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Stiffness degradation of natural fine grained soils during cyclic loading
ARTICLE IN PRESS
Soil Dynamics and Earthquake Engineering 27 (2007) 843–854 www.elsevier.com/locate/soildyn
Stiffness degradation of natural fine grained soils during cyclic loading D.V. Okura, A. Ansalb, Department of Civil Engineering, Eskis- ehir Osmangazi University, Mes- elik, 26480 Eskis- ehir, Turkey Kandilli Observatory and Earthquake Research Institute, Bog˘azic- i University, 34684 C - engelko¨y, Istanbul, Turkey a
b
Received 12 March 2006; received in revised form 7 November 2006; accepted 28 January 2007
Abstract Cyclic behavior of natural fine grained soils under a broad range of strains were investigated considering the effects of plasticity index and changes in confining pressures based on cyclic triaxial tests. A total of 98 stress controlled cyclic triaxial tests were conducted on normally consolidated and slightly overconsolidated samples. The investigation was divided into two parts. The first part consists of stress controlled cyclic triaxial tests under different stress amplitudes that were conducted to estimate the modulus reduction and the thresholds between nonlinear elastic, elasto-plastic and viscoplastic behavior. The second part involves the investigation of the undrained stress–strain behavior of fine grained soils under irregular cyclic loadings. The results showed that the elastic threshold is approximately equal to 90% of Gmax. Another transition point was defined as the flow threshold where the value of tangent of shear modulus ratio changes for the second time. Simple empirical relationships to estimate the dynamic shear modulus and damping ratio was formulated and compared with the similar empirical relationships proposed in the literature. The results provide useful guidelines for preliminary estimation of dynamic shear modulus and damping ratio values for fine grained soils based on laboratory tests. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fine grained soils; Cyclic behavior; Dynamic shear modulus; Damping ratio; Modulus reduction
1. Introduction During seismic loading, soil layers are subjected to cyclic shear stresses with different amplitudes and frequencies that will induce transient and permanent deformations. The structures located on and the infrastructures located in these layers are going to be affected from these deformations and may be damaged. In addition, the change in the stress–strain-strength properties namely ‘‘dynamic properties’’ of soil layers during cyclic loading may have a significant influence on the stability of earth dams, embankments, retaining structures and natural slopes. Therefore, it is necessary to take into account the changes in dynamic characteristics of natural soil layers to decrease the vulnerability. The objective of this research is to gain an understanding of the possible mechanisms responsible for the dynamic Corresponding author.
E-mail address: [email protected] (A. Ansal). 0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.01.005
behavior of natural fine grained soils by performing cyclic triaxial tests and by comparing the results with the results reported in the literature. Based on the results of previous experimental investigations, it is well recognized that during cyclic loading with a stress or strain amplitude above a specific value, permanent loss of strength and stiffness is observed in fine grained soils with the increase in the number of cycles, because of the excess pore water pressure accumulation and particle structure breakdown [1–6]. Dynamic material properties are needed in order to evaluate response and the stability of the surface layers. The basic dynamic soil characteristics needed for dynamic analyses are the small strain shear modulus, shear modulus reduction and the damping ratio increase with shear strain amplitude. The small strain shear modulus, Gmax (the value of shear modulus at very small strains gp5 104%) is one of the important characteristic of soil deformability and plays a significant role in dynamic response analysis. Gmax is generally estimated by using laboratory or field
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tests related to seismic wave propagation and reduction curves are obtained based cyclic tests conducted in the laboratory. Widespread laboratory studies by using resonant-column, cyclic simple shear, cyclic torsional shear and cyclic triaxial devices have shown that the dynamic properties of fine grained soils are more difficult to understand than coarse grained soils and tests are time consuming as well. However, there are large number of studies on the dynamic shear modulus and damping factors of fine grained soils and several experimentally determined relationships for evaluating these parameters have been reported in the literature. The well-known study by Hardin and Black [7,8], and Hardin and Drnevich [9,10] presented empirical equations to estimate the maximum dynamic shear modulus and damping ratio for clays. Kim and Novak [11], Kagawa [12], Teachavorasinskun et al. [13] conducted different types of cyclic tests on many types of soil samples taken from different sites. Kokusho et al. [14], Vucetic and Dobry [15] suggested that the plasticity index (PI) is the primary parameter influencing the dynamic shear modulus and damping ratio in fine grained soils. The increase of the plasticity index causes an increase on the normalized dynamic shear modulus and a decrease on the damping ratio. By combining different sets of test results from the literature, Vucetic and Dobry [15] also plotted the curves showing the influence of the PI on the normalized dynamic modulus and damping ratio with respect to shear strain. 