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An experimental research on damping ratio and dynamic shear modulus ratio of frozen silty clay of the Qinghai-Tibet engineering corridor
Transportation Geotechnics 21 (2019) 100269
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An experimental research on damping ratio and dynamic shear modulus ratio of frozen silty clay of the Qinghai-Tibet engineering corridor
T
⁎
Zhijian Wua,b, , Dan Zhangc, Tao Zhaod, Jinlian Mab, Duoyin Zhaob a
College of Transportation Science and Engineering, Nanjing Tech University, Nanjing 210009, China Key Laboratory of Loess Earthquake Engineering, Lanzhou Institute of Seismology, China Earthquake Administration, Lanzhou 730000, China c Gansu Construction Investment (Holdings) Group Corporation Steel Structure Co. Ltd, Lanzhou, Gansu 730000, China d Shaanxi Railway Institute, Weinan 714000, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Frozen silty clay Qinghai-Tibet engineering corridor Damping ratio Dynamic shear modulus Confining pressure Temperature Water content Frequency
Strong earthquakes occurred frequently in Qinghai-Tibet Plateau. Major national infrastructure facilities, including existing and proposed projects, are located densely in the permafrost engineering corridor of QinghaiTibet Plateau. Based on dynamic triaxial test under the effect of cycle loading for remolded frozen soil taken from Qinghai-Tibet engineering corridor, experimental studies on the influence of confining pressure, temperature, moisture content and frequency on damping ratio and dynamic shear modulus ratio have been carried out systematically. The results are summarized as follows: The damping ratio of frozen Qinghai-Tibet silty clay increases with increasing in confining pressure, temperature and water content, and decreasing in frequency. In addition, the dynamic shear modulus ratio reduces with increasing in confining pressure, temperature and water content, and decreasing in frequency. The relationship equation between the maximum dynamic shear modulus of frozen silty clay and temperature, water content and confining pressure is obtained by using multivariate nonlinear fitting method, which shows a better fitting effect. The results of this research can provide basic data for the design of major engineering and seismic fortification along Qinghai-Tibet engineering corridor.
Introduction The Qinghai-Tibet Engineering Corridor passing through Qinghai Province, is a strategic channel that connects Tibet with the mainland. It is about 550 km in length through a large area of continuous permafrost and 82 km in length through an island-like discontinuous permafrost. Its maximum width is only a few kilometers, while its minimum width is only hundreds of meters. Recently, many linear and major infrastructure projects such as the Qinghai-Tibet Highway, the Qinghai-Tibet Railway, the Gelat Oil Product Pipeline, the Lansila Cable Communication Project, and the high-voltage transmission line have been built within this limited corridor [9]. The Qinghai-Tibet Railway has been open to traffic for 12 years, and the Tibet Autonomous Region has also ushered in a new round of economic development. Major linear projects including the Qinghai-Tibet Expressway and the Railway Double Line have been incorporated into the relevant national development plans. For instance, the construction of the Qinghai-Tibet Expressway is imminent. These new projects will still be concentrated in the narrow corridor of the Qinghai-Tibet project. Frozen soil is a controlled environmental
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geological problem in the corridor. The construction of major projects will also have significant environmental effects while being constrained by the frozen environment. With the increasingly intensive layout of corridor engineering, the mutual influence of structure groups and disturbance to the permafrost geological environment of the corridor will become increasingly prominent [8]. At the same time, due to the active tectonic movements and the frequent occurrence of strong earthquakes on the Qinghai-Tibet Plateau, the normal operation of the major engineering facilities in the corridor will face greater potential risks under superposition of seismic effects (Fig. 1). Due to the important geographical location of the Qinghai-Tibet engineering corridor and the special engineering mechanical properties of permafrost, the research on the dynamic characteristics of plateau frozen soil has attracted more and more attention from domestic scholars. Based on several dynamic triaxial tests and resonant column tests with negative temperature, Hunaidi et al. (1996), Wang et al. (2007), Wu et al. (2003), Zhao et al. (2003) and Ling et al. (2014) studied the effects of temperature, water content, vibration frequency and confining pressure on the dynamic elastic modulus and damping ratio of frozen soil under cyclic loads or seismic loads, and obtained the
Corresponding author at: College of Transportation Science and Engineering, Nanjing Tech University, Nanjing 210009, China. E-mail address: [email protected] (Z. Wu).
https://doi.org/10.1016/j.trgeo.2019.100269 Received 19 February 2019; Received in revised form 7 July 2019; Accepted 17 August 2019 Available online 20 August 2019 2214-3912/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Distribution of permafrost and strong earthquake epicenters at and around the Qinghai-Tibet Engineering Corridor between January 1st, 1980 and December 31 st, 2018.
variation law of dynamic elastic modulus and damping ratio with different dynamic strain amplitudes [3,14–15,18,5]. Christ et al. (2009) proposed different methods to determine the unfrozen water content in frozen soils [1]. Liu et al. (2018) explored the influence of capillary cohesion and ice cementation on the strength and deformation characteristics of frozen soil [6]. Ling et al. (2009), Zhu et al. (2011), Luo et al. (2015) and Niu et al. (2016) respectively studied the influence law of dynamic parameters of frozen soil under graded loading through indoor dynamic triaxial test [4,20,7,11]. Yu et al. (2018) compared the variation rules and modes of nonlinear curves of initial shear modulus, shear modulus ratio and damping ratio of frozen soil at different negative temperatures [16]. At present, based on the consideration of engineering background, the seismic response analysis of strong earthquake activities and complex site effects on the Qinghai-Tibet Plateau is still rare, and the shear modulus ratio and damping ratio of the soil in the corridor are indispensable and important parameters for the seismic response analysis of the soil layer. In this paper, the experimental study on the influence of confining pressure, negative temperature, water content and frequency on the shear modulus ratio, damping ratio and maximum dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor is carried out. It can provide important basic information for seismic fortification of major projects in the area.
Ip = 14.9. Based on the empirical relationship between the plasticity index of the soil and the specific gravity of the particles, the particle density was calculated to be 2.73 g/cm3.
