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Improved Hardin-Drnevich model for the dynamic modulus and damping ratio of frozen soil
Cold Regions Science and Technology 153 (2018) 64–77
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Improved Hardin-Drnevich model for the dynamic modulus and damping ratio of frozen soil
T
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Xiaobo Yua,b, Huabei Liua, , Rui Sunb, Xiaoming Yuanb a b
School of civil engineering & mechanics, Huazhong University of Science & Technology, Wuhan 430074, China Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Hardin-Drnevich model Frozen soil Resonant column test Empirical correction factor Seismic ground response
The Hardin-Drnevich model is frequently employed in the seismic ground response analysis, but the original model cannot be directly applied to frozen soils, the dynamic properties of which are functions of the soil temperature. Resonant column tests were carried out and empirical correction factors were proposed to revise the model. The improved model was applied to the seismic ground response analysis of a permafrost site, and the influences of the variations of the correction factors were analyzed. The resonant column test results indicated that soil temperature had large influences on the shear modulus and damping ratio, while in the range investigated in this study, the effect of the degree of saturation was not significant. The proposed correction factors properly fitted the test results in the present and previous studies, but the correction factors varied considerably even for the same category of clayey or silty or sandy soils. Compared with the correction factors for the modulus reduction curve and damping curve, the correction factor for the maximum shear modulus had a larger effect on the seismic ground response of a permafrost site. In the seismic response analyses, the response spectral values at the ground surface were generally larger if the investigated site was assumed to be non-frozen.
1. Introduction Hardin et al. (1972) proposed a hyperbolic model to depict the shear modulus reduction and damping evolution, which was widely used with various modifications in seismic ground response analyses (Hardin and Drnevich, 1972; Stokoe et al., 1999; Wichtmann et al., 2015; Senetakis et al., 2013; Subramaniam and Banerjee, 2013; Wang and Kuwano, 1999). With the increasing infrastructure development in cold regions, it is necessary to quantitatively describe the dynamic properties of frozen soils, and a model similar to that by Hardin et al. (1972) may provide an important tool for this purpose. However, most of the previous modifications of the Hardin's model have been focused on non-frozen soils. A modified Hardin's model for frozen soils accounting for the important factors that influence the dynamic properties is therefore called for in order to satisfy the need for the constructions in cold regions. Previous studies have shown that the strain amplitude, temperature and moisture content are several factors that may influence the dynamic modulus and damping ratio of the frozen soil (Li et al., 2012). The influence of strain amplitude reflects the nonlinear properties of soil, which is already accounted for in the Hardin's model (Ling et al., 2009; Al Hunaidi et al., 1996; Kaplar, 1969; Yang et al., 2010).
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Corresponding author. E-mail address: [email protected] (H. Liu).
https://doi.org/10.1016/j.coldregions.2018.05.004 Received 14 May 2017; Received in revised form 26 April 2018; Accepted 18 May 2018 Available online 23 May 2018 0165-232X/ © 2018 Elsevier B.V. All rights reserved.
In the frozen soil, the liquid phase transfers into the ice phase, the extent of which is dependent on the soil temperature. Previous studies (Qi and Ma, 2010; Ma and Wang, 2012; Zhao et al., 2002) have concluded that, the dynamic modulus of the frozen soil is larger than that of the unfrozen soil and it increases with a decrease in the soil temperature. In contrast, the damping ratio of the frozen soil decreases with a decrease in the temperature. Previous studies were mostly focused on the maximum shear modulus Gmax and its relationship with the frozensoil temperature, while few attempts have been directed to investigate the evolutions of the G/Gmax (shear modulus ratio) and D (damping ratio) curves of the frozen soil. However, the G/Gmax and D curves are as important as Gmax in seismic ground response analyses. Limited studies have shown that moisture content might also influence the maximum shear modulus Gmax (Zhao et al., 2002; Zhu et al., 2011; Huang et al., 2013; Wilson, 1982; Wang et al., 2002). Some studies showed that there existed an optimal moisture content at which Gmax attained its peak value (Huang et al., 2013). Some studies showed that beyond certain threshold value, the influence of the moisture content was small (Zhu et al., 2011). Compared to the studies on shear modulus, fewer studies can be found on the influence of moisture content on the damping ratio (Zhu et al., 2011; Huang et al., 2013; Ling et al., 2015), and a definite conclusion on the influence cannot be obtained from these studies.
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Fig. 1. Photos of the RCA, (a)specimen, cooling helical tube, and basement, (b) electromagnetism vibrator, (c) pressure chamber, (d) heat shield. Table 1 Parameters of the resonant column tests. Soil type Confining pressure (kPa)
Temperature (°C)
Clay Silt Sand
20, −9, −14, −18 18, 28, 36 20, −5, −10, −15 13,18,21 20, −11, −15, −18 8, 13, 20
100 100 100
Moisture content (%)
Saturation degree Sr 0.52, 0.82, 1.0 0.63, 0.87, 1.0 0.29, 0.47, 0.73
This study is focused on the testing and application of the dynamic shear modulus and damping ratio of frozen soils. A series of resonant column tests were carried out, in which the effects of the soil temperature and moisture content on the dynamic modulus and damping ratio were investigated. The conventional Hardin-Drnevich model was revised with correction factors to model the dynamic responses of frozen soils. The revised Hardin-Drnevich model was applied in the seismic ground response analysis of a site in Alaska in the United States. The influences of the variation of the modulus and damping curves were analyzed.
Fig. 2. Particle-size distributions of the tested soils.
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Fig. 3. Parts of the resonant frequency and damping ratio data for the clay (with a moisture content of 28% at −18 °C).
Fig. 4. Typical test results and their fitting by Eq. 1 and Eq. 2.
Fig. 5. Hardin model parameters for the clay.
2. Resonant column tests on frozen soils
corresponding strain amplitude. Accuracy of the RCA had been validated before the tests by extensive calibrations. A temperature-control cold bath manufactured by the Thermo Fisher Scientific Inc., Shanghai, China, served as the main component of the temperature-control module. A thermometer was installed inside the pressure cell of the RCA, 2 cm away from the specimen. And the data were recorded in real time using a computer code. The pressure cell was packaged with insulation cotton. The temperature in the cell can be controlled between 0 °C to −25 °C, with an error of 0.1 °C. All
2.1. Apparatus The low-temperature experiments were carried out with a resonant column apparatus (RCA) as shown in Fig. 1, which was manufactured by the Global Digital Systems Ltd. Instrument, Hampshire, United Kingdom. The device is able to capture the first-order resonant frequency of a cylindrical specimen as well as the damping ratio at the 66
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Table 2 Values of αG from literature. Identification
Type
Classification (ASTM-D2487)
Ip
D50
Moisture content
Reference
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16
Sand Clay Sand Sand Silt Silt Silt Silt Clay Silt Silt Clay Sand Clay Clay Sand
SP, Poorly graded sands CL, Lean clay SP, Poorly graded sand SP, Poorly graded sand SC, Clayey sand CL-ML, Silty clay ML-OL, Organic silt CL-ML, Silty clay CL, Lean clay CH-OH, Organic clay PT, Peat CL, Lean clay SP, Poorly graded sand CL, Lean clay CL, Lean clay SP, Poorly graded sand
– – – – 14 21 24 19 20 22 – 18.5 – 10 8.4 –
2.667 0.001 1.769 0.915 0.093 0.013 0.026 0.012 8.6e-4 0.002 0.183 0.004 0.669 0.010 0.035 0.102
9.5% 23.5% – 6.8% 8.1% 15.6% 15.7% 12.8% – – – 13–23% 8.5–10.5% 10–19% 10–22% 9–18%
Erik Simonsen et al Erik Simonsen et al Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Zhu Z Zhang F Li et al Li et al Li et al
Table 3 The values of αD from literature. Identification
Type
Classification (ASTM-D2487)
Ip
D50
Moisture content
Reference
S12 S13 S17 S18 S19
Clay Sand Clay Silt Sand
CL, Lean clay SP, Poorly graded sand – – –
18.5 – – 5–20 –
0.004 0.669 – 0.029 –
13–23% 8.5–10.5% – – –
Zhu Z Zhang F Al-Hunaidi M.O Wilson C.R Vinson T.S
Table 4 Information of earthquake events. ID
Time
EQ1 EQ2 EQ3 EQ4 EQ5 EQ6
2006-07-27 2009-06-22 2010-04-07 2013-06-27 2015-11-06 2013-09-11
13:18:01 19:28:05 16:19:16 11:40:46 14:26:50 01:03:00
(UTC) (UTC) (UTC) (UTC) (UTC) (UTC)
Magnitude
Depth (km)
Distance (km)
Direction
4.70 5.4 4.6 4.40 4.4 4
35.99 91.49 42.73 46.40 87.17 24.54
13.26 64.59 35.32 11.88 46.1 36.9
EW EW EW EW EW EW
soils originally were 2 m below the ground surface, and were disturbed during sampling. In the laboratory, the soils were sieved, and soil fractions with particle size lower than 2 mm were saved for the tests. Standard sand manufactured in Fujian, China was used as the sandy soil in the tests. Fig. 2 shows the particle-size-distribution of the tested soils. Some parameters of the tested soils are as follows: The dry densities of the soils are 1.45 g/cm3, 1.64 g/cm3, and 1.55 g/cm3 for the clay, silt, sand, respectively; the plastic limit of the clay is 17%, and its liquid limit is 51%; the plastic limit of the silt is 21.2%, and its liquid limit is 30.5%. The remolded soil specimens were prepared according to the Chinese Standard for Soil Test Methods (GB/T50123–1999) (Chinese Ministry of Construction, 1999). For the clay and silt, they were dried in an electric oven for 8 h before specimen preparation. Then 350 g of dry soil was taken to a salver, and a certain quantity of distilled water, which was calculated based on the required moisture content, was sprayed onto the soils. The soils were then left undisturbed in a sealed condition for 12 h, so that moisture in the soil diffused uniformly. Then the moist soil was poured into a Ф50 mold and compacted. All the clay and silt specimens were preserved in a moisturized chamber before use. For the sand, specimens were prepared directly on the RCA pedestal. The moisturizing procedure was similar to those of the clay and silt. The specimen mold with a rubber membrane attached inside was placed directly on the RCA pedestal. The air between the membrane and mold was eliminated by a vacuum pump. Then the sandy soil was poured into the mold and compacted.