2. Basic relations The dynamic shear modulus varies with cyclic shear strain amplitude. At low strain levels, there is reduction in the dynamic shear modulus and the reduction accelerates as the strain amplitude increases. During the duration of the deformation process due to the cyclic loading, energy is dissipated through hysteretic damping. For small values of shear starin, (gp5 104%), the dynamic shear modulus Gsec remains practically constant and this is equal to its initial value, Gmax. In spite of number of cycles, no accumulation of plastic strains or excess pore pressure is observed during this range of cyclic strains. Significant numbers of experimental studies have shown that for fine grained soils, Gmax is mainly affected by void ratio, and effective confining stress [16,6]. The influence of the effective confining pressure, particularly is significant for soils of low plasticity [5]. The effects of confining pressures and plasticity index on shear modulus ratio were formulated by Ishibashi and Zhang [17] in the form s0m , g and PI. Theoretically, no dissipation of energy takes place at strains below the elastic threshold strain amplitude. On the other hand, a number of experimental studies indicated that some energy is dissipated even at very low strain levels [18], thus the damping ratio is never zero. Beyond the linear strain threshold, the increase in the size of hysterisis loops indicates that the damping ratio increases with the increasing cyclic strain amplitude.
The dissipated energy in each cycle is commonly represented by the area of the hysteresis loop for that cycle. The mathematical models defining the shape of the stress–strain loops may be considered as bi-linear, hyperbolic or Ramberg–Osgood type [19]. Just as dynamic shear modulus is influenced by soil plasticity, so is damping ratio [14,20]. Damping is also influenced by effective confining pressure particularly for low plastic soils as shown by Ishibashi and Zhang [17] who also developed empirical relationships for damping ratio for plastic and nonplastic soils. 3. Testing apparatus The cyclic loading of soil samples was carried out by using a servo-controlled pneumatic triaxial apparatus similar to the one described by Kokusho [5]. The apparatus is capable of carrying out stress controlled cyclic and strain-controlled static loading tests. The maximum cell pressure, the maximum axial load, the maximum axial displacement are 10 kgf/cm2, 200 kgf and 20 mm, respectively. The frequency can be changed between 0.001 and 10 Hz. To have reliable small strains (103%) measurements, two non contact displacement transducers were attached on the top cap of the sample. Additional displacement transducers were also attached outside of the cell in order to measure large axial deformations. The load cell for monitoring the axial load is encased in the triaxial cell to eliminate the piston friction. The support columns are inside the cell to assure vertical alignment and to minimize the sample disturbance during sample preparation. 4. Sample and test procedure The soil samples used in this research were taken from various sites in Turkey after the 1999 Kocaeli Earthquake as a part of post earthquake investigations. The physical properties of the samples are summarized in Table 1. A total of 98 samples taken from 56 different boreholes were used in this investigation. The soil samples were taken with thin walled shelby tubes of 75 mm in diameter and were extruded by cutting the tubes longitudinal in order to minimize the disturbance. The samples were prepared according to the specifications of the Japanese Standard [21]. The trimmed solid samples had a diameter of 50 mm and a height of 105. Eight strips of filter paper were placed around the sample in order to maintain the quick completion of primary consolidation and to balance of pore water pressure throughout the sample during the test. The samples were isotropically consolidated in the triaxial cell with a given effective pressure and a back pressure. The effective cell pressures were approximately 50% higher than the preconsolidation pressures determined separately by conventional odometer tests to minimize the effect of sample disturbance as suggested by Ladd and Foott [22]. The cyclic axial load was
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Table 1 Summary of the properties of tested samples Depth range of samples (m)
Water content, wo (%)
Void ratio, eo
Liquid limit, wL (%)
Plasticity index, PI (%)
Soil type
2.50–23.55
25–52
0.68–1.40
38–70
9–40
ML, MH, CL, CH
0.04 S1
Cyclic stress ratio, σd / 2σc
0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
0
5
10
15
20 25 30 Number of cycles, N
35
40
45
50
Fig. 1. Actual loading sequence used during first group of tests.