Test plans
Loading conditions
Preparation of soil samples
The test was carried out in the State Key Laboratory of Frozen Soil Engineering of the Chinese Academy of Sciences, and the test equipment used MTS-810 low temperature dynamic triaxial apparatus. The equipment mainly includes circulating refrigeration equipment, high pressure three-axis pressure chamber, numerical control equipment for testing machine and automatic data acquisition system. This equipment’s maximum axial pressure is 250 kN, the maximum frequency is 20 Hz, the deformation rate ranges from 0.001 to 1000 mm/min, the minimum control temperature is −70 °C, and the temperature control
The collected soil samples were crushed with iron mill, passed through a 2 mm sieve, and the fine-grained soil under the sieve was collected for future use. After a small amount of soil samples were dried in an oven for 8 h, the initial water content of the soil samples was obtained. According to the initial water content of soil samples, the dispersed soil with water content of 13%, 15% and 17% were configured. In order to keep the soil moisture uniform, the dispersed soil should be placed for 24 h with limited evaporation. In order to ensure the preparation of samples with good uniformity, Zheng et al. (2008) proposed the method of two-end compaction on the standard sample making machine to make samples with a diameter of 61.8 mm and a height of 125 mm. In this way, the influence of friction on the test results could be effectively overcome [19]. Put compacted samples into a copper mold, then place them in a −30 °C low temperature cold storage for rapid freezing. After 48 h, the mold was released. Finally, the temperature was set in the incubator according to the test requirements. And the test was carried out after keeping constant temperature for 24 h.
Taken from the Beilu River section of the Qinghai-Tibet Engineering Corridor, the soil samples are silty clay with a dry density of 1.80 g/ cm3. According to relevant test specifications, three types of soil samples with different water contents were respectively configured. After limited evaporation and being placed for 24 h, the plastic limit of the clay was measured by the liquid-plastic limit combined with measuring instrument ωP = 15.9% , the liquid limit ωL = 30.8%, plasticity index 2
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Table 1 Load conditions under different confining pressures. σ3/MPa
σ0/MPa
σmin/MPa
σ△/MPa
0.4 0.8 1.2
0.450 0.850 1.250
0.433 0.833 1.233
0.017 0.017 0.017
Note: σ3 is confining pressure, σ0 is initial static stress, σmin is minimum stress, σΔ is increment of dynamic stress amplitude between adjacent loading stages.
accuracy is ± 0.1 °C. The loading process of the samples includes 1 h confining consolidation process, axial static load application process, axial static load retention for 30 s process and axial dynamic load vibration process. The axial dynamic load is applied to the samples step by step according to the method suggested by Seed and Idriss [12]. Each sample kept the minimum stress unchanged under the loading conditions of the test. The dynamic stress amplitude of each stage increased with the order of loading stages, and the increase of the dynamic stress amplitude between adjacent loading stages was equal. There are three kinds of loading conditions according to different confining pressure environments. The specific loading conditions are shown in Table 1. The next stage load is applied after the dynamic load of each stage reaches 10 times.
Fig. 2. Schematic diagram of the typical stress-strain hysteresis curve of frozen soil.
curve is defined as the dynamic shear modulus Gd ; the dynamic shear stress amplitude (τm ) and dynamic shear strain amplitude (γm ) can be converted by formula (1) and formula (2) [2,13,17].
Test conditions For the frozen silty clay in the Qinghai-Tibet engineering corridor, a series of orthogonal tests were carried out to investigate the influence of three temperatures, three water contents, three confining pressures and four frequencies on the dynamic constitutive relationship, dynamic shear modulus and damping ratio of the frozen clay. The specific test contents are as follows:
τm = σm/2
(1)
γm = εm (1 + μ)
(2)
In the formula, σm is the dynamic stress amplitude, μ is the poisson ratio, εm is the dynamic strain amplitude. The test results show that the relationship between the dynamic shear stress amplitude (τm ) and the dynamic shear strain amplitude (γm ) of soil samples conforms to the Hardin-Drnevich model and can be expressed as:
(1) Orthogonal test of temperature (T), water content (ω) and confining pressure (σ3). a. Freezing temperature (T): There are three kinds of freezing temperatures for samples, i.e., −1.5 °C, −3.0 °C and −5.0 °C. b. Water content (ω): Three kinds of soil samples with different water content were allocated to simulate frozen soil with different ice content under negative temperature. The water content was 13%, 15% and 17% respectively. c. Confining pressure (σ3): Three confining pressures are 0.4 MPa, 0.8 MPa and 1.2 MPa respectively. (2) The influence test of frequency (f). When the water content ω = 17\% , confining pressure σ3 = 0.4 MPa and the temperature T = −1.5 °C, the test loading condition of four frequencies, i.e., 1 Hz, 5 Hz, 8 Hz and 10 Hz is carried out.
γm a + bγm
(3)
1 = a + bγm Gd
(4)
τm =
In the formula, a and b are the test parameters of the soil. Where, a 1 is the intercept of the G ~γm relation curve, and its physical meaning is d as follows:
a=
1 Gd max
(5)
In the formula, Gd max is the maximum dynamic shear modulus, and its value refers to the corresponding dynamic shear modulus (Gd ) when the dynamic strain amplitude (γm ) of soil mass approaches 0. 1 b is the slope of the G ~γm relation curve, whose physical sigd nificance is as follows:
The test condition parameters are shown in Table 2. Analysis of test results Dynamic shear modulus ratio and damping ratio of frozen soil
b= According to the dynamic triaxial tests with negative temperature, the stress-strain hysteresis curve of frozen soil under dynamic load is shown in Fig. 2 [10,20]. In the figure, the mean slope of the hysteresis
1 τmult
(6)
In the formula, τmult is the final stress amplitude, which is equal to the dynamic shear stress amplitude (τm ) when the dynamic strain
Table 2 Parameters of test conditions. Type of soil
T/°C
ω/%
σ3/MPa
f/Hz
Number of groups
Frozen Silty Clay of the Qinghai-Tibet
−1.5, −3.0, −5.0 −5.0
13, 15, 17 17
0.4, 0.8, 1.2 0.4
2 1, 5, 8, 10
27 4
3
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amplitude (γm ) of soil mass approaches ∞. Substituting the formula (5) and the formula (6) into the formula (4). In addition, γr is the reference shear strain amplitude, formulas are as follows:
τmult Gd max
γr =
Gd max 1 + γm/ γr
Gd =
(7)
(8)
The dynamic shear modulus ratio (αG ) can be expressed as,
αG =
Gd Gd max
(9)
The stress-strain hysteresis energy dissipation effect of the soil can be described by equivalent hysteresis damping ratio (λ ), which is determined according to the stress-strain hysteresis curve, as shown in formula (10).