Table 5 Parameters of the MCE spectrum of Anchorage, Alaska. Seimic hazard
PGA(g)
Ss(g)
Sl(g)
MCE (2% PE in 50 Years)
0.6
1.5
0.678
the cables and connectors were maintained waterproof in the tests. For soil testing, it is suggested that the mass polar moment of inertia of the base should be enhanced as much as possible (Yuan et al., 2006; Clayton et al., 2009). Therefore the RCA used in this study was connected to a stainless steel workbench, the mass polar moment of inertia of which is about 46.4 (kg⋅m2). Theory of wave propagation can be employed to determine the shear modulus of soil from the resonant frequency (Novak and Kim, 1981). The natural attenuation vibration method, with the assistance of the logarithmic decrement, was used to determine the damping ratio. The tests were carried out with increasing input excitation, so that the shear modulus reduction curve and the damping ratio curve were obtained using a single specimen. 2.2. Soil specimens In the present study, clay, silt and sand were used in the tests. The clay and silt were taken from a highway construction site in Harbin, China. The 67
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Fig. 6. Hardin model parameters for the silt.
Fig. 7. Hardin model parameters for the sand.
Fig. 8. Correction factors of the clay.
Fig. 9. Correction factors of the silt.
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Fig. 10. Correction factors of the sand.
Fig. 11. Error propagation in the model for the clay.
Fig. 14. Values of αD for the soils from literature.
Fig. 12. Gradation curves of the soils from literature.
Fig. 13. Values of αG for the soils from literature: (a) clay, (b) silt, (c) sand. 69
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Fig. 15. Typical profile of site in permafrost region, Anchorage (Yang et al., 2008).
Fig. 16. Selected EW original record of earthquake time-histories.
10 min. Then the confining pressure was increased to 100 kPa, followed by 8 h of consolidation for the clay and silt specimens, or 1 h for the sand specimen. Afterwards, the specimens were kept in freezing temperature for 24 h, while the 100 kPa confining pressure was maintained during this step. The freezing occurred in a closed system without water supply from the drainage valves.
2.3. Consolidation and freezing After a specimen was mounted on the RCA, the bottom drainage valves were opened, and 20 kPa negative pressure was applied to the specimen by vacuum pump to prevent disturbance on the specimen. When the RCA was completely installed, a confining pressure of 50 kPa was applied on the specimen, and the negative pressure was removed via the bottom drainage valves. The pressure was maintained for 70
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Fig. 17. Response spectra of the bedrock motions with 5% damping, (a) original record, (b) after normalization.
Fig. 18. Nonlinear curves applied for the soil layers. The 5th layer uses G/Gmax of “Clay (Seed & Sun 1989)” and D of “Clay (Idriss 1990)”. The 6th layer uses G/Gmax of “Clay (PI=5-10 Sun et al.)” and D of “Clay (Idriss 1990)”.
3. Test results
soil, but the influence was not large. It is necessary to mention that an abrupt change existed at about 0 °C, which was the phase transformation point (Bing and Ma, 2011; Cui and Li, 1994). When the temperature was above 0 °C, the parameters were independent of the temperature, which is consistent with previous experimental findings (Zhang and Hulsey, 2014; Simonsen et al., 2002).
Table.1 shows the testing parameters, which were assumed based on the ground conditions in some frozen sites in China. Resonant column tests were carried out according to the parameters in Table 1 with varying magnitudes of strain. Eq. 1 and Eq. 2 are commonly used to approximate the modulus reduction (G/Gmax) and damping curves of soils, as proposed by Hardin et al. (Hardin and Drnevich, 1972). These two equations were used to fit the test data.
4. Modified Hardin's model for the frozen soil 4.1. Empirical correction factors
G 1 = Gmax 1 + γ / γr
(1)
It is noticed that in the range tested in this study the moisture content had much smaller influence on the G/Gmax and D curves than the temperature. Considering only the temperature effects, empirical correction factors can be introduced into Eq. 1 and Eq. 2. The relationships between the curve parameters and temperature are given in Eq. 3 to Eq. 5.
n
G ⎞ D = Dmax ⎛1 − Gmax ⎠ ⎝ ⎜
⎟
(2)
In Eq. 1 and Eq. 2, γ is the shear strain, and γr is the reference strain. G is the dynamic shear modulus (MPa). Gmax is the maximum shear modulus (MPa), which is the modulus corresponding to negligible shear strain; D is the damping ratio; Dmax is the asymptotic value of the damping ratio as the shear strain increases; and n is a fitting parameter. Fig. 3 presents parts of the test results on the clay. All the test results of G and D were fitted by Eq. 1 and Eq. 2. Fig. 4 is the curve fitting results of the typical specimens in the frozen states. It can be seen that these two equations work well with the test data. Fig. 5 to Fig. 7 show the main parameter values of the fitting curves for the tested soils. From Fig. 5 to Fig. 7, it can be seen that Gmax, γr and Dmax were very sensitive to the soil temperature, but in the tested range, the degree of saturation did not affect the model parameters considerably. With a decrease of temperature from 0 °C to −18 °C, Gmax increases significantly, and γr and Dmax decrease considerably. In the tested range, the degree of saturation mainly affected Gmax and Dmax of the sandy
GF max (T ) = αG (T ) × Gmax (0)
(3)
γFr (T ) = α γ (T ) × γr (0)
(4)
DF max (T ) = αD (T ) × Dmax (0)
(5)
In Eq. 3 to Eq. 5, αG, αγ and αD are the empirical correction factors for Gmax, γr and Dmax, respectively. GFmax, γFr and DFmax are the initial modulus, reference strain amplitude, and maximum damping ratio of the frozen soil, respectively. Gmax(0), γr(0) and Dmax(0) are the maximum shear modulus, reference strain amplitude, and maximum damping ratio of the soil in an unfrozen state, respectively. The subscript ‘F’ represents the frozen state. According to the test results, it was found that an exponential-type function was able to depict the temperature effects, as shown in Eq. 6. 71
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Fig. 19. Nonlinear curves of layers 1–4, (a)(b) 1st, (c)(d) 2nd, (e)(f) 3rd, (g)(h) 4th.
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ΔG Fmax ΔαG = G Fmax αG
(8)
γ Δ(G/GFmax ) = ⋅Δα γ (G/GFmax ) α γ (α γ γr + γ )
(9)
nγr ΔD ΔαD = − ⋅Δα γ D αD α γ γr + γ
It can be seen that the error of GFmax induced by the error ΔαG is linear, while the errors of G/GFmax and D induced by Δαγ and ΔαD are nonlinear. The errors of G/GFmax and D are presented in Fig. 11 for the clay. In Fig. 11(a), the RE's of αγ were set to 20%, 40%, and 60%, with a temperature range of −18 °C ~ −2 °C. The RE's propagation to the G/ GFmax is less than those of αγ, although varying with shear strain and temperature. Similar phenomenon can be observed for the RE of damping in Fig.11(b). There existed a lower limit for the RE of αD propagating to the D curve, as shown in Eq.(11).
Fig. 20. Variation of Vs in the analysis.