then applied to the sample under undrained conditions with frequency of 0.5 Hz. The investigation program consisted of two sets of tests with different loading stages. The first set was composed of multi stage tests that were used where the number of available test samples is limited. In these tests, samples were first consolidated and later were subjected five cycles of cyclic axial stresses starting from quite low stress amplitudes under undrained conditions. The samples were cyclically loaded step by step with gradually increasing stress amplitudes covering a wide strain range of 103–101% under a constant confining pressure. For evaluation of the stress–strain properties, the 3rd cycle was selected as the representative cycle for that strain range. The second test type was again a multi stage but this time the sample was cyclically loaded to some level of virgin strain and then cyclically loaded at a considerably smaller level of strain. The application of small cyclic strains with increasing and decreasing amplitudes gives no prestraining effects on the dynamic properties for a range of strain as explained in the following paragraphs. The dynamic shear modulus and the damping ratio were again calculated from the stress vs. strain hysterisis loop for the 3rd cycle of every five cycles. On the both group of tests, the failure was defined corresponding to 10% double strain amplitude. The actual
test records of loading sequences for the first group tests are shown in Fig 1. 5. Test results For the first group of tests conducted on normally consolidated samples, an empirical expression for the small strain dynamic shear modulus, Gmax, suggested by Ishihara [23] Gmax ¼ A
ðB eÞ2 0 C ðs Þ ð1 þ eÞ c
(1)
was adopted, where Gmax is in MPa, e ¼ void ratio, and s0c ¼ effective confining pressure in kPa. The constants in Eq. (1) are calculated using a nonlinear curve fitting algorithm based on laboratory measured values as A ¼ 466, B ¼ 3.4, and C ¼ 0.66. In Fig. 2, small strain dynamic shear modulus normalized by the void ratio function, F(e) are plotted against the confining stress, s0c , where F ðeÞ ¼
ð3:4 eÞ2 ð1 þ eÞ
(2)
As can be observed from Fig. 2, the correlation coefficient for the Gmax/F(e) vs. s0c relationship is relatively high indicating well established correlation and validating the adopted Gmax formulation.
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30
Gmax/F(e) (MPa)
20
10 9 8 7
Correlation Coefficient = 0.9099
6 5
60
70
80 90 100
200
300
400
500
Effective Confining Pressure, σc′
Fig. 2. Relationship between measured values of Gmax/F(e) and s0c . 400 300
Ha rdin and Black. 1968. Kagaw a. 1992 Kallioglou et all. 1999. Equation (1) Laboratory measured
Dynamic shear modulus, Gmax
200
100 70 60 50 40 30 20
0.55
0.57
0.64
0.66
0.76
0.79
0.81
0.84
0.89
0.97
1.02
1.06
1.07
1.08
1.11
1.17
1.27
1.27
1.40
1.50
10
Void ratio of the samples tested, e
Fig. 3. Comparison of small strain dynamic shear modulus calculated by Eq (1) with the previously suggested relationships and laboratory measured values for different samples.