λ=
1 AL 4π AT
(10)
In the formula, AL is the area of the ellipse enclosed by the hysteresis curve; AT is the area of triangle OCE. Influence of confining pressure on damping ratio and dynamic shear modulus ratio Fig. 3 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under confining pressures of 0.4 MPa, 0.8 MPa and 1.2 MPa when the temperature T = −1.5 °C and water content are 13%, 15% and 17% respectively. It can be seen from the figure that when the dynamic shear strain amplitude is between 0.02% and 0.12%, the damping ratio shows a tendency to increase slowly; the dynamic shear modulus ratio changes significantly with the increase of the strain amplitude and tends to decrease gradually. Under the same dynamic shear strain amplitude, the damping ratio increases slightly with the increase of confining pressure, but the change is not obvious, and change amplitude remains about 0.03. This is probably because in the range of testing confining pressure, sliding and directional arrangement between soil particles, soil-ice particles and ice-ice particles are still in a stable state within the test confining pressure range, and the damping ratio does not change significantly with the increase of confining pressure. As the dynamic shear modulus ratio decreases with the increase of confining pressure, it can be seen that the increase of confining pressure can induce the swelling and softening phenomenon of frozen soil, which leads to the gradual decrease of the strength of frozen soil. The dynamic shear modulus ratio decreases with the increase of the influence degree of dynamic load. Influence of temperature on damping ratio and dynamic shear modulus ratio Fig. 4 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under three different temperatures of −1.5 °C, −3.0 °C and −5.0 °C when the water content ω = 13\% and confining pressure are 0.4 MPa, 0.8 MPa and 1.2 MPa respectively. It can be seen from Fig. 4 that with the increase of dynamic shear strain amplitude, the damping ratio presents a slow increasing trend, and a relatively flat curve. While the dynamic shear modulus ratio shows a decreasing trend. Under the same dynamic shear strain amplitude, the damping ratio increases with increasing in temperature, and the dynamic shear modulus ratio reduces with increasing in temperature, which indicates that when the temperature is at −1.5 °C the ice crystal content in the frozen soil is less, and the unfrozen water content is relatively more and the ratio of the melting amount of the ice crystal to the content of
Fig. 3. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different confining pressures.
unfrozen water is relatively small under the action of the dynamic load. Moreover, with the decreasing of the temperature, the amount of unfrozen water in the soil reduces, the ice crystal content increases, the cementation effect of the soil enhances, the strength of the frozen soil 4
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Fig. 5. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different water contents.
increases, the damping ratio decreases, and the dynamic shear modulus ratio increases. Influence of water content on damping ratio and dynamic shear modulus ratio Fig. 5 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under three different water content conditions of 13%, 15% and 17% when the confining pressure is 0.4 MPa and temperatures are −1.5 °C, −3 °C and −5 °C respectively. The figure indicates that when the amplitude of the dynamic shear strain increases, the damping ratio increases slowly and gradually stabilizes, and the dynamic shear modulus ratio gradually decreases, which is similar to the variation of the damping ratio and the dynamic shear modulus ratio under different confining pressures and different negative temperatures. Under the same dynamic shear strain amplitude, the damping ratio increases with the increase of water content, and dynamic shear modulus ratio decreases with the increase of water content. While the dynamic shear strain amplitude is small (< 0.04%), the dynamic shear modulus ratio decreases with the change of water content. With the increase of water content, the content of unfrozen water in the soil increases, the viscosity and damping ratio of the frozen soil increase while the strength decrease, and the dynamic shear modulus ratio decreases gradually under the influence of dynamic load. Influence of frequency on damping ratio and dynamic shear modulus ratio Fig. 6 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under four different frequencies, i.e., 1 Hz, 5 Hz, 8 Hz and 10 Hz , when the water content ω = 17\% , confining pressure σ3 = 0.4 MPa and the temperature T = −1.5 °C .As is presented in the figure, with the increase of dynamic shear strain amplitude, the damping ratio shows a gradual increase trend, while the dynamic shear modulus ratio decreases continuously. Under the same dynamic shear strain amplitude, the damping ratio decreases gradually with the increase of loading frequency, which indicates that in the process of applying the vibration load, the load causes the plastic flow of the ice body in the frozen soil and the reorientation of ice crystals in the soil. At the same time, when the loading frequency changes between 1 Hz and 10 Hz, its influence on the shear modulus ratio is not significant, but showing a slight increase trend with the increase of the loading frequency. Due to the increase of
Fig. 4. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different temperatures.
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Fig. 7. Relationship between maximum dynamic shear modulus and water content and confining pressure. Fig. 6. Relationship between damping ratio, shear modulus ratio and dynamic shear strain amplitude under different frequencies.
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Fig. 8. Relationship between maximum dynamic shear modulus and water content and negative temperature.
Fig. 9. Relationship between maximum dynamic shear modulus and temperature and confining pressure.
loading frequency, more vibration energy is converted into heat energy which increases the temperature, resulting in partial melting of ice crystals and softening of frozen soil, and the increase of modulus tends to be consistent.
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Table 3 Statistical table of test constants.
