERIA From Fig. 4, it can be seen that the maximal values of (s 1 -s 3) appear in the stress-strain curves of eº15z when the conˆning pressure s 3 =0 MPa. For such cases, the peak value of (s 1 -s 3) was taken as the failure

6 50 YUANMING ET AL. Fig. 8. Comparisons between the experimental data of sandy soil and the calculated results of the improved Duncan-Chang model with dišerent conˆning pressures at -69C Fig. 9. Comparison between Mohr criteria and the corresponding experimental results at -69C strength of the frozen sandy clay tested in this paper. With conˆning pressures s 3Æ3MPa,nopeakvalueexists in the stress-strain curves of the frozen sandy clay within the range of e 1º15.0z, so,thevaluesof(s 1 -s 3)ata strain of 15.0z is taken as the failure strength of frozen sandyclayinthispaper. According to the experimental results, the stress circles with various conˆning pressures at -69C areplottedin Fig. 9(a), in which the dotted straight line is the linear Mohr-Coulomb criteria and the solid envelope represents the experimental results. Figure 9(a) also shows that the shear strength increases with increasing pressure until about 1.0 MPa, for further increases in conˆning pressure, the shear strength decreases due to pressure melting. If the shear strength is determined by the Mohr-Coulomb criteria, t=c+s tan q, a linear equation, t M = s, shouldbeobtained.itcanbeseeninfig.9(a) that for the shear failure of frozen sandy clay at -6.09C there are large errors between the Mohr-Coulomb criteria and the corresponding experimental results, and thus, a new nonlinear failure criteria is proposed in this paper. According to the test results and data regression, the relationship between conˆning pressure, s 3, and axial stress, s 1, can be expressed by the following equation, s 1 =(k 0) s3/po0a s c Ø 1+s3 s T» b0 Where s c and s T are the uni-axial compressive and tensile strengths, respectively, and K 0 and b 0 are experimental parameters. For the frozen sandy clay at -6.09C, s c =.35 MPa, s T =0.60 MPa, K 0 =0.98, b 0 =0.791, and P a = MPa, is normal atmospheric pressure. For the frozen sandy clay at -4.09C, s c =1.639 MPa, s T = MPa, K 0 =0.985, and b 0 =0.80. The equation of the Mohr circle, expressed using s 3 (in our test, s =s 3)ands 1,is +s Ø 3 s-s1» +t =Ø s1 -s 3» (5) Where s and t are the normal and shear stresses on the element failure plane, s 1 and s 3 are maximum and minimum principal stresses, respectively. From Eq. (5), we can obtain the following formulation of function f (4)

7 NONLINEAR MOHR'S CRITERION 51 +s f=ø 3 s-s1» +t -Ø s1 -s 3» =0 (6) Utilizing a chain rule for dišerentiating the implicit function and Eq. (6), the following expressions can be obtained, &f =- =- -s+s1 &f -s+s 3 From the equation above, the following formula can be obtained s=s 3 + s1 -s 3 (7) +1 Substituting Eq. (7) into Eq. (5), the following equation is obtained t = (s1 -s 3) From the formulation above, the expression of shear stress t is given by Fig. 10. t= s1 -s 3 +1 According to Fig. 10, tan s deˆned by s 1 +s 3 tan q= dt -s ds = t (8) Substituting Eqs. (7) and (8) into Eq. (9), the following equation is obtained Angle n plane of Mohr stress circle (9) -1 tan q= (10) According to Eq. (4), the following expression can be obtained by dišerentiating for s 3, ln k0 = (k 0) s3/pa s c P a Ø +(k 0) s3/pa s c Ø 1+s3 1+s3 b s T» b s T» b s 3 +s T (11) Eqs. (7), (8), (10) and (11) are mathematic formulae of nonlinear Mohr criteria of the frozen sandy clay. When s 3ª-s T, ª/, s=s 3, t=0, f= p. When s 1»0, Eq. (11) can be simpliˆed as following expression according to Eq. (4) ln k0 =Ø + b P a s 3 +s T» s1 (1) Substituting Eq. (1) into Eqs. (7) and (8), respectively, the mathematic formulae of the nonlinear Mohr criteria can be obtained as follows, (s 1 -s 3)P a(s 3 +s T) s=s 3 + s 1(s 3 +s T)lnk 0 +P a(bs 1 +s 3 +s T) t= (s1 -s 3) P as 1(s 3 +s T)[(s 3 +s T)lnk 0 +P ab] s 1 (s 3 +s T )lnk 0 +P a (bs 1 +s 3 +s T ) (13a) (13b) In order to examine the precision of the nonlinear Mohr criteria expressed by Eqs. (11) (13), the shear failure stress circles of the frozen sandy clay under dišerent con- ˆning pressures are shown in Fig. 9(b), in which a dotted curve is described by the nonlinear Mohr criteria, and a solid envelope curve is described by the corresponding experimental results. The envelope curve of all stress circles is ˆtted by the following method. First, the ˆtting curve equation is assumed as the following expression: t E =f(s`a 1, b 1, c 1, d 1)=a 1 +b 1s+c 1s +d 1s 3 (14) Where a 1, b 1, c 1 and d 1 are parameters without dimension to be determined. According to Eq. (5) and Fig. 10, we can obtain the following formulae: s i =p i - sin, t i = cos (15) p i = s1 +s 3, = s1 -s 3 (16) Table 3. The variation of vs s 1 and s 3 at -69C s 1 (MPa) s 3 (MPa) (rad)