The small strain dynamic shear moduli determined for the samples tested in the cyclic triaxial system were also calculated using Eq. (1), and relationships proposed by Hardin and Black [7,12] and Kallioglou et al. [24] as shown in Fig. 3 with respect to each other and in comparison to laboratory measured values. The formulation suggested by Hardin and Black [7] is generally valid for stiff kaolinite clay and was based on resonant column tests that yield higher Gmax values when compared to dynamic triaxial test as observed in Fig. 3. The differences between the relationship proposed by Kagawa [12] and this study are limited and most likely due to the testing scheme adopted by [12] where the cyclic tests were conducted right after the completion of the primary consolidation. A very good agreement is observed with the relationship proposed by Kallioglou et al. [24]. The expression proposed by
Kallioglou et al. [24] is for undisturbed clay samples which have similar physical properties used in this investigation. As shown in Fig. 3, the agreement among the Gmax calculated from Eq. (1) and laboratory measured values are relatively good. However, it is interesting to note that plasticity index appears to have no significant effect on the small strain dynamic shear moduli obtained experimentally and modeled relatively well using Eq. (1). Okur and Ansal [25] presented results of the multi-stage loading program to show the trends between plasticity index and the shear modulus ratio and the damping ratio curves based on several additional tests conducted. Fig. 4 shows the dynamic shear modulus reduction and damping ratio increase with increasing shear strain amplitude. The results confirm that especially plasticity index is the governing factor both in dynamic shear modulus reduction
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1.2 C16, PI=40
Dynamic shear modulus ratio, G/Gmax
S3, PI=35
1
C15, PI=27 C12, PI=25 C11, PI=20
0.8
C13, PI=18 S2, PI=15 C10, PI=12
0.6
C9, PI=10 C5, PI=9
0.4
0.2
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) 25 C16, PI=40 C15, PI=27
20
C12, PI=25 C11, PI=20
Damping ratio, D (%)
C13, PI=18 C10, PI=12
15
C9, PI=10 C5, PI=9
10
5
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%)
Fig. 4. Variation of dynamic shear modulus and damping ratio with shear strain amplitude for samples with different plasticity index.
and damping ratio increase as stated in [14,26]. The effect of void ratio is much less significant than plasticity as shown in Fig. 5, most likely because it has much less effect on inter particle forces that govern the fine grained soil behavior.
A simple relationship is adapted to model dynamic shear modulus reduction with cyclic shear strain amplitude using a hyperbolic model
6. Dynamic shear modulus and damping ratio
where ga is cyclic shear strain amplitude, gr is reference strain defined as a function of plasticity index, PI. The empirical relationship for the reference strain, gr was estimated as an exponential function with respect to PI and the constants in this formulation was calculated using
Dynamic shear modulus and damping ratio are the two important parameters in the evaluation of cyclic behavior of soil elements as well as in site response analysis.
G 1 ¼ Gmax 1 þ ga =gr
(3)
ARTICLE IN PRESS D.V. Okur, A. Ansal / Soil Dynamics and Earthquake Engineering 27 (2007) 843–854
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Dynamic shear modulus ratio, G/Gmax
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
IT, PI=12, e=0.83, σ 'c=150 kPa C10, PI=12, e=0.96, σ 'c=200 kPa C1, PI=12, e=1.42, σ'c=300 kPa
0.1 0 0.0001 0.0002
0.0005 0.001
0.002
0.005
0.01
0.02 0.03 0.05
0.1
0.2 0.3
0.5 0.7 1
2
3 4 5
Cyclic shear strain amplitude, γa (%) 24 22 20
IT, PI=12, e=0.83, σ'c=150 kPa C10, PI=12, e=0.96, σ'c=200 kPa C1, PI=12, e=1.42, σ'c=300 kPa
Damping ratio, D (%)
18 16 14 12 10 8 6 4 2 0 0.0001 0.0002
0.0005 0.001
0.002
0.005
0.01
0.02 0.03 0.05
0.1
0.2 0.3
0.5 0.7 1
2
3 4 5
Cyclic shear strain amplitude, γa (%)
Fig. 5. Effects of void ratio on dynamic shear modulus reduction and damping ratio increase with cyclic shear strain amplitude.
a nonlinear regression analysis as gr ¼
1 21 1 expð106 PI1:585 Þ
(4)
A similar empirical expression was also used to model the increase in damping ratio with cyclic shear strain amplitude based on plasticity index: D ¼ C0 þ C1 C2PI C3 expðC4 ga C5PI Þ (5) where C0–C5 are empirical constants that were determined using a nonlinear curve fitting algorithm based on the cyclic triaxial test results as C0 ¼ 6.3, C1 ¼ 19.2, C2 ¼ 0.9976, C3 ¼ 1.54, C4 ¼ 8.73, and C5 ¼ 0.99.