αopt = y0 + A0 eT / r , (opt = G, γ , D)
Δα γ ΔD ΔαD = − ⋅n, (γ = 0) D αD αγ
(6)
Here, T is temperature (°C). y0, A0, r are the curve-fitting parameters. The subscript ‘opt’ represents the optional term G, γ and D, where αG stands for the initial modulus correction factor, αγ stands for the reference strain correction factor, and αD stands for the damping correction factor. Genetic Algorithm (GA), which is widely employed to calculate undetermined coefficients in curve-fitting (Rokonuzzaman and Sakai, 2010), was used to estimate the coefficients in Eq. 6. Fig. 8 to Fig. 10 show the fitted results with Eq. 6 for the tested soils, in which all the Rsquares of the results are larger than 0.99. It can be seen that αG increased significantly with a decrease in the temperature, but the influence of temperature on αγ was much smaller. αD decreased with a decrease in the temperature.
5. Analysis of the data from literature 5.1. Maximum shear modulus Previous studies on the dynamic properties of frozen soils were mostly focused on the maximum shear modulus, while few results are available for the modulus reduction curves and damping curves. The test data on the maximum shear modulus were therefore analyzed with the model in section 3, and the values of αG of 16 kinds of soils available in the literature (Ling et al., 2009; Kaplar, 1969; Ling et al., 2015; Simonsen et al., 2002; Li et al., 2016) are shown in Table 2. Gradation curves of these soils are presented in Fig. 12. According to ASTM-D2487 (ASTM, 2006), all the soils in Table 2 can be classified into three categories: clay, silt and sand. The values of GFmax were all obtained at a strain magnitude of about 10−6 as reported in the references, which were employed to determine the values of αG in Eq. 3 and are shown in Fig. 13. Fig. 13 shows that the value of αG is dependent on the specific type of soil, even for the same category according to ASTM-D2487 (ASTM, 2006). It can also be seen that in some cases, the moisture content played an important role in the test results. This is not difficult to understand, as frozen soils with very low degree of saturation and very high degree of saturation would certainly behave differently. But beyond certain threshold value, which is mostly the case in the soils tested
The errors of the empirical correction factors would propagate to the G/GFmax and D curves, which may be described by the first order Taylor expansion. N
∑ 1
∂lnf ⋅Δxi ∂xi
(11)
It can thus be concluded that from the accuracy point of view, the importance order of the correction factors, from high to low, is αG, αD and αγ.
4.2. Error propagation
Δf = f
(10)
(7)
The left-hand-side of Eq. 7 is the relative errors (RE) of the G/GFmax and D, where f represents the G/GFmax or D function containing the correction factors. The xi represents the three correction factors, αG, αγ or αD. The Δf and Δxi represent the absolute errors, which are given as the following:
Fig. 21. Some spectral accelerations at the ground surface with 0.2 g input and varying αG, (a) δ= − 50%, (b) δ=0%, (c) δ=50%. 73
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Fig. 22. Average spectra at the ground surface with varying αG, (a) 0.05 g, (b) 0.2 g, (c) 0.4 g.
2011). However, previous studies were mostly restricted to the analyses using limited shear modulus and damping data of frozen soils, and the influences of the variations of the modulus and damping curves have seldom been attempted (Sun et al., 2009). The revised Hardin-Drnevich model can be widely used in the seismic ground response analysis of permafrost sites. However as shown in section 4, the correction factors may vary in a wide range even for a similar soil category. Therefore, a case study was carried out on a permafrost site. The one-dimensional equivalent-linear computer code Proshake was employed for the seismic ground response analysis (Systems, 2017).
in this study in last section, the moisture content would not be very important to the frozen soil behavior. 5.2. Maximum damping ratio Fewer data are available on the maximum damping ratio in the literature (Ling et al., 2009; Al Hunaidi et al., 1996; Wilson, 1982; Ling et al., 2015; Vinson et al., 1978), but they were also analyzed by the model in section 3, and the results of αD are presented in Table 3 and Fig. 14. Similarly to the value of αG, the value of αD also varied considerably for different specific types of soil, even if they belong to similar soil category according to ASTM-D2487 (ASTM, 2006).
6.1. Study site
6. Model application and sensitivity analysis
A permafrost site in Alaska of the United States was analyzed for the seismic response. The Delaney Park Alaska Digital Array (DPK, NEES@ UCSB) is located in downtown Anchorage, Alaska. Yang et al. (Xu et al., 2011; Yang et al., 2008) studied the effects of frozen soils on the seismic response of the site. This site was employed as a reference for this study, at which the soil profile with permafrost was established, as shown in Fig. 15.
The influence of the frozen soil would be larger for the seismic ground response of permafrost sites than that of seasonally frozen sites (Qi et al., 2006; Xu et al., 2011). Some studies found that in the worst case scenario, the average values of spectral accelerations for the periods between 0.4–1.0 s might be much larger than the seismic design spectral values that neglected the effects of frozen soil (Yang et al., 74
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Fig. 23. Average spectra at the ground surface with varying αγ, αD, (a) 0.05 g, (b) 0.2 g, (c) 0.4 g.
Here αi refers to αG, αγ or αD. αio is the base-line value of αi according to Fig. 8–10. δi is the fluctuation of αi, which is set to ± 10%, ± 30% and ± 50% artificially. It is to be noted that αD is fixed at 1.0 when their value are beyond 1.0. Fig. 18 shows the G/Gmax and D curves of the unfrozen layers according to the manual of Proshake, while the curves corrected by αi are present in Fig. 19. It can be seen that the G/Gmax and D curves vary considerably. The effects of confining stress on the correction factors were neglected in this study according to previous findings (Zhu et al., 2011). Fig. 20 presents the varying shear wave velocity used in the analysis, which was obtained using Gmax and soil density. The Vs below -30 m was identical, as this range was out of the frozen region.
6.2. Input motion The DPK array has recorded about 2000 seismic events within the range of 100 km until the end of 2016. Six of these bedrock records were selected in the present study, with the moment magnitude Mw ranging from 4.0 to 6.0. The information of these recorded motions are presented in Fig. 16, Fig. 17 and Table 4. The time-histories of the recorded motions were pre-processed with baseline correction and amplitude modulation. Their peak accelerations were scaled to 0.05 g, 0.2 g and 0.4 g to investigate the influence of input intensity. 6.3. Nonlinear properties of soils
6.4. Results Ground temperature was determined based on the studies by Osterkamp (Osterkamp, 2005) and Hulsey (Hulsey et al., 2011), as shown in Fig. 15. In this study, the correction factors from Fig. 8–10 were employed as the bases. As shown in Section 4, similar category of soil may have very different correction factors. To take into account the influence of the uncertainty, variations of the correction factors were carried out according to Eq. 12:
αi = (1 + δi ) αio
Two sets of results are presented in the following. The first set was obtained by varying αG (δ = ± 10%, ± 30, ± 50%) but fixing αγ and αD (δ = 0%), while the other one was obtained by varying αγ and αD but fixing αG. The response spectra at the ground surface are presented in Fig. 21 to Fig. 21. It can be seen in Fig. 21 that the spectral accelerations at the ground surface were different for different seismic inputs. Average spectra, which may grasp their main characteristics in the period between 0.1 s
(12) 75
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Acknowledgements
and 1.0 s, were therefore employed in the following to discuss the influence of the correction factors. As shown in Fig. 22, the effect of αG was significant for the studied site. The result shows that it may be necessary to take into account the different shear wave velocities of frozen soils in the seismic site classification in a permafrost region. The finding is different from some previous ones (Xu et al., 2011; Yang et al., 2011), which may be explained by the differences in the sites and the input seismic motions. The thickness of frozen soils, their shear wave velocities, and the input motions in the previous studies (Xu et al., 2011; Yang et al., 2011) were all different from the ones in this study. Fig. 22 also presents the spectral response assuming that the soils were unfrozen. The difference is very significant, further pinpointing the importance of considering frozen soils in the seismic site classification of permafrost regions. Uncertainties in the estimation of αγ and αD also commonly exist in frozen soils as discussed in Section 4. Fig. 21 shows their influences on the average seismic spectral responses at the ground surface. The average spectra were obtained similarly to those in Fig. 22. Although similar to the effects of varying αG, the variations of αγ and αD had smaller influences on the response spectra. The influences also increased with an increase in the input acceleration. This is understandable, since with small seismic input, the ground response was close to linear, and the effect of shear wave velocity would dominate. The frozen spectral accelerations were mostly lower than the unfrozen ones and the MCE (Maximum credible earthquake) spectrum in the region according to NEHRP (Building Seismic Safety Council (BSSC), 2015). The unfrozen site belongs to Site Class D. The parameters of the MCE spectrum were listed in Table 5. It is suggested that the seismic response analysis without considering frozen layers may be conservative in some cold regions. However, this conclusion may not be universal, depending on the frequency characteristics of the input motions (See Fig. 23)