−1.5 −3 −5 0.4 0.8 1.2 13 15 17
Temperature (°C)
Confining pressure (MPa)
Water content (%)
h
i
j
k
l
m
R
1104.77 665.84 −201.15 2274.87 753.28 8.87 232.98 28.04 174.33
−148.01 −55.49 −13.56 −265.45 61.82 4.67 −3.762 −23.70 48.28
54.751 −346.69 561.82 280.08 256.98 73.31 −472.74 39.43 −76.72
5.37 1.69 4.07 7.89 1.13 0.09 2.55 6.13 27.97
−9.49 33.98 330.68 12.43 10.23 20.52 218.12 −54.44 82.93
1.38 28.45 51.58 19.85 20.34 6.63 −92.35 −65.59 −26.44
0.97 0.99 0.97 0.95 0.99 0.99 0.96 0.99 0.99
confining pressure and temperature on the maximum dynamic shear modulus, multivariate nonlinear fitting is adopted to construct the relational expression Gd max = f (T , ω, σ3) , and the maximum dynamic shear modulus estimation model based on these three factors and is obtained. The fitted regression formula is as follows:
Table 4 Statistical table of test constants and fitting results. Temperature/°C
Water content /%
Confining pressure/MPa
Test modulus/ MPa
Calculated modulus/MPa
−1.5 −3 −5 −1.5
15 15 13 17
1.2 0.8 1.2 0.4
165.51 292.23 654.95 195.32
161.06 288.42 663.54 201.22
2
Gd max = x1 T + x2 T 2 + x3 σ3 + x 4 σ32 + x5 Tσ3 + e x 6 ω
The maximum dynamic shear modulus under any temperature, water content and confining pressure can be obtained through formula (14). Iterative calculation is performed by nonlinear least squares method, and the parameter values of x1 ~ x6 in formula (14) are obtained, i.e., x1 = −12.711, x2 = 9.725, x3 = −71.793, x4 = 30.220, x5 = −63.508, x6 = 0.017. The theoretical value obtained by calculating the fitting formula and the experimental value are shown in Table 4. The correlation coefficient is R = 0.97, the fitting effect is good, which shows that the dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor can be calculated according to this formula.
Maximum dynamic shear modulus Variation law of the maximum dynamic shear modulus The maximum dynamic shear modulus refers to the corresponding shear modulus when the dynamic strain amplitude of soil mass approaches 0. According to the method proposed by Zhang and Xie [17], the maximum dynamic shear modulus of frozen silt clay can be calculated by Hardin-Drnevich model. The relationship between the maximum dynamic shear modulus of frozen silty clay and water content, confining pressure and temperature is shown in Figs. 7–9, Fig. 7 shows the relationship between maximum dynamic shear modulus and water contents and confining pressures at three different temperatures; Fig. 8 shows the relationship between maximum dynamic shear modulus and temperatures and confining pressures at three different confining pressures; Fig. 9 shows the relationship between the maximum dynamic shear modulus and temperatures and confining pressures at three different water contents. It can be seen that under these three different conditions, the maximum dynamic shear modulus always increases in the concave shape with the two influencing factors. Data fitting was carried out for Figs. 7–9 respectively. The empirical formula for fitting are shown in formulas (11)–(13), and the relevant parameters of the formulas are shown in Table 3.
Gd max = h + iω + jσ3 + kω2 + lσ32 + mωσ3
(11)
Gd max = h + iω + jT + kω2 + lT 2 + mωT
(12)
Gd max = h + iT + jσ3 + kT 2 + lσ32 + mTσ3
(13)
(14)
Conclusions (1) Damping ratio and shear modulus ratio of Qinghai-Tibet frozen silty clay in the Qinghai-Tibet Engineering Corridor are significantly affected by shear strain amplitude. With the increase of shear strain amplitude, the damping ratio presents a slow increasing trend while the shear modulus ratio shows a gradual decreasing trend. This change law is affected by factors such as confining pressure, temperature, water content and frequency. The damping ratio of frozen Qinghai-Tibet silty clay increases with increasing in confining pressure, temperature and water content, and with decreasing in frequency. The dynamic shear modulus ratio reduces with increasing in confining pressure, temperature, water content and decreasing in frequency. (2) The relationship between the maximum dynamic shear modulus and the two parameters of Qinghai-Tibet frozen silty clay is obtained by taking the parameters of temperature and confining pressure, temperature and water content, water content and confining pressure as variables respectively. (3) The relationship between the maximum dynamic shear modulus of frozen silty clay and three factors, i.e., temperature, water content and confining pressure, is obtained by using the multi-element nonlinear fitting method, and the fitting result is good, which shows that the dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor can be obtained according to the fitted regression formula.
In these formulas, Gd max is the maximum dynamic shear modulus (MPa), ω is water content (%), σ3 is confining pressure (MPa), T is temperature (°C). h, i, j, k, l and m are all constant coefficients, which can be seen from Table 3 that the correlation coefficients obtained by the fitting formula are all above 0.95. It shows that the fitting effect of using multivariate polynomial equation to fit the relationship between the maximum dynamic shear modulus Gd max and any two factors of water content ω , confining pressure σ3 and temperature T is ideal.
Acknowledgements This study is financially supported by the Scientific Research Foundation for Introducing Talent of Nanjing Tech University, the National Natural Science Foundation of China (No. 41472297) and the
Multiple nonlinear fitting formula of Gd max In order to comprehensively analyze the influence of water content, 8
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Key Project of Natural Science Foundation of China (No. 41630636).
Qinghai-Tibet Plateau. Adv Earth Sci 2012;27(11):1185–91. [10] Ma W, Wang DY. The mechanics of frozen ground. Beijing: Science Press; 2014. [11] Niu YQ, Lai YM, Wang X, et al. Research on influences of initial water content on deformation and strength behaviors of frozen silty clay. Rock Soil Mech 2016;37(2):500–6. [12] Seed HB, Idriss IM. Simplified procedure for evaluation soil liquefaction potential. J Soil Mech Found Div 1971;97(9):1249–73. [13] Seed HB, Idriss IM. Soil moduli and damping factors for dynamic response analyses. Earthq Eng Res Center, Univ California Berkeley; 1970. [14] Wang LX, Hu QL, Ling XZ, et al. Experimental study on dynamic shear modulus of remolded frozen silty clay for Qinghai-Xizang railway. Earthq Eng Eng Vibr 2007;27(2):177–80. [15] Wu ZJ, Ma W, Wang LM, et al. Laboratory study on the effect of temperature and confining pressure on strength of frozen soil under seismic dynamic lading. J Glaciol Geocryol 2003;25(6):648–52. [16] Yu XB, Liu HB, Sun R, et al. Improved Hardin-Drnevich model for the dynamic modulus and damping ratio of frozen soil. Cold Reg Sci Technol 2018;153:64–77. [17] Zhang KX, Xie JF. Soil dynamics. Beijing: Seismological Press; 1989. [18] Zhao SP, Zhu YL, He P, et al. Testing study on dynamic mechanics parameters of frozen soil. Chin J Rock Mech Eng 2003;22(Suppl 2):2677–81. [19] Zheng JF, Ma W, Zhao SP, et al. Development of the specimen-preparing technique for remoulded soil samples. J Glaciol Geocryol 2008;30(3):494–500. [20] Zhu ZY, Ling XZ, Wang ZY, et al. Experimental investigation of the dynamic behavior of frozen clay from the Beiluhe subgrade along the QTR. Cold Reg Sci Technol 2011;69:91–7.