The modeling achieved by Eqs (4) and (5) are presented with respect to test data in Fig. 6 and as can be observed, the proposed empirical models for the modulus reduction and damping ratio increase with respect to cyclic shear strain amplitude are yielding relatively very good fits for the given test results. Hardin and Drnevich [10,27] have studied the relationships between dynamic shear modulus and damping ratio and suggested to adopt G D¼f (6) Gmax for sandy soils. Based on the results obtained in this study as shown in Fig. 7, it was possible to establish by simple
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a
849
1.2
Dynamic shear modulus ratio, G/Gmax
C16, PI=40 1
S3, PI=35 C12, PI=25 C11, PI=20
0.8
S2, PI=15 C5, PI=9
0.6
0.4
0.2
0 0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%)
b
25 C16, PI=40
Damping ratio, D (%)
20
C15, PI=27 C10, PI=12
15
10
5
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) Fig. 6. Comparison between modeled and measured: (a) Dynamic shear modulus reduction and (b) damping ratio increase with cyclic shear strain amplitude.
regression analyses the following relationship: G D ¼ 18:114 þ 20:033 G max
(7)
The purpose is to develop a simple approach to estimate the damping ratio and damping ratio increase with cyclic shear strain amplitude as a function of to dynamic shear modulus ratio. Since it is easier and less time consuming and less susceptible to possible errors, one may need to
calculate damping ratio based on dynamic shear modulus. The correlation coefficient (R ¼ 0.9799) being relatively high would justify such a simplification. 7. Dynamic behavior The amplitude of cyclic shear strain at which a clear degradation in stiffness begins to occur has been known to depend on several factors such as effective confining stress
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850
25
D = -18.114(G/Gmax) + 20.033
Damping ratio (%)
20
R2 = 0.9602
15
10
5
0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dynamic shear modulus Ratio, G/Gmax
0.8
0.9
1
Fig. 7. Relationship between dynamic shear modulus ratio and damping ratio.
Cyclic stress ratio amplitude, σd/2σc
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1
0
20
40
60
80
100
120
Number of cycles, N Fig. 8. The loading scheme for the second group of tests.
and number of cycles. If the effective confining stress is small, dynamic shear modulus ratio, G/Gmax and damping ratio, D start to change at lower levels of cyclic shear strain amplitudes for saturated fine grained soils [26,28–31]. It was observed that there is a lower limit below which cyclic degradation will be negligible and that this lower limit threshold would be dependent on the pore water pressure buildup. The studies on the influence of plasticity index also showed that the value of threshold generally increases with the plasticity index. The second group of dynamic triaxial tests was conducted to determine the threshold between nonlinear
elastic and elasto-plastic behavior and the undrained stress–strain behavior of fine grained soils under irregular cyclic loading. The samples were consolidated to effective stresses equal to the calculated in situ mean effective stress from consolidation tests. At the initial stage, cyclic loads were applied from small stress levels and increased after every five cycles. When the pore water pressure began to build up, the stress level once again decreased to the initial value, then increased again as shown in Fig. 8. Loading and unloading cycles showed that the nonlinear elastic threshold is approximately equal to strain levels corresponding to the 90% of G/Gmax (Fig. 9) that appears
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1.1
A
Dynamic shear modulus ratio,G/GMAX
1
B Sample Y8, PI=28
C
0.9
E 0.8 0.7 0.6
G D
0.5 0.4 0.3
H
F
0.2 0.1 0 0.002 0.003
0.005
0.01
0.02 0.03
0.05 0.07 0.1
0.2
0.3 0.4 0.5 0.7
1
2
3
4 5
Cyclic shear strain amplitude, γa (%)
Fig. 9. Variation of shear modulus ratio in the second group of tests.