The authors are appreciated to the support from the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (Grant No.2016A02). The records of seismic motions were obtained from NEES@UCSB (http://www.nees.ucsb. edu). Reference Al Hunaidi, M.O., Chen, P.A., Rainer, J.H., et al., 1996. Shear moduli and damping in frozen and unfrozen clay by resonant column test. Can. Geotech. J. 33, 510–514. http://dx.doi.org/10.1139/t96-073. ASTM, 2006. Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System). http://www.dres.ir/fanni/khak/DocLib4/D %202487%20%E2%80%93%2098%20%20;RDI0ODCTUKVE.pdf. Bing, H., Ma, W., 2011. Laboratory investigation of the freezing point of saline soil. Cold Reg. Sci. Technol. 67 (1), 79–88. http://dx.doi.org/10.1016/j.coldregions.2011.02. 008. Building Seismic Safety Council (BSSC), 2015. NEHRP recommended provisions for new buildings and other structures, Volume I: Part 1 (Provisions) and Part 2(Commentary). Federal Emergency Management Agency, Washington D.C. https:// www.fema.gov/media-library/assets/documents/107646. Chinese Ministry of Construction, 1999. Standard for Soil Test Method (GB/ T50123–1999). China Planning Press, Beijing (In Chinese). Clayton, C.R.I., Zervos, A., Kim, S.G., et al., 2009. The Stokoe resonant column apparatus: effects of stiffness, mass and specimen fixity. Géotechnique 59 (5), 429–437. http:// dx.doi.org/10.1680/geot.2007.00096. Cui, G., Li, Y., 1994. The study on freezing point of wet sand under load. J. Glaciol. Geocryol., vol. 16 (4), 320–326. (In Chinese with English Abstract). http://bcdt. westgis.ac.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=2023. Hardin, B.O., Drnevich, V.P., 1972. Shear modulus and damping in soil: measurement and parameter effects. ASCE Soil Mech. Found. Div. J. 98 (6), 603–624. Huang, X., Li, D., Ming, F., et al., 2013. An experimental study on the relationship between acoustic parameters and mechanical properties of frozen silty clay. Sci. Cold Arid Reg. 5 (5), 596–602. http://www.scar.ac.cn/hhkxen/ch/reader/create_pdf. aspx?file_no=2013514&flag=1&journal_id=hhkxen&year_id=2013. Hulsey, J.L., Horzdovsky, J.E., Davis, D., et al., 2011. Seasonally Frozen Soil Effects on the Seismic Performance of Highway Bridges. http://tundra.ine.uaf.edu/wp-content/ uploads/2012/11/107014.-Hulsey-Yang.Dec_.2011.pdf. Kaplar, C.W., 1969. Laboratory Determination of Dynamic Moduli of Frozen Soils and of Ice. Cold Regions Research And Engineering Lab, Hanover NH. Li, S., Gao, L., Chai, S., 2012. Significance and interaction of factors on mechanical properties of frozen soil. Rock Soil Mech., vol. 33 (4), 1173–1177. (In Chinese with English Abstract). 10.16285/j.rsm.2012.04.017. Li, D., Xing, H., Feng, M., et al., 2016. The impact of unfrozen water content on ultrasonic wave velocity in frozen soils. Procedia Eng. 143, 1210–1217. http://dx.doi.org/10. 1016/j.proeng.2016.06.114. Ling, X., Zhu, Z., Zhang, F., et al., 2009. Dynamic elastic modulus for frozen soil from the embankment on Beiluhe Basin along the Qinghai–Tibet Railway. Cold Reg. Sci. Technol. 57 (1), 7–12. http://dx.doi.org/10.1016/j.coldregions.2009.01.004. Ling, X., Zhang, F., Li, Q., et al., 2015. Dynamic shear modulus and damping ratio of frozen compacted sand subjected to freeze–thaw cycle under multi-stage cyclic loading. Soil Dyn. Earthq. Eng. 76, 111–121. http://dx.doi.org/10.1016/j.soildyn. 2015.02.007. Ma, W., Wang, D., 2012. Studies on frozen soil mechanics in China in past 50 years and their prospect. Chin. J. Geotech. Eng., vol. 34 (4), 625–640. (In Chinese with English Abstract). http://manu31.magtech.com.cn/Jwk_ytgcxb/CN/Y2012/V34/I4/625. Novak, M., Kim, T.C., 1981. Resonant column technique for dynamic testing of cohesive soils. Can. Geotech. J. 18 (3), 448–455. http://dx.doi.org/10.1139/t81-049. Osterkamp, T.E., 2005. The recent warming of permafrost in Alaska. Glob. Planet. Chang. 49 (3–4), 187–202. http://dx.doi.org/10.1016/j.gloplacha.2005.09.001. Qi, J., Ma, W., 2010. State-of-art of research on mechanical properties of frozen soils. Rock Soil Mech., vol. 31 (1), 133–143. (In Chinese with English Abstract). 10.16285/ j.rsm.2010.01.036. Qi, J., Ma, W., Sun, C., et al., 2006. Ground motion analysis in seasonally frozen regions. Cold Reg. Sci. Technol. 44 (2), 111–120. http://dx.doi.org/10.1016/j.coldregions. 2005.09.003. Rokonuzzaman, M., Sakai, T., 2010. Calibration of the parameters for a hardening–softening constitutive model using genetic algorithms. Comput. Geotech. 37 (4), 573–579. http://dx.doi.org/10.1016/j.compgeo.2010.02.007. Senetakis, K., Anastasiadis, A., Pitilakis, K., 2013. Normalized shear modulus reduction and damping ratio curves of quartz sand and rhyolitic crushed rock. Soils Found. 53 (6), 879–893. http://dx.doi.org/10.1016/j.sandf.2013.10.007. Simonsen, E., Janoo, V.C., Isacsson, U., 2002. Resilient properties of unbound road materials during seasonal frost conditions. J. Cold Reg. Eng. 16 (1), 28–50. http://dx. doi.org/10.1061/(ASCE)0887-381X(2002)16:1(28). Stokoe, K.H., Darendeli, M.B., Andrus, R.D., Brown, L.T., 1999. Dynamic soil properties: laboratory, field and correlation studies. In: In Proceedings of the 2nd International Conference on Earthquake Geotechnical Engineering, pp. 811–846. Subramaniam, P., Banerjee, S., 2013. Shear modulus degradation model for cohesive soils. Soil Dyn. Earthq. Eng. 53 (10), 210–216. http://dx.doi.org/10.1016/j.soildyn. 2013.07.003.
7. Summary and conclusions The goal of this study is to propose a revised Hardin-Drnevich model for the dynamic shear modulus and damping ratio of frozen soils. A series of low temperature RC tests were carried out considering the influence of the soil temperature and moisture content. Based on the test results, empirical correction factor αG, αγ and αD were proposed to revise the Hardin-Drnevich model. The αG and αD for other frozen soils were also collected from previous studies. The influences of the correction factors and their variations on the seismic response of a permafrost site were analyzed. The following conclusions may be obtained from this study. (1) The test results show that soil temperature had large effects on the maximum shear modulus, modulus reduction curves and damping curves of sand, silt and clay. With a decrease in the soil temperature, GFmax increased, while γFr and DFmax decreased. The effects of the moisture content were small and could be neglected for the range of degree of saturation investigated in this study. (2) The proposed correction factors matched preferably with the experimental data and those from previous studies. With the correction factors, the revised Hardin-Drnevich model may be employed to analyze shear modulus and damping ratio of frozen soils at different soil temperatures. However, unique functions of correction factors could not be found for one category of frozen soil, for example clay. They varied considerable for different specific frozen soils. (3) Compared with the correction factors for the modulus reduction curve and damping curve, the correction factor for the maximum shear modulus or shear wave velocity had a larger effect on the seismic ground response of a permafrost site. The response spectral values at the ground surface were generally larger if the investigated site was assumed to be non-frozen. 76
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response. J. Cold Reg. Eng. 25 (2), 53–70. http://dx.doi.org/10.1061/(ASCE)CR. 1943-5495.0000022. Yang, Z., Dutta, U., Xiong, F., et al., 2008. Seasonal frost effects on the dynamic behavior of a twenty-story office building. Cold Reg. Sci. Technol. 51 (1), 76–84. http://dx.doi. org/10.1016/j.coldregions.2007.05.001. Yang, Z.J., Dutta, U., Xu, G., 2010. Effects of Permafrost and Seasonally Frozen Ground on the Seismic Response of Transportation Infrastructure Sites. University of Alaska Fairbanks. http://tundra.ine.uaf.edu/wp-content/uploads/2012/09/107017Frozen-Ground-Permafrost-yang.hazirbaba-use-INEAUTC1103.pdf. Yang, Z., Dutta, U., Xu, G., et al., 2011. Numerical analysis of permafrost effects on the seismic site response. Soil Dyn. Earthq. Eng. 31 (3), 282–290. http://dx.doi.org/10. 1016/j.soildyn.2010.08.004. Yuan, X., Sun, J., Sun, R.A., 2006. Modified approach for calculating dynamic shear modulus of stiff specimens by resonant column tests. Earthq. Eng. Eng. Vib. 5 (1), 143–150. http://dx.doi.org/10.1007/s11803-006-0472-x. Zhang, Y., Hulsey, J.L., 2014. Temperature and freeze-thaw effects on dynamic properties of fine-grained soils. J. Cold Reg. Eng. 29 (2), 04014012. http://dx.doi.org/10.1061/ (ASCE)CR.1943-5495.0000077. Zhao, S., Zhu, Y., He, P., et al., 2002. Recent progress and suggestion in the research on dynamic response of frozen soil. J. Glaciol. Geocryol. 24 (5), 681–685. http:// research.iarc.uaf.edu/NICOP/DVD/ICOP%202003%20Permafrost/Pdf/Chapter_228. pdf. Zhu, Z., Ling, X., Wang, Z., et al., 2011. Experimental investigation of the dynamic behavior of frozen clay from the Beiluhe subgrade along the QTR. Cold Reg. Sci. Technol. 69 (1), 91–97. https://doi.org/10.1016/j.coldregions.2011.07.007.