References [1] Christ M, Park JB. Ultrasonic technique as tool for determining physical and mechanical properties of frozen soils. Cold Reg Sci Technol 2009;58(3):136–42. [2] Hardin BO, Drnevich VP. Shear modulus and damping in soils: design equations and curves. J Soil Mech Found Div 1972;98(SM7):667–92. [3] Hunaidi MA, Chen PA, Rainer JH, et al. Shear moduli and damping in frozen and unfrozen clay by resonant column tests. Can Geotech J 1996;33(3):510–4. [4] Ling XZ, Zhu ZY, Zhang F, et al. Dynamic elastic modulus for frozen soil from the embankment on Beiluhe basin along the Qinghai-Tibet railway. Cold Reg Sci Technol 2009;57:7–12. [5] Ling ZC, Gao ML, Yi TH, et al. Status analysis of the frozen soil's dynamics parameter study. Adv Mater Res 2014;941:2626–30. [6] Liu ZY, Liu JK, Li X, et al. Effect of capillary cohesion and ice cementation on strength and deformation of unsaturated frozen silty clay. Chin J Rock Mech Eng 2018;37(6):1551–9. [7] Luo F, He YT, Zhao SP, et al. Experimental study of damping ratio of frozen soil under stepwise loading. Rock Soil Mech 2015;36(1):3143–9. [8] Ma W, Mu YH, Xie SB, et al. Thermal-mechanical influences and environmental effects of expressway construction on the Qinghai-Tibet permafrost engineering corridor. Adv Earth Sci 2017;31(5):459–64. [9] Ma W, Niu FJ, Mu YH. Basic research on the major permafrost projects in the
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An experimental research on damping ratio and dynamic shear modulus ratio of frozen silty clay of the Qinghai-Tibet engineering corridor
T
⁎
Zhijian Wua,b, , Dan Zhangc, Tao Zhaod, Jinlian Mab, Duoyin Zhaob a
College of Transportation Science and Engineering, Nanjing Tech University, Nanjing 210009, China Key Laboratory of Loess Earthquake Engineering, Lanzhou Institute of Seismology, China Earthquake Administration, Lanzhou 730000, China c Gansu Construction Investment (Holdings) Group Corporation Steel Structure Co. Ltd, Lanzhou, Gansu 730000, China d Shaanxi Railway Institute, Weinan 714000, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Frozen silty clay Qinghai-Tibet engineering corridor Damping ratio Dynamic shear modulus Confining pressure Temperature Water content Frequency
Strong earthquakes occurred frequently in Qinghai-Tibet Plateau. Major national infrastructure facilities, including existing and proposed projects, are located densely in the permafrost engineering corridor of QinghaiTibet Plateau. Based on dynamic triaxial test under the effect of cycle loading for remolded frozen soil taken from Qinghai-Tibet engineering corridor, experimental studies on the influence of confining pressure, temperature, moisture content and frequency on damping ratio and dynamic shear modulus ratio have been carried out systematically. The results are summarized as follows: The damping ratio of frozen Qinghai-Tibet silty clay increases with increasing in confining pressure, temperature and water content, and decreasing in frequency. In addition, the dynamic shear modulus ratio reduces with increasing in confining pressure, temperature and water content, and decreasing in frequency. The relationship equation between the maximum dynamic shear modulus of frozen silty clay and temperature, water content and confining pressure is obtained by using multivariate nonlinear fitting method, which shows a better fitting effect. The results of this research can provide basic data for the design of major engineering and seismic fortification along Qinghai-Tibet engineering corridor.
Introduction The Qinghai-Tibet Engineering Corridor passing through Qinghai Province, is a strategic channel that connects Tibet with the mainland. It is about 550 km in length through a large area of continuous permafrost and 82 km in length through an island-like discontinuous permafrost. Its maximum width is only a few kilometers, while its minimum width is only hundreds of meters. Recently, many linear and major infrastructure projects such as the Qinghai-Tibet Highway, the Qinghai-Tibet Railway, the Gelat Oil Product Pipeline, the Lansila Cable Communication Project, and the high-voltage transmission line have been built within this limited corridor [9]. The Qinghai-Tibet Railway has been open to traffic for 12 years, and the Tibet Autonomous Region has also ushered in a new round of economic development. Major linear projects including the Qinghai-Tibet Expressway and the Railway Double Line have been incorporated into the relevant national development plans. For instance, the construction of the Qinghai-Tibet Expressway is imminent. These new projects will still be concentrated in the narrow corridor of the Qinghai-Tibet project. Frozen soil is a controlled environmental
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geological problem in the corridor. The construction of major projects will also have significant environmental effects while being constrained by the frozen environment. With the increasingly intensive layout of corridor engineering, the mutual influence of structure groups and disturbance to the permafrost geological environment of the corridor will become increasingly prominent [8]. At the same time, due to the active tectonic movements and the frequent occurrence of strong earthquakes on the Qinghai-Tibet Plateau, the normal operation of the major engineering facilities in the corridor will face greater potential risks under superposition of seismic effects (Fig. 1). Due to the important geographical location of the Qinghai-Tibet engineering corridor and the special engineering mechanical properties of permafrost, the research on the dynamic characteristics of plateau frozen soil has attracted more and more attention from domestic scholars. Based on several dynamic triaxial tests and resonant column tests with negative temperature, Hunaidi et al. (1996), Wang et al. (2007), Wu et al. (2003), Zhao et al. (2003) and Ling et al. (2014) studied the effects of temperature, water content, vibration frequency and confining pressure on the dynamic elastic modulus and damping ratio of frozen soil under cyclic loads or seismic loads, and obtained the
Corresponding author at: College of Transportation Science and Engineering, Nanjing Tech University, Nanjing 210009, China. E-mail address: [email protected] (Z. Wu).
https://doi.org/10.1016/j.trgeo.2019.100269 Received 19 February 2019; Received in revised form 7 July 2019; Accepted 17 August 2019 Available online 20 August 2019 2214-3912/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Distribution of permafrost and strong earthquake epicenters at and around the Qinghai-Tibet Engineering Corridor between January 1st, 1980 and December 31 st, 2018.