1.2
25
1
N=15
N=150 20
N=75
0.8
N=115
15
0.6 10
0.4
5
0.2 0 0.0001
Damping ratio, D (%)
Shear modulus ratio, G/Gmax
SB1
0.001
0.01 0.1 Cyclic shear strain amplitude, γa (%)
1
10
0
Fig. 10. Variation of shear modulus and damping ratio for different cycles.
to be in the range of 0.01–0.02%. If the stress amplitude is reduced prior to point B (Fig. 9), dynamic shear modulus returns back to its initial value, but when the stress amplitude level exceeds the limit defined by point B, particle structure deformations starts to take place and upon unloading a reduction in the dynamic shear modulus is observed depending on the order exceedance as shown by point C and later by point E that was unloaded from point D (Fig. 9). The modulus reduction observed is due to the rapidly increasing levels of cyclic strains leading to particle structure breakdown and related softening. However, below this threshold point B, engineering properties of soil layers remain practically unaffected. Point F represents approximately the flow threshold, the unloading before this
threshold would lead to recovery of the approximately 60% of the initial Gmax (point G), however once this threshold is exceeded, the recovery is only in the order of 20% as shown by point H in Fig. 9. As pointed out before, the cyclic stress ratio were increased after every five cycles in the first group of tests. Fig. 10 shows the variation of dynamic shear modulus ratio and damping ratio against cyclic shear strain amplitude for different cycles for test SB1. The hysterisis loops corresponding to different stages of loading are given in Fig. 11. As expected, the shape of the hysterisis loops for the cycles 10–15 have the same area for every cycle that indicates nonlinear elastic response of the soil element. The dynamic shear modulus ratio is at its 100% value. The nonlinear
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0.06
SB1 Cyclic stress ratio amplitude σd/2σc
Cyclic stress ratio amplitude σd /2σc
0.01 0.008 0.006 0.004 0.002 0 -0.002
N=11-15
-0.004
Gmax=34 MPa
-0.006
SB1
0.04
0.02
0
-0.02
N=71-75 G=26.8 MPa
-0.04
G/Gmaks=76 %
G/Gmax=98 %
-0.008 -0.002 -0.0015 -0.001 -0.0005
D=5.6 %
D=3.6 %
0
-0.06 -0.02
0.0005 0.001 0.0015 0.002 0.0025 0.003
-0.015 -0.01
Cyclic shear strain amplitude, γa (%) 0.5
SB1
Cyclic stress ratio amplitude σd /2σc
Cyclic stress ratio amplitude σd /2σc
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
N=116-120 G=12.9 MPa G/Gmaks=36.6% D=12.6%
-0.15 -0.2 -0.25 -0.15
-0.005
0
0.005
0.01
0.015
0.02
0.025
Cyclic shear strain amplitude, γ (%) a
SB1
0.4 0.3 0.2 0.1 0 -0.1 -0.2
N=146-150 G=1.5 MPa G/Gmax=4% D=20.3%
-0.3 -0.4 -0.5
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-2
0.25
Cyclic shear strain amplitude, γa (%)
-1
0
1
2
3
4
Cyclic shear strain amplitude, γa (%)
Fig. 11. Shapes of hysterisis curves for different cyclic shear stress amplitudes and cycles.
elastic response begins to change at cycle 65–70 corresponding to point B in Fig. 9. It must be underlined that there is no exact transformation point between nonlinear elastic and elasto-plastic response of cohesive soils. But it can be seen from Fig. 9 that after decreasing of stress from point B back to the initial value, the curve returns to point C with a reduction of the dynamic shear modulus ratio. The area of the loops begin to change at the cycles between 115 and 120 despite the same cyclic stress ratio which means the plastic alteration began in the particle structure of the sample. Between cycles 145 and 150, the change is very evident. This transformation point is defined the flow threshold, gP, where the sample is behaving in a viscoplastic mode since it rigidity is approximately 10% of it is initial
value and it has reached the steady state phase where the residual strength is mobilized. Even though it is very empirical, one conventional procedure that can be adopted is to draw three tangent straight lines to three parts of the modulus reduction curve as shown in Fig. 12 and estimate the elastic and flow threshold strain levels from the intersection of these three straight lines. Similar response patterns were observed for all samples tested and correlations were established for the elastic and flow strain thresholds in terms of plasticity index, where gE ¼
0:035 1 þ 11:92 expð0:1PIÞ
(8)
ARTICLE IN PRESS Dynamic shear modulus ratio, G/Gmax
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1.2 1 0.8 0.6 γE
0.4
γP
0.2 0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) Fig. 12. Tangents for nonlinear elastic, elasto-plastic and viscoplastic phases of the dynamic modulus reduction response and estimation of the elastic and flow thresholds.
gP ¼
1 1:39 0:33PI 0:28
(9)
thresholds were determined using an empirical approach and simple approximate correlations were proposed based on the plasticity index.