8 5 YUANMING ET AL. Where p i and are the center and radius of i-th Mohr-circle, respectively. is the angle between the vertical line and the line through the circle center and a tangent point at which the envelope is tangent with the i-th Mohr-circle. s i and t i depend on p i, and. and the parameter a 1, b 1, c 1 and d 1 can be determined by the least square method (Bardet, 1997), i.e., Letting &P & =0, P= K S i=1 [t i -(a 1 +b 1s i +c 1s i +d 1s 3 i )] = K S i=1 s cos ()-[a 1 +b 1( p i - sin ) +c 1( p i - sin ) +d 1( p i - sin ) 3 ]t (17) &P &a 1 =0, &P &b 1 =0, &P =0, and &P =0, &c 1 &d 1 Table 4. Relative errors between the linear, the nonlinear Mohr- Coulomb criteria and the envelope equation at -69C s (MPa) t E (MPa) t M (MPa) t NM (MPa) d M (z) d NM (z) parameter a 1, b 1, c 1, d 1 and can be determined. The values of a 1, b 1, c 1 and d 1 are: a 1 =0.6667, b 1 =0.690, c 1 = and d 1 =0.0009, and the values of are listed in Table 3 when the sandy clay is at -69C. According to the parameter values obtained, the equation of the envelope line is given by, d M t E =f(s`a 1, b 1, c 1, d 1) = s-0.043s s 3 (18) stands for relative error between the linear Mohr- Coulomb criteria, t M, and the data, t E, from Eq. (18). The relative errors between the nonlinear Mohr criteria, t NM, and the data, t E, from Eq. (18), are denoted by d NM. Their values are listed in Table 4. FromTable4,itcanbeseenthatwhens= (MPa), d M =18.3z, d NM =-10.3z; when s=7.057 (MPa), d M =-9.7z, d NM =-4.9z; and when s= (MPa), d M =5.z, d NM =-.1z. The errors between the nonlinear Mohr criteria and the envelope equation are smaller than those between the linear Mohr- Coulomb criteria and the envelope equation. Therefore, the nonlinear Mohr criteria is more precise, and can describes the shear strength of the frozen sandy clay more accurately than the linear Mohr-Coulomb criteria does. In addition, the improved Duncan-Chang model proposed in this paper is also suitable for silty clay with a Fig. 11. Comparisons between the experimental data of silty clay and the calculated results of the improved Duncan-Chang model with dišerent conˆning pressures at -49C and-69c

9 NONLINEAR MOHR'S CRITERION 53 high precision under the condition of strain rate of /s -1 at -4.09C and-6.09c (Fig. 11). Based on the present experimental data studied in this paper, it can be found that the scope of material, temperature, and con- ˆning pressure most suitable for the application of the proposed model is sandy clay and silty clay, -4-69C, and 0 18 MPa, respectively. When the model is applied to the numerical analysis, the following tests must be done to determine the parameters: uniaxial compressive and tensile strength tests, and a series of triaxial compressive tests under dišerent conˆning pressures, respectively to ascertain the exact nature of the material, the frozen temperature and the strain rate. After the test results are obtained, a model on frozen soil can be obtained by following the procedure provided in this paper. Equation (3) is used to evaluate the stress-strain behavior before the failure of foundation soil in actual engineering issues, and Eq. (13) is mainly used to evaluate the stress state when the foundation soil is found to be in critical condition. CONCLUSIONS An improved Duncan-Chang model is proposed. This model can describe not only the strain softening behavior of frozen sandy soils but also the strain hardening behavior of frozen sandy soils. This model solves the problem that the generally hyperbolic model can not describe in the strain hardening behavior, and that the Duncan- Chang model can not describe in the strain softening behavior of the frozen sandy soils. Moreover, the modeling precision of the improved Duncan-Chang model is better than that of Duncan-Chang model and the generally hyperbolic model. A set of nonlinear Mohr criterion of frozen sandy soils is presented. It has higher precision and describes the shear strength of frozen sandy soils more accurately than the linear Mohr-Coulomb criterion does. ACKNOWLEDGEMENTS We would like to thank very much the two anonymous reviewers whose constructive comments are helpful for this paper revision. This research was supported by National Natural Science Foundation of China ( ), the National Hi-Tech Research and Development Plan (008AA11Z103), the Western Project Program of the Chinese Academy of Sciences (KZCX-XB-10), the Program for Innovative Research Group of Natural Science Foundation of China (No ), and the foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZY-03). REFERENCES 1) Aas, G. (1981): Laboratory determination of strength properties of frozen salt marine clay, Engineering Geology, 18, ) Arenson, L. U. and Springman, S. M. (005): Triaxial constant stress and constant strain rate tests on ice-rich permafrost samples, Canadian Geotechnical Journal, 4, ) Arenson, L. U. and Springman, S. M. 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