variation law of dynamic elastic modulus and damping ratio with different dynamic strain amplitudes [3,14–15,18,5]. Christ et al. (2009) proposed different methods to determine the unfrozen water content in frozen soils [1]. Liu et al. (2018) explored the influence of capillary cohesion and ice cementation on the strength and deformation characteristics of frozen soil [6]. Ling et al. (2009), Zhu et al. (2011), Luo et al. (2015) and Niu et al. (2016) respectively studied the influence law of dynamic parameters of frozen soil under graded loading through indoor dynamic triaxial test [4,20,7,11]. Yu et al. (2018) compared the variation rules and modes of nonlinear curves of initial shear modulus, shear modulus ratio and damping ratio of frozen soil at different negative temperatures [16]. At present, based on the consideration of engineering background, the seismic response analysis of strong earthquake activities and complex site effects on the Qinghai-Tibet Plateau is still rare, and the shear modulus ratio and damping ratio of the soil in the corridor are indispensable and important parameters for the seismic response analysis of the soil layer. In this paper, the experimental study on the influence of confining pressure, negative temperature, water content and frequency on the shear modulus ratio, damping ratio and maximum dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor is carried out. It can provide important basic information for seismic fortification of major projects in the area.
Ip = 14.9. Based on the empirical relationship between the plasticity index of the soil and the specific gravity of the particles, the particle density was calculated to be 2.73 g/cm3.
Test plans
Loading conditions
Preparation of soil samples
The test was carried out in the State Key Laboratory of Frozen Soil Engineering of the Chinese Academy of Sciences, and the test equipment used MTS-810 low temperature dynamic triaxial apparatus. The equipment mainly includes circulating refrigeration equipment, high pressure three-axis pressure chamber, numerical control equipment for testing machine and automatic data acquisition system. This equipment’s maximum axial pressure is 250 kN, the maximum frequency is 20 Hz, the deformation rate ranges from 0.001 to 1000 mm/min, the minimum control temperature is −70 °C, and the temperature control
The collected soil samples were crushed with iron mill, passed through a 2 mm sieve, and the fine-grained soil under the sieve was collected for future use. After a small amount of soil samples were dried in an oven for 8 h, the initial water content of the soil samples was obtained. According to the initial water content of soil samples, the dispersed soil with water content of 13%, 15% and 17% were configured. In order to keep the soil moisture uniform, the dispersed soil should be placed for 24 h with limited evaporation. In order to ensure the preparation of samples with good uniformity, Zheng et al. (2008) proposed the method of two-end compaction on the standard sample making machine to make samples with a diameter of 61.8 mm and a height of 125 mm. In this way, the influence of friction on the test results could be effectively overcome [19]. Put compacted samples into a copper mold, then place them in a −30 °C low temperature cold storage for rapid freezing. After 48 h, the mold was released. Finally, the temperature was set in the incubator according to the test requirements. And the test was carried out after keeping constant temperature for 24 h.
Taken from the Beilu River section of the Qinghai-Tibet Engineering Corridor, the soil samples are silty clay with a dry density of 1.80 g/ cm3. According to relevant test specifications, three types of soil samples with different water contents were respectively configured. After limited evaporation and being placed for 24 h, the plastic limit of the clay was measured by the liquid-plastic limit combined with measuring instrument ωP = 15.9% , the liquid limit ωL = 30.8%, plasticity index 2
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Table 1 Load conditions under different confining pressures. σ3/MPa
σ0/MPa
σmin/MPa
σ△/MPa
0.4 0.8 1.2
0.450 0.850 1.250
0.433 0.833 1.233
0.017 0.017 0.017
Note: σ3 is confining pressure, σ0 is initial static stress, σmin is minimum stress, σΔ is increment of dynamic stress amplitude between adjacent loading stages.
accuracy is ± 0.1 °C. The loading process of the samples includes 1 h confining consolidation process, axial static load application process, axial static load retention for 30 s process and axial dynamic load vibration process. The axial dynamic load is applied to the samples step by step according to the method suggested by Seed and Idriss [12]. Each sample kept the minimum stress unchanged under the loading conditions of the test. The dynamic stress amplitude of each stage increased with the order of loading stages, and the increase of the dynamic stress amplitude between adjacent loading stages was equal. There are three kinds of loading conditions according to different confining pressure environments. The specific loading conditions are shown in Table 1. The next stage load is applied after the dynamic load of each stage reaches 10 times.
Fig. 2. Schematic diagram of the typical stress-strain hysteresis curve of frozen soil.
curve is defined as the dynamic shear modulus Gd ; the dynamic shear stress amplitude (τm ) and dynamic shear strain amplitude (γm ) can be converted by formula (1) and formula (2) [2,13,17].
Test conditions For the frozen silty clay in the Qinghai-Tibet engineering corridor, a series of orthogonal tests were carried out to investigate the influence of three temperatures, three water contents, three confining pressures and four frequencies on the dynamic constitutive relationship, dynamic shear modulus and damping ratio of the frozen clay. The specific test contents are as follows:
τm = σm/2
(1)
γm = εm (1 + μ)
(2)
In the formula, σm is the dynamic stress amplitude, μ is the poisson ratio, εm is the dynamic strain amplitude. The test results show that the relationship between the dynamic shear stress amplitude (τm ) and the dynamic shear strain amplitude (γm ) of soil samples conforms to the Hardin-Drnevich model and can be expressed as:
(1) Orthogonal test of temperature (T), water content (ω) and confining pressure (σ3). a. Freezing temperature (T): There are three kinds of freezing temperatures for samples, i.e., −1.5 °C, −3.0 °C and −5.0 °C. b. Water content (ω): Three kinds of soil samples with different water content were allocated to simulate frozen soil with different ice content under negative temperature. The water content was 13%, 15% and 17% respectively. c. Confining pressure (σ3): Three confining pressures are 0.4 MPa, 0.8 MPa and 1.2 MPa respectively. (2) The influence test of frequency (f). When the water content ω = 17\% , confining pressure σ3 = 0.4 MPa and the temperature T = −1.5 °C, the test loading condition of four frequencies, i.e., 1 Hz, 5 Hz, 8 Hz and 10 Hz is carried out.