8. Conclusion References Cyclic behavior of saturated fine grained soils were studied in detail based on a set of undrained stress controlled dynamic triaxial tests conducted on low to medium plastic fine grained soil samples obtained from various sites in Turkey. The idea was to confirm the effecting soil parameters and to develop semi-empirical correlations to evaluate the dynamic properties of cohesive soil layers. The results confirm the strong influence of both plasticity and the effective confining stress on the behavior of the low-medium fine grained soils. An empirical expression was developed to calculate the maximum dynamic shear modulus, Gmax based on void ratio and confining pressure. The strain-dependent change of dynamic shear modulus ratio, G/Gmax and damping ratio, D, of fine grained soils is highly related with the plasticity index. Since the plasticity index is easily determined parameter in geotechnical engineering, empirical relationships were formulated for fine grained soil modulus reduction and damping increase curves. The curves determined for the shear modulus ratio agrees very well with the test data and the literature. One of the important features of fine grained soil layers subjected to cyclic excitations is the reduction of stiffness. The results obtained from a series of tests with changing stress amplitudes on undisturbed soil samples have indicated that the reduction of stiffness starts to become effective once a critical cyclic shear strain level defined as elastic threshold is exceeded. It was also possible to define a second critical strain level defined as flow threshold where the soil sample has reached the steady state conditions and starts to behave as viscoplastic material. These two critical
[1] Ansal A, Erken A, Rate dependent dynamic behavior of normally consolidated clay. In: Seventh European conference on earthquake engineering, Athens, Greece, 20–25 September 1982.p. 329–36. [2] Ansal A, Erken A. Undrained behavior of clay under cyclic shear stresses. J Geotech Eng ASCE 1989;115:968–83. [3] Azzouz AS, Malek AM, Baligh MM. Cyclic behavior of clays in undrained simple shear. J Geotech Eng ASCE 1989;115(5):637–57. [4] Yasuhara L, Hirao K, Hyde A. Effects of cyclic loading on undrained strength and compressibility of clay. Soils Found 1992;32:100–16. [5] Kokusho T. Cyclic triaxial test of dynamic soil properties for wide strain range. Soils Found 1980;20:45–60. [6] Koutsoftas DC, Fischer JA. Dynamic properties of two marine clays. J Geotech Eng Div ASCE 1980;GT6:645–57. [7] Hardin BO, Black WL. Vibration modulus of normally consolidated clay. J Soil Mech Found ASCE 1968;94:353–69. [8] Hardin BO, Black WL. Closure to vibration modulus of normally consolidated clay. J Soil Mech Found ASCE 1969;95:1531–7. [9] Hardin BO, Drnevich VP. Shear modulus and damping in soils: measurement and parameter effects. J Soil Mech Found ASCE 1972;98:603–24. [10] Hardin BO, Drnevich VP. Shear modulus and damping in soils: design equation and curves. J Soil Mech Found ASCE 1972;98:667–92. [11] Kim TC, Novak M. Dynamic properties of some cohesive soils of Ontario. Can Geotech J 1981;18:371–89. [12] Kagawa T. Moduli and damping factors of soft marine clays. J Geotech Eng 1992;118:1360–75. [13] Teachavorasinskun S, Thongchim P, Lukkunappasit P. Shear modulus and damping ratio of a clay during undrained cyclic loading. Geotechnique 2001;51(5):467–70. [14] Kokusho T, Yoshida Y, Esashi Y. Dynamic properties of soft clay for wide strain range. Soils Found 1982;22:1–18. [15] Vucetic M, Dobry R. Effect of soil plasticity on cyclic response. J Geotech Eng Div ASCE 1991;117:89–107. [16] Humphries WK, Wahls HE. Stress history effects on dynamic modulus of clay. J Soil Mech Found ASCE 1968;94:371–89.