Sun, R., Yuan, X., Liu, X., 2009. Effects of dynamic shear modulus ratio and velocity on surface ground motion and their equivalent relations. Chin. J. Geotech. Eng., vol. 31 (8), 1267–1274. (In Chinese with English Abstract). http://www.cgejournal.com/ CN/Y2009/V31/I8/1267. Systems, EduPro Civil, 2017. Proshake: Ground Response Analysis Program Version 2.0 user's Manual, Sammamish, WA. http://www.proshake.com/proshake_2.0/User %20Manual.pdf. Vinson, T., Chaichanavong, T., Li, J., 1978. Dynamic Testing of Frozen Soils under Simulated Earthquake Loading Conditions. In: Silver, M., Tiedemann, D. (Eds.), STP35678S Dynamic Geotechnical Testing, pp. 196–227. http://dx.doi.org/10.1520/ STP35678S. Wang, G.X., Kuwano, J., 1999. Modeling of strain dependency of shear modulus and damping of clayey sand. Soil Dyn. Earthq. Eng. 18 (6), 463–471. http://dx.doi.org/ 10.1016/S0267-7261(99)00010-X. Wang, D., Zhu, Y., Zhao, S., et al., 2002. Study on experimental determination of the dynamic elastic mechanical parameters of frozen soil by ultrasonic technique. Chin. J. Geotech. Eng., vol. 24 (5), 612–615. (In Chinese with English Abstract). http:// manu31.magtech.com.cn/Jwk_ytgcxb/CN/Y2002/V24/I5/612. Wichtmann, T., Hernández, M.A.N., Triantafyllidis, T., 2015. On the influence of a noncohesive fines content on small strain stiffness, modulus degradation and damping of quartz sand. Soil Dyn. Earthq. Eng. 69 (2), 103–114. http://dx.doi.org/10.1016/j. soildyn.2014.10.017. Wilson, C.R., 1982. Dynamic Properties of Naturally Frozen Fairbanks Silt. Oregon State University, Corvallis, Oregon, United States (Masters Thesis). http://ir.library. oregonstate.edu/xmlui/bitstream/1957/9497/1/Wilson_Charles_R_1982.pdf. Xu, G., Dutta, U., Yang, Z., et al., 2011. Seasonally frozen soil effects on the seismic site
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Contents lists available at ScienceDirect
Cold Regions Science and Technology journal homepage: www.elsevier.com/locate/coldregions
Improved Hardin-Drnevich model for the dynamic modulus and damping ratio of frozen soil
T
⁎
Xiaobo Yua,b, Huabei Liua, , Rui Sunb, Xiaoming Yuanb a b
School of civil engineering & mechanics, Huazhong University of Science & Technology, Wuhan 430074, China Key Laboratory of Earthquake Engineering and Engineering Vibration, Institute of Engineering Mechanics, China Earthquake Administration, Harbin 150080, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Hardin-Drnevich model Frozen soil Resonant column test Empirical correction factor Seismic ground response
The Hardin-Drnevich model is frequently employed in the seismic ground response analysis, but the original model cannot be directly applied to frozen soils, the dynamic properties of which are functions of the soil temperature. Resonant column tests were carried out and empirical correction factors were proposed to revise the model. The improved model was applied to the seismic ground response analysis of a permafrost site, and the influences of the variations of the correction factors were analyzed. The resonant column test results indicated that soil temperature had large influences on the shear modulus and damping ratio, while in the range investigated in this study, the effect of the degree of saturation was not significant. The proposed correction factors properly fitted the test results in the present and previous studies, but the correction factors varied considerably even for the same category of clayey or silty or sandy soils. Compared with the correction factors for the modulus reduction curve and damping curve, the correction factor for the maximum shear modulus had a larger effect on the seismic ground response of a permafrost site. In the seismic response analyses, the response spectral values at the ground surface were generally larger if the investigated site was assumed to be non-frozen.
1. Introduction Hardin et al. (1972) proposed a hyperbolic model to depict the shear modulus reduction and damping evolution, which was widely used with various modifications in seismic ground response analyses (Hardin and Drnevich, 1972; Stokoe et al., 1999; Wichtmann et al., 2015; Senetakis et al., 2013; Subramaniam and Banerjee, 2013; Wang and Kuwano, 1999). With the increasing infrastructure development in cold regions, it is necessary to quantitatively describe the dynamic properties of frozen soils, and a model similar to that by Hardin et al. (1972) may provide an important tool for this purpose. However, most of the previous modifications of the Hardin's model have been focused on non-frozen soils. A modified Hardin's model for frozen soils accounting for the important factors that influence the dynamic properties is therefore called for in order to satisfy the need for the constructions in cold regions. Previous studies have shown that the strain amplitude, temperature and moisture content are several factors that may influence the dynamic modulus and damping ratio of the frozen soil (Li et al., 2012). The influence of strain amplitude reflects the nonlinear properties of soil, which is already accounted for in the Hardin's model (Ling et al., 2009; Al Hunaidi et al., 1996; Kaplar, 1969; Yang et al., 2010).
⁎
Corresponding author. E-mail address: [email protected] (H. Liu).
https://doi.org/10.1016/j.coldregions.2018.05.004 Received 14 May 2017; Received in revised form 26 April 2018; Accepted 18 May 2018 Available online 23 May 2018 0165-232X/ © 2018 Elsevier B.V. All rights reserved.
In the frozen soil, the liquid phase transfers into the ice phase, the extent of which is dependent on the soil temperature. Previous studies (Qi and Ma, 2010; Ma and Wang, 2012; Zhao et al., 2002) have concluded that, the dynamic modulus of the frozen soil is larger than that of the unfrozen soil and it increases with a decrease in the soil temperature. In contrast, the damping ratio of the frozen soil decreases with a decrease in the temperature. Previous studies were mostly focused on the maximum shear modulus Gmax and its relationship with the frozensoil temperature, while few attempts have been directed to investigate the evolutions of the G/Gmax (shear modulus ratio) and D (damping ratio) curves of the frozen soil. However, the G/Gmax and D curves are as important as Gmax in seismic ground response analyses. Limited studies have shown that moisture content might also influence the maximum shear modulus Gmax (Zhao et al., 2002; Zhu et al., 2011; Huang et al., 2013; Wilson, 1982; Wang et al., 2002). Some studies showed that there existed an optimal moisture content at which Gmax attained its peak value (Huang et al., 2013). Some studies showed that beyond certain threshold value, the influence of the moisture content was small (Zhu et al., 2011). Compared to the studies on shear modulus, fewer studies can be found on the influence of moisture content on the damping ratio (Zhu et al., 2011; Huang et al., 2013; Ling et al., 2015), and a definite conclusion on the influence cannot be obtained from these studies.