γm a + bγm
(3)
1 = a + bγm Gd
(4)
τm =
In the formula, a and b are the test parameters of the soil. Where, a 1 is the intercept of the G ~γm relation curve, and its physical meaning is d as follows:
a=
1 Gd max
(5)
In the formula, Gd max is the maximum dynamic shear modulus, and its value refers to the corresponding dynamic shear modulus (Gd ) when the dynamic strain amplitude (γm ) of soil mass approaches 0. 1 b is the slope of the G ~γm relation curve, whose physical sigd nificance is as follows:
The test condition parameters are shown in Table 2. Analysis of test results Dynamic shear modulus ratio and damping ratio of frozen soil
b= According to the dynamic triaxial tests with negative temperature, the stress-strain hysteresis curve of frozen soil under dynamic load is shown in Fig. 2 [10,20]. In the figure, the mean slope of the hysteresis
1 τmult
(6)
In the formula, τmult is the final stress amplitude, which is equal to the dynamic shear stress amplitude (τm ) when the dynamic strain
Table 2 Parameters of test conditions. Type of soil
T/°C
ω/%
σ3/MPa
f/Hz
Number of groups
Frozen Silty Clay of the Qinghai-Tibet
−1.5, −3.0, −5.0 −5.0
13, 15, 17 17
0.4, 0.8, 1.2 0.4
2 1, 5, 8, 10
27 4
3
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amplitude (γm ) of soil mass approaches ∞. Substituting the formula (5) and the formula (6) into the formula (4). In addition, γr is the reference shear strain amplitude, formulas are as follows:
τmult Gd max
γr =
Gd max 1 + γm/ γr
Gd =
(7)
(8)
The dynamic shear modulus ratio (αG ) can be expressed as,
αG =
Gd Gd max
(9)
The stress-strain hysteresis energy dissipation effect of the soil can be described by equivalent hysteresis damping ratio (λ ), which is determined according to the stress-strain hysteresis curve, as shown in formula (10).
λ=
1 AL 4π AT
(10)
In the formula, AL is the area of the ellipse enclosed by the hysteresis curve; AT is the area of triangle OCE. Influence of confining pressure on damping ratio and dynamic shear modulus ratio Fig. 3 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under confining pressures of 0.4 MPa, 0.8 MPa and 1.2 MPa when the temperature T = −1.5 °C and water content are 13%, 15% and 17% respectively. It can be seen from the figure that when the dynamic shear strain amplitude is between 0.02% and 0.12%, the damping ratio shows a tendency to increase slowly; the dynamic shear modulus ratio changes significantly with the increase of the strain amplitude and tends to decrease gradually. Under the same dynamic shear strain amplitude, the damping ratio increases slightly with the increase of confining pressure, but the change is not obvious, and change amplitude remains about 0.03. This is probably because in the range of testing confining pressure, sliding and directional arrangement between soil particles, soil-ice particles and ice-ice particles are still in a stable state within the test confining pressure range, and the damping ratio does not change significantly with the increase of confining pressure. As the dynamic shear modulus ratio decreases with the increase of confining pressure, it can be seen that the increase of confining pressure can induce the swelling and softening phenomenon of frozen soil, which leads to the gradual decrease of the strength of frozen soil. The dynamic shear modulus ratio decreases with the increase of the influence degree of dynamic load. Influence of temperature on damping ratio and dynamic shear modulus ratio Fig. 4 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under three different temperatures of −1.5 °C, −3.0 °C and −5.0 °C when the water content ω = 13\% and confining pressure are 0.4 MPa, 0.8 MPa and 1.2 MPa respectively. It can be seen from Fig. 4 that with the increase of dynamic shear strain amplitude, the damping ratio presents a slow increasing trend, and a relatively flat curve. While the dynamic shear modulus ratio shows a decreasing trend. Under the same dynamic shear strain amplitude, the damping ratio increases with increasing in temperature, and the dynamic shear modulus ratio reduces with increasing in temperature, which indicates that when the temperature is at −1.5 °C the ice crystal content in the frozen soil is less, and the unfrozen water content is relatively more and the ratio of the melting amount of the ice crystal to the content of
Fig. 3. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different confining pressures.
unfrozen water is relatively small under the action of the dynamic load. Moreover, with the decreasing of the temperature, the amount of unfrozen water in the soil reduces, the ice crystal content increases, the cementation effect of the soil enhances, the strength of the frozen soil 4
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Fig. 5. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different water contents.
increases, the damping ratio decreases, and the dynamic shear modulus ratio increases. Influence of water content on damping ratio and dynamic shear modulus ratio Fig. 5 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under three different water content conditions of 13%, 15% and 17% when the confining pressure is 0.4 MPa and temperatures are −1.5 °C, −3 °C and −5 °C respectively. The figure indicates that when the amplitude of the dynamic shear strain increases, the damping ratio increases slowly and gradually stabilizes, and the dynamic shear modulus ratio gradually decreases, which is similar to the variation of the damping ratio and the dynamic shear modulus ratio under different confining pressures and different negative temperatures. Under the same dynamic shear strain amplitude, the damping ratio increases with the increase of water content, and dynamic shear modulus ratio decreases with the increase of water content. While the dynamic shear strain amplitude is small (< 0.04%), the dynamic shear modulus ratio decreases with the change of water content. With the increase of water content, the content of unfrozen water in the soil increases, the viscosity and damping ratio of the frozen soil increase while the strength decrease, and the dynamic shear modulus ratio decreases gradually under the influence of dynamic load. Influence of frequency on damping ratio and dynamic shear modulus ratio Fig. 6 shows the relationship between damping ratio λ , dynamic shear modulus ratio α and dynamic shear strain amplitude γm of frozen silty clay under four different frequencies, i.e., 1 Hz, 5 Hz, 8 Hz and 10 Hz , when the water content ω = 17\% , confining pressure σ3 = 0.4 MPa and the temperature T = −1.5 °C .As is presented in the figure, with the increase of dynamic shear strain amplitude, the damping ratio shows a gradual increase trend, while the dynamic shear modulus ratio decreases continuously. Under the same dynamic shear strain amplitude, the damping ratio decreases gradually with the increase of loading frequency, which indicates that in the process of applying the vibration load, the load causes the plastic flow of the ice body in the frozen soil and the reorientation of ice crystals in the soil. At the same time, when the loading frequency changes between 1 Hz and 10 Hz, its influence on the shear modulus ratio is not significant, but showing a slight increase trend with the increase of the loading frequency. Due to the increase of
Fig. 4. Relationship between damping ratio, dynamic shear modulus ratio and dynamic shear strain amplitude under different temperatures.