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[17] Ishibashi I, Zhang X. Unified dynamic shear moduli and damping ratios of sand and clay. Soils Found 1993;33:182–91. [18] Lanzo G, Vucetic M. Effect of soil plasticity on damping ratio at small cyclic strains. Soils Found 1999;39:131–41. [19] Idriss IM, Seed HB. Seismic response of horizontal soil layers. J Soil Mech Found Div ASCE 1968;94(4):1003–31. [20] Dobry R, Vucetic M, Dynamic properties and seismic response of soft clay deposits. In: Proceeding of the international symposium on geotechnical engineering on soft soils, Mexico City, Mexico, 13–15 September 1987. p. 51–87. [21] JGS T 520-1990, Practice for preparing triaxial test specimens. Japanese Society of Soil Mechanics and Foundation Engineering 1990. [22] Ladd CC, Foott R. New design procedure for stability of soft clays. J Geotech Eng Div ASCE 1974;GT7:763–86. [23] Ishihara K. Soil behavior in earthquake geotechnics. Oxford: Clarendon Press; 1996. [24] Kallioglou, P., Tika, T.H., Pitilakis, K. Dynamic characteristics of natural cohesive soils. J Geotech Eng 1999:17;113–117.
[25] Okur, D.V., Ansal, A.M. Dynamic characteristics of clays, In: Proceedings of the third Japan–Turkey workshop on earthquake engineering, Istanbul, 21–25 February 2000, Turkey. p. 45–52. [26] Vucetic M. Cyclic threshold shear strains in soils. J Geotech Eng Div ASCE 1994;120:2208–28. [27] Tatsuoka F, Siddique MS, Park CS, Sakamato M, Abe F. Modeling stress strain relations of sand. Soils Found 1993;33:60–81. [28] Georgiannou VN, Hight DW, Burland JB. Behavior of clayey sands under undrained cyclic triaxial loading. Geotechnique 1991;41: 383–93. [29] Mancuso, C., Silvestri, F., Vinale, F. Soil properties relevant to seismic microzonation. In: First Japan–Turkey workshop on earthquake engineering, Istanbul, Turkey, 15–18 March 1997. p. 82–113. [30] Presti, D.L., Lai, C., Foti, S. Geological and geotechnical investigations for ground response analyses. In: Geotechnical earthquake engineering and microzonation seminar, Istanbul, Turkey, 25–26 August 2001. [31] Shibuya S, Tanaka H. Estimate of elastic shear modulus in Holocene soil deposits. Soils Found 1996;36:45–55.
Soil Dynamics and Earthquake Engineering 27 (2007) 843–854 www.elsevier.com/locate/soildyn
Stiffness degradation of natural fine grained soils during cyclic loading D.V. Okura, A. Ansalb, Department of Civil Engineering, Eskis- ehir Osmangazi University, Mes- elik, 26480 Eskis- ehir, Turkey Kandilli Observatory and Earthquake Research Institute, Bog˘azic- i University, 34684 C - engelko¨y, Istanbul, Turkey a
b
Received 12 March 2006; received in revised form 7 November 2006; accepted 28 January 2007
Abstract Cyclic behavior of natural fine grained soils under a broad range of strains were investigated considering the effects of plasticity index and changes in confining pressures based on cyclic triaxial tests. A total of 98 stress controlled cyclic triaxial tests were conducted on normally consolidated and slightly overconsolidated samples. The investigation was divided into two parts. The first part consists of stress controlled cyclic triaxial tests under different stress amplitudes that were conducted to estimate the modulus reduction and the thresholds between nonlinear elastic, elasto-plastic and viscoplastic behavior. The second part involves the investigation of the undrained stress–strain behavior of fine grained soils under irregular cyclic loadings. The results showed that the elastic threshold is approximately equal to 90% of Gmax. Another transition point was defined as the flow threshold where the value of tangent of shear modulus ratio changes for the second time. Simple empirical relationships to estimate the dynamic shear modulus and