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Fig. 1. Photos of the RCA, (a)specimen, cooling helical tube, and basement, (b) electromagnetism vibrator, (c) pressure chamber, (d) heat shield. Table 1 Parameters of the resonant column tests. Soil type Confining pressure (kPa)
Temperature (°C)
Clay Silt Sand
20, −9, −14, −18 18, 28, 36 20, −5, −10, −15 13,18,21 20, −11, −15, −18 8, 13, 20
100 100 100
Moisture content (%)
Saturation degree Sr 0.52, 0.82, 1.0 0.63, 0.87, 1.0 0.29, 0.47, 0.73
This study is focused on the testing and application of the dynamic shear modulus and damping ratio of frozen soils. A series of resonant column tests were carried out, in which the effects of the soil temperature and moisture content on the dynamic modulus and damping ratio were investigated. The conventional Hardin-Drnevich model was revised with correction factors to model the dynamic responses of frozen soils. The revised Hardin-Drnevich model was applied in the seismic ground response analysis of a site in Alaska in the United States. The influences of the variation of the modulus and damping curves were analyzed.
Fig. 2. Particle-size distributions of the tested soils.
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Fig. 3. Parts of the resonant frequency and damping ratio data for the clay (with a moisture content of 28% at −18 °C).
Fig. 4. Typical test results and their fitting by Eq. 1 and Eq. 2.
Fig. 5. Hardin model parameters for the clay.
2. Resonant column tests on frozen soils
corresponding strain amplitude. Accuracy of the RCA had been validated before the tests by extensive calibrations. A temperature-control cold bath manufactured by the Thermo Fisher Scientific Inc., Shanghai, China, served as the main component of the temperature-control module. A thermometer was installed inside the pressure cell of the RCA, 2 cm away from the specimen. And the data were recorded in real time using a computer code. The pressure cell was packaged with insulation cotton. The temperature in the cell can be controlled between 0 °C to −25 °C, with an error of 0.1 °C. All
2.1. Apparatus The low-temperature experiments were carried out with a resonant column apparatus (RCA) as shown in Fig. 1, which was manufactured by the Global Digital Systems Ltd. Instrument, Hampshire, United Kingdom. The device is able to capture the first-order resonant frequency of a cylindrical specimen as well as the damping ratio at the 66
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Table 2 Values of αG from literature. Identification
Type
Classification (ASTM-D2487)
Ip
D50
Moisture content
Reference
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16
Sand Clay Sand Sand Silt Silt Silt Silt Clay Silt Silt Clay Sand Clay Clay Sand
SP, Poorly graded sands CL, Lean clay SP, Poorly graded sand SP, Poorly graded sand SC, Clayey sand CL-ML, Silty clay ML-OL, Organic silt CL-ML, Silty clay CL, Lean clay CH-OH, Organic clay PT, Peat CL, Lean clay SP, Poorly graded sand CL, Lean clay CL, Lean clay SP, Poorly graded sand
– – – – 14 21 24 19 20 22 – 18.5 – 10 8.4 –
2.667 0.001 1.769 0.915 0.093 0.013 0.026 0.012 8.6e-4 0.002 0.183 0.004 0.669 0.010 0.035 0.102
9.5% 23.5% – 6.8% 8.1% 15.6% 15.7% 12.8% – – – 13–23% 8.5–10.5% 10–19% 10–22% 9–18%
Erik Simonsen et al Erik Simonsen et al Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Chester W. Kaplar Zhu Z Zhang F Li et al Li et al Li et al
Table 3 The values of αD from literature. Identification
Type
Classification (ASTM-D2487)
Ip
D50
Moisture content
Reference
S12 S13 S17 S18 S19
Clay Sand Clay Silt Sand
CL, Lean clay SP, Poorly graded sand – – –
18.5 – – 5–20 –
0.004 0.669 – 0.029 –
13–23% 8.5–10.5% – – –
Zhu Z Zhang F Al-Hunaidi M.O Wilson C.R Vinson T.S
Table 4 Information of earthquake events. ID
Time
EQ1 EQ2 EQ3 EQ4 EQ5 EQ6
2006-07-27 2009-06-22 2010-04-07 2013-06-27 2015-11-06 2013-09-11
13:18:01 19:28:05 16:19:16 11:40:46 14:26:50 01:03:00
(UTC) (UTC) (UTC) (UTC) (UTC) (UTC)
Magnitude
Depth (km)
Distance (km)
Direction
4.70 5.4 4.6 4.40 4.4 4
35.99 91.49 42.73 46.40 87.17 24.54
13.26 64.59 35.32 11.88 46.1 36.9
EW EW EW EW EW EW
soils originally were 2 m below the ground surface, and were disturbed during sampling. In the laboratory, the soils were sieved, and soil fractions with particle size lower than 2 mm were saved for the tests. Standard sand manufactured in Fujian, China was used as the sandy soil in the tests. Fig. 2 shows the particle-size-distribution of the tested soils. Some parameters of the tested soils are as follows: The dry densities of the soils are 1.45 g/cm3, 1.64 g/cm3, and 1.55 g/cm3 for the clay, silt, sand, respectively; the plastic limit of the clay is 17%, and its liquid limit is 51%; the plastic limit of the silt is 21.2%, and its liquid limit is 30.5%. The remolded soil specimens were prepared according to the Chinese Standard for Soil Test Methods (GB/T50123–1999) (Chinese Ministry of Construction, 1999). For the clay and silt, they were dried in an electric oven for 8 h before specimen preparation. Then 350 g of dry soil was taken to a salver, and a certain quantity of distilled water, which was calculated based on the required moisture content, was sprayed onto the soils. The soils were then left undisturbed in a sealed condition for 12 h, so that moisture in the soil diffused uniformly. Then the moist soil was poured into a Ф50 mold and compacted. All the clay and silt specimens were preserved in a moisturized chamber before use. For the sand, specimens were prepared directly on the RCA pedestal. The moisturizing procedure was similar to those of the clay and silt. The specimen mold with a rubber membrane attached inside was placed directly on the RCA pedestal. The air between the membrane and mold was eliminated by a vacuum pump. Then the sandy soil was poured into the mold and compacted.
Table 5 Parameters of the MCE spectrum of Anchorage, Alaska. Seimic hazard
PGA(g)
Ss(g)
Sl(g)
MCE (2% PE in 50 Years)
0.6
1.5
0.678
the cables and connectors were maintained waterproof in the tests. For soil testing, it is suggested that the mass polar moment of inertia of the base should be enhanced as much as possible (Yuan et al., 2006; Clayton et al., 2009). Therefore the RCA used in this study was connected to a stainless steel workbench, the mass polar moment of inertia of which is about 46.4 (kg⋅m2). Theory of wave propagation can be employed to determine the shear modulus of soil from the resonant frequency (Novak and Kim, 1981). The natural attenuation vibration method, with the assistance of the logarithmic decrement, was used to determine the damping ratio. The tests were carried out with increasing input excitation, so that the shear modulus reduction curve and the damping ratio curve were obtained using a single specimen. 2.2. Soil specimens In the present study, clay, silt and sand were used in the tests. The clay and silt were taken from a highway construction site in Harbin, China. The 67
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Fig. 6. Hardin model parameters for the silt.
Fig. 7. Hardin model parameters for the sand.
Fig. 8. Correction factors of the clay.
Fig. 9. Correction factors of the silt.
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Fig. 10. Correction factors of the sand.
Fig. 11. Error propagation in the model for the clay.
Fig. 14. Values of αD for the soils from literature.
Fig. 12. Gradation curves of the soils from literature.
Fig. 13. Values of αG for the soils from literature: (a) clay, (b) silt, (c) sand. 69
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Fig. 15. Typical profile of site in permafrost region, Anchorage (Yang et al., 2008).
Fig. 16. Selected EW original record of earthquake time-histories.
10 min. Then the confining pressure was increased to 100 kPa, followed by 8 h of consolidation for the clay and silt specimens, or 1 h for the sand specimen. Afterwards, the specimens were kept in freezing temperature for 24 h, while the 100 kPa confining pressure was maintained during this step. The freezing occurred in a closed system without water supply from the drainage valves.
2.3. Consolidation and freezing After a specimen was mounted on the RCA, the bottom drainage valves were opened, and 20 kPa negative pressure was applied to the specimen by vacuum pump to prevent disturbance on the specimen. When the RCA was completely installed, a confining pressure of 50 kPa was applied on the specimen, and the negative pressure was removed via the bottom drainage valves. The pressure was maintained for 70
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Fig. 17. Response spectra of the bedrock motions with 5% damping, (a) original record, (b) after normalization.
Fig. 18. Nonlinear curves applied for the soil layers. The 5th layer uses G/Gmax of “Clay (Seed & Sun 1989)” and D of “Clay (Idriss 1990)”. The 6th layer uses G/Gmax of “Clay (PI=5-10 Sun et al.)” and D of “Clay (Idriss 1990)”.
3. Test results
soil, but the influence was not large. It is necessary to mention that an abrupt change existed at about 0 °C, which was the phase transformation point (Bing and Ma, 2011; Cui and Li, 1994). When the temperature was above 0 °C, the parameters were independent of the temperature, which is consistent with previous experimental findings (Zhang and Hulsey, 2014; Simonsen et al., 2002).