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Fig. 7. Relationship between maximum dynamic shear modulus and water content and confining pressure. Fig. 6. Relationship between damping ratio, shear modulus ratio and dynamic shear strain amplitude under different frequencies.
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Fig. 8. Relationship between maximum dynamic shear modulus and water content and negative temperature.
Fig. 9. Relationship between maximum dynamic shear modulus and temperature and confining pressure.
loading frequency, more vibration energy is converted into heat energy which increases the temperature, resulting in partial melting of ice crystals and softening of frozen soil, and the increase of modulus tends to be consistent.
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Table 3 Statistical table of test constants.
−1.5 −3 −5 0.4 0.8 1.2 13 15 17
Temperature (°C)
Confining pressure (MPa)
Water content (%)
h
i
j
k
l
m
R
1104.77 665.84 −201.15 2274.87 753.28 8.87 232.98 28.04 174.33
−148.01 −55.49 −13.56 −265.45 61.82 4.67 −3.762 −23.70 48.28
54.751 −346.69 561.82 280.08 256.98 73.31 −472.74 39.43 −76.72
5.37 1.69 4.07 7.89 1.13 0.09 2.55 6.13 27.97
−9.49 33.98 330.68 12.43 10.23 20.52 218.12 −54.44 82.93
1.38 28.45 51.58 19.85 20.34 6.63 −92.35 −65.59 −26.44
0.97 0.99 0.97 0.95 0.99 0.99 0.96 0.99 0.99
confining pressure and temperature on the maximum dynamic shear modulus, multivariate nonlinear fitting is adopted to construct the relational expression Gd max = f (T , ω, σ3) , and the maximum dynamic shear modulus estimation model based on these three factors and is obtained. The fitted regression formula is as follows:
Table 4 Statistical table of test constants and fitting results. Temperature/°C
Water content /%
Confining pressure/MPa
Test modulus/ MPa
Calculated modulus/MPa
−1.5 −3 −5 −1.5
15 15 13 17
1.2 0.8 1.2 0.4
165.51 292.23 654.95 195.32
161.06 288.42 663.54 201.22
2
Gd max = x1 T + x2 T 2 + x3 σ3 + x 4 σ32 + x5 Tσ3 + e x 6 ω
The maximum dynamic shear modulus under any temperature, water content and confining pressure can be obtained through formula (14). Iterative calculation is performed by nonlinear least squares method, and the parameter values of x1 ~ x6 in formula (14) are obtained, i.e., x1 = −12.711, x2 = 9.725, x3 = −71.793, x4 = 30.220, x5 = −63.508, x6 = 0.017. The theoretical value obtained by calculating the fitting formula and the experimental value are shown in Table 4. The correlation coefficient is R = 0.97, the fitting effect is good, which shows that the dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor can be calculated according to this formula.
Maximum dynamic shear modulus Variation law of the maximum dynamic shear modulus The maximum dynamic shear modulus refers to the corresponding shear modulus when the dynamic strain amplitude of soil mass approaches 0. According to the method proposed by Zhang and Xie [17], the maximum dynamic shear modulus of frozen silt clay can be calculated by Hardin-Drnevich model. The relationship between the maximum dynamic shear modulus of frozen silty clay and water content, confining pressure and temperature is shown in Figs. 7–9, Fig. 7 shows the relationship between maximum dynamic shear modulus and water contents and confining pressures at three different temperatures; Fig. 8 shows the relationship between maximum dynamic shear modulus and temperatures and confining pressures at three different confining pressures; Fig. 9 shows the relationship between the maximum dynamic shear modulus and temperatures and confining pressures at three different water contents. It can be seen that under these three different conditions, the maximum dynamic shear modulus always increases in the concave shape with the two influencing factors. Data fitting was carried out for Figs. 7–9 respectively. The empirical formula for fitting are shown in formulas (11)–(13), and the relevant parameters of the formulas are shown in Table 3.
Gd max = h + iω + jσ3 + kω2 + lσ32 + mωσ3
(11)
Gd max = h + iω + jT + kω2 + lT 2 + mωT
(12)
Gd max = h + iT + jσ3 + kT 2 + lσ32 + mTσ3
(13)
(14)
Conclusions (1) Damping ratio and shear modulus ratio of Qinghai-Tibet frozen silty clay in the Qinghai-Tibet Engineering Corridor are significantly affected by shear strain amplitude. With the increase of shear strain amplitude, the damping ratio presents a slow increasing trend while the shear modulus ratio shows a gradual decreasing trend. This change law is affected by factors such as confining pressure, temperature, water content and frequency. The damping ratio of frozen Qinghai-Tibet silty clay increases with increasing in confining pressure, temperature and water content, and with decreasing in frequency. The dynamic shear modulus ratio reduces with increasing in confining pressure, temperature, water content and decreasing in frequency. (2) The relationship between the maximum dynamic shear modulus and the two parameters of Qinghai-Tibet frozen silty clay is obtained by taking the parameters of temperature and confining pressure, temperature and water content, water content and confining pressure as variables respectively. (3) The relationship between the maximum dynamic shear modulus of frozen silty clay and three factors, i.e., temperature, water content and confining pressure, is obtained by using the multi-element nonlinear fitting method, and the fitting result is good, which shows that the dynamic shear modulus of the frozen silty clay in the Qinghai-Tibet Engineering Corridor can be obtained according to the fitted regression formula.
In these formulas, Gd max is the maximum dynamic shear modulus (MPa), ω is water content (%), σ3 is confining pressure (MPa), T is temperature (°C). h, i, j, k, l and m are all constant coefficients, which can be seen from Table 3 that the correlation coefficients obtained by the fitting formula are all above 0.95. It shows that the fitting effect of using multivariate polynomial equation to fit the relationship between the maximum dynamic shear modulus Gd max and any two factors of water content ω , confining pressure σ3 and temperature T is ideal.
Acknowledgements This study is financially supported by the Scientific Research Foundation for Introducing Talent of Nanjing Tech University, the National Natural Science Foundation of China (No. 41472297) and the
Multiple nonlinear fitting formula of Gd max In order to comprehensively analyze the influence of water content, 8
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Key Project of Natural Science Foundation of China (No. 41630636).
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