Table.1 shows the testing parameters, which were assumed based on the ground conditions in some frozen sites in China. Resonant column tests were carried out according to the parameters in Table 1 with varying magnitudes of strain. Eq. 1 and Eq. 2 are commonly used to approximate the modulus reduction (G/Gmax) and damping curves of soils, as proposed by Hardin et al. (Hardin and Drnevich, 1972). These two equations were used to fit the test data.
4. Modified Hardin's model for the frozen soil 4.1. Empirical correction factors
G 1 = Gmax 1 + γ / γr
(1)
It is noticed that in the range tested in this study the moisture content had much smaller influence on the G/Gmax and D curves than the temperature. Considering only the temperature effects, empirical correction factors can be introduced into Eq. 1 and Eq. 2. The relationships between the curve parameters and temperature are given in Eq. 3 to Eq. 5.
n
G ⎞ D = Dmax ⎛1 − Gmax ⎠ ⎝ ⎜
⎟
(2)
In Eq. 1 and Eq. 2, γ is the shear strain, and γr is the reference strain. G is the dynamic shear modulus (MPa). Gmax is the maximum shear modulus (MPa), which is the modulus corresponding to negligible shear strain; D is the damping ratio; Dmax is the asymptotic value of the damping ratio as the shear strain increases; and n is a fitting parameter. Fig. 3 presents parts of the test results on the clay. All the test results of G and D were fitted by Eq. 1 and Eq. 2. Fig. 4 is the curve fitting results of the typical specimens in the frozen states. It can be seen that these two equations work well with the test data. Fig. 5 to Fig. 7 show the main parameter values of the fitting curves for the tested soils. From Fig. 5 to Fig. 7, it can be seen that Gmax, γr and Dmax were very sensitive to the soil temperature, but in the tested range, the degree of saturation did not affect the model parameters considerably. With a decrease of temperature from 0 °C to −18 °C, Gmax increases significantly, and γr and Dmax decrease considerably. In the tested range, the degree of saturation mainly affected Gmax and Dmax of the sandy
GF max (T ) = αG (T ) × Gmax (0)
(3)
γFr (T ) = α γ (T ) × γr (0)
(4)
DF max (T ) = αD (T ) × Dmax (0)
(5)
In Eq. 3 to Eq. 5, αG, αγ and αD are the empirical correction factors for Gmax, γr and Dmax, respectively. GFmax, γFr and DFmax are the initial modulus, reference strain amplitude, and maximum damping ratio of the frozen soil, respectively. Gmax(0), γr(0) and Dmax(0) are the maximum shear modulus, reference strain amplitude, and maximum damping ratio of the soil in an unfrozen state, respectively. The subscript ‘F’ represents the frozen state. According to the test results, it was found that an exponential-type function was able to depict the temperature effects, as shown in Eq. 6. 71
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Fig. 19. Nonlinear curves of layers 1–4, (a)(b) 1st, (c)(d) 2nd, (e)(f) 3rd, (g)(h) 4th.
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ΔG Fmax ΔαG = G Fmax αG
(8)
γ Δ(G/GFmax ) = ⋅Δα γ (G/GFmax ) α γ (α γ γr + γ )
(9)
nγr ΔD ΔαD = − ⋅Δα γ D αD α γ γr + γ
It can be seen that the error of GFmax induced by the error ΔαG is linear, while the errors of G/GFmax and D induced by Δαγ and ΔαD are nonlinear. The errors of G/GFmax and D are presented in Fig. 11 for the clay. In Fig. 11(a), the RE's of αγ were set to 20%, 40%, and 60%, with a temperature range of −18 °C ~ −2 °C. The RE's propagation to the G/ GFmax is less than those of αγ, although varying with shear strain and temperature. Similar phenomenon can be observed for the RE of damping in Fig.11(b). There existed a lower limit for the RE of αD propagating to the D curve, as shown in Eq.(11).
Fig. 20. Variation of Vs in the analysis.
αopt = y0 + A0 eT / r , (opt = G, γ , D)
Δα γ ΔD ΔαD = − ⋅n, (γ = 0) D αD αγ
(6)
Here, T is temperature (°C). y0, A0, r are the curve-fitting parameters. The subscript ‘opt’ represents the optional term G, γ and D, where αG stands for the initial modulus correction factor, αγ stands for the reference strain correction factor, and αD stands for the damping correction factor. Genetic Algorithm (GA), which is widely employed to calculate undetermined coefficients in curve-fitting (Rokonuzzaman and Sakai, 2010), was used to estimate the coefficients in Eq. 6. Fig. 8 to Fig. 10 show the fitted results with Eq. 6 for the tested soils, in which all the Rsquares of the results are larger than 0.99. It can be seen that αG increased significantly with a decrease in the temperature, but the influence of temperature on αγ was much smaller. αD decreased with a decrease in the temperature.
5. Analysis of the data from literature 5.1. Maximum shear modulus Previous studies on the dynamic properties of frozen soils were mostly focused on the maximum shear modulus, while few results are available for the modulus reduction curves and damping curves. The test data on the maximum shear modulus were therefore analyzed with the model in section 3, and the values of αG of 16 kinds of soils available in the literature (Ling et al., 2009; Kaplar, 1969; Ling et al., 2015; Simonsen et al., 2002; Li et al., 2016) are shown in Table 2. Gradation curves of these soils are presented in Fig. 12. According to ASTM-D2487 (ASTM, 2006), all the soils in Table 2 can be classified into three categories: clay, silt and sand. The values of GFmax were all obtained at a strain magnitude of about 10−6 as reported in the references, which were employed to determine the values of αG in Eq. 3 and are shown in Fig. 13. Fig. 13 shows that the value of αG is dependent on the specific type of soil, even for the same category according to ASTM-D2487 (ASTM, 2006). It can also be seen that in some cases, the moisture content played an important role in the test results. This is not difficult to understand, as frozen soils with very low degree of saturation and very high degree of saturation would certainly behave differently. But beyond certain threshold value, which is mostly the case in the soils tested
The errors of the empirical correction factors would propagate to the G/GFmax and D curves, which may be described by the first order Taylor expansion. N
∑ 1
∂lnf ⋅Δxi ∂xi
(11)
It can thus be concluded that from the accuracy point of view, the importance order of the correction factors, from high to low, is αG, αD and αγ.
4.2. Error propagation
Δf = f
(10)
(7)
The left-hand-side of Eq. 7 is the relative errors (RE) of the G/GFmax and D, where f represents the G/GFmax or D function containing the correction factors. The xi represents the three correction factors, αG, αγ or αD. The Δf and Δxi represent the absolute errors, which are given as the following:
Fig. 21. Some spectral accelerations at the ground surface with 0.2 g input and varying αG, (a) δ= − 50%, (b) δ=0%, (c) δ=50%. 73
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Fig. 22. Average spectra at the ground surface with varying αG, (a) 0.05 g, (b) 0.2 g, (c) 0.4 g.
2011). However, previous studies were mostly restricted to the analyses using limited shear modulus and damping data of frozen soils, and the influences of the variations of the modulus and damping curves have seldom been attempted (Sun et al., 2009). The revised Hardin-Drnevich model can be widely used in the seismic ground response analysis of permafrost sites. However as shown in section 4, the correction factors may vary in a wide range even for a similar soil category. Therefore, a case study was carried out on a permafrost site. The one-dimensional equivalent-linear computer code Proshake was employed for the seismic ground response analysis (Systems, 2017).
in this study in last section, the moisture content would not be very important to the frozen soil behavior. 5.2. Maximum damping ratio Fewer data are available on the maximum damping ratio in the literature (Ling et al., 2009; Al Hunaidi et al., 1996; Wilson, 1982; Ling et al., 2015; Vinson et al., 1978), but they were also analyzed by the model in section 3, and the results of αD are presented in Table 3 and Fig. 14. Similarly to the value of αG, the value of αD also varied considerably for different specific types of soil, even if they belong to similar soil category according to ASTM-D2487 (ASTM, 2006).
6.1. Study site
6. Model application and sensitivity analysis
A permafrost site in Alaska of the United States was analyzed for the seismic response. The Delaney Park Alaska Digital Array (DPK, NEES@ UCSB) is located in downtown Anchorage, Alaska. Yang et al. (Xu et al., 2011; Yang et al., 2008) studied the effects of frozen soils on the seismic response of the site. This site was employed as a reference for this study, at which the soil profile with permafrost was established, as shown in Fig. 15.
The influence of the frozen soil would be larger for the seismic ground response of permafrost sites than that of seasonally frozen sites (Qi et al., 2006; Xu et al., 2011). Some studies found that in the worst case scenario, the average values of spectral accelerations for the periods between 0.4–1.0 s might be much larger than the seismic design spectral values that neglected the effects of frozen soil (Yang et al., 74
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Fig. 23. Average spectra at the ground surface with varying αγ, αD, (a) 0.05 g, (b) 0.2 g, (c) 0.4 g.
Here αi refers to αG, αγ or αD. αio is the base-line value of αi according to Fig. 8–10. δi is the fluctuation of αi, which is set to ± 10%, ± 30% and ± 50% artificially. It is to be noted that αD is fixed at 1.0 when their value are beyond 1.0. Fig. 18 shows the G/Gmax and D curves of the unfrozen layers according to the manual of Proshake, while the curves corrected by αi are present in Fig. 19. It can be seen that the G/Gmax and D curves vary considerably. The effects of confining stress on the correction factors were neglected in this study according to previous findings (Zhu et al., 2011). Fig. 20 presents the varying shear wave velocity used in the analysis, which was obtained using Gmax and soil density. The Vs below -30 m was identical, as this range was out of the frozen region.
6.2. Input motion The DPK array has recorded about 2000 seismic events within the range of 100 km until the end of 2016. Six of these bedrock records were selected in the present study, with the moment magnitude Mw ranging from 4.0 to 6.0. The information of these recorded motions are presented in Fig. 16, Fig. 17 and Table 4. The time-histories of the recorded motions were pre-processed with baseline correction and amplitude modulation. Their peak accelerations were scaled to 0.05 g, 0.2 g and 0.4 g to investigate the influence of input intensity. 6.3. Nonlinear properties of soils
6.4. Results Ground temperature was determined based on the studies by Osterkamp (Osterkamp, 2005) and Hulsey (Hulsey et al., 2011), as shown in Fig. 15. In this study, the correction factors from Fig. 8–10 were employed as the bases. As shown in Section 4, similar category of soil may have very different correction factors. To take into account the influence of the uncertainty, variations of the correction factors were carried out according to Eq. 12:
αi = (1 + δi ) αio
Two sets of results are presented in the following. The first set was obtained by varying αG (δ = ± 10%, ± 30, ± 50%) but fixing αγ and αD (δ = 0%), while the other one was obtained by varying αγ and αD but fixing αG. The response spectra at the ground surface are presented in Fig. 21 to Fig. 21. It can be seen in Fig. 21 that the spectral accelerations at the ground surface were different for different seismic inputs. Average spectra, which may grasp their main characteristics in the period between 0.1 s
(12) 75
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Acknowledgements
and 1.0 s, were therefore employed in the following to discuss the influence of the correction factors. As shown in Fig. 22, the effect of αG was significant for the studied site. The result shows that it may be necessary to take into account the different shear wave velocities of frozen soils in the seismic site classification in a permafrost region. The finding is different from some previous ones (Xu et al., 2011; Yang et al., 2011), which may be explained by the differences in the sites and the input seismic motions. The thickness of frozen soils, their shear wave velocities, and the input motions in the previous studies (Xu et al., 2011; Yang et al., 2011) were all different from the ones in this study. Fig. 22 also presents the spectral response assuming that the soils were unfrozen. The difference is very significant, further pinpointing the importance of considering frozen soils in the seismic site classification of permafrost regions. Uncertainties in the estimation of αγ and αD also commonly exist in frozen soils as discussed in Section 4. Fig. 21 shows their influences on the average seismic spectral responses at the ground surface. The average spectra were obtained similarly to those in Fig. 22. Although similar to the effects of varying αG, the variations of αγ and αD had smaller influences on the response spectra. The influences also increased with an increase in the input acceleration. This is understandable, since with small seismic input, the ground response was close to linear, and the effect of shear wave velocity would dominate. The frozen spectral accelerations were mostly lower than the unfrozen ones and the MCE (Maximum credible earthquake) spectrum in the region according to NEHRP (Building Seismic Safety Council (BSSC), 2015). The unfrozen site belongs to Site Class D. The parameters of the MCE spectrum were listed in Table 5. It is suggested that the seismic response analysis without considering frozen layers may be conservative in some cold regions. However, this conclusion may not be universal, depending on the frequency characteristics of the input motions (See Fig. 23)
The authors are appreciated to the support from the Scientific Research Fund of the Institute of Engineering Mechanics, China Earthquake Administration (Grant No.2016A02). The records of seismic motions were obtained from NEES@UCSB (http://www.nees.ucsb. edu). Reference Al Hunaidi, M.O., Chen, P.A., Rainer, J.H., et al., 1996. Shear moduli and damping in frozen and unfrozen clay by resonant column test. Can. Geotech. J. 33, 510–514. http://dx.doi.org/10.1139/t96-073. ASTM, 2006. Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification System). http://www.dres.ir/fanni/khak/DocLib4/D %202487%20%E2%80%93%2098%20%20;RDI0ODCTUKVE.pdf. Bing, H., Ma, W., 2011. Laboratory investigation of the freezing point of saline soil. Cold Reg. Sci. Technol. 67 (1), 79–88. http://dx.doi.org/10.1016/j.coldregions.2011.02. 008. Building Seismic Safety Council (BSSC), 2015. 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7. Summary and conclusions The goal of this study is to propose a revised Hardin-Drnevich model for the dynamic shear modulus and damping ratio of frozen soils. A series of low temperature RC tests were carried out considering the influence of the soil temperature and moisture content. Based on the test results, empirical correction factor αG, αγ and αD were proposed to revise the Hardin-Drnevich model. The αG and αD for other frozen soils were also collected from previous studies. The influences of the correction factors and their variations on the seismic response of a permafrost site were analyzed. The following conclusions may be obtained from this study. (1) The test results show that soil temperature had large effects on the maximum shear modulus, modulus reduction curves and damping curves of sand, silt and clay. With a decrease in the soil temperature, GFmax increased, while γFr and DFmax decreased. The effects of the moisture content were small and could be neglected for the range of degree of saturation investigated in this study. (2) The proposed correction factors matched preferably with the experimental data and those from previous studies. With the correction factors, the revised Hardin-Drnevich model may be employed to analyze shear modulus and damping ratio of frozen soils at different soil temperatures. However, unique functions of correction factors could not be found for one category of frozen soil, for example clay. They varied considerable for different specific frozen soils. (3) Compared with the correction factors for the modulus reduction curve and damping curve, the correction factor for the maximum shear modulus or shear wave velocity had a larger effect on the seismic ground response of a permafrost site. The response spectral values at the ground surface were generally larger if the investigated site was assumed to be non-frozen. 76
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Sun, R., Yuan, X., Liu, X., 2009. Effects of dynamic shear modulus ratio and velocity on surface ground motion and their equivalent relations. Chin. J. Geotech. Eng., vol. 31 (8), 1267–1274. (In Chinese with English Abstract). http://www.cgejournal.com/ CN/Y2009/V31/I8/1267. Systems, EduPro Civil, 2017. Proshake: Ground Response Analysis Program Version 2.0 user's Manual, Sammamish, WA. http://www.proshake.com/proshake_2.0/User %20Manual.pdf. Vinson, T., Chaichanavong, T., Li, J., 1978. Dynamic Testing of Frozen Soils under Simulated Earthquake Loading Conditions. In: Silver, M., Tiedemann, D. (Eds.), STP35678S Dynamic Geotechnical Testing, pp. 196–227. http://dx.doi.org/10.1520/ STP35678S. Wang, G.X., Kuwano, J., 1999. Modeling of strain dependency of shear modulus and damping of clayey sand. Soil Dyn. Earthq. Eng. 18 (6), 463–471. http://dx.doi.org/ 10.1016/S0267-7261(99)00010-X. Wang, D., Zhu, Y., Zhao, S., et al., 2002. Study on experimental determination of the dynamic elastic mechanical parameters of frozen soil by ultrasonic technique. Chin. J. Geotech. Eng., vol. 24 (5), 612–615. (In Chinese with English Abstract). http:// manu31.magtech.com.cn/Jwk_ytgcxb/CN/Y2002/V24/I5/612. Wichtmann, T., Hernández, M.A.N., Triantafyllidis, T., 2015. On the influence of a noncohesive fines content on small strain stiffness, modulus degradation and damping of quartz sand. Soil Dyn. Earthq. Eng. 69 (2), 103–114. http://dx.doi.org/10.1016/j. soildyn.2014.10.017. Wilson, C.R., 1982. Dynamic Properties of Naturally Frozen Fairbanks Silt. Oregon State University, Corvallis, Oregon, United States (Masters Thesis). http://ir.library. oregonstate.edu/xmlui/bitstream/1957/9497/1/Wilson_Charles_R_1982.pdf. Xu, G., Dutta, U., Yang, Z., et al., 2011. Seasonally frozen soil effects on the seismic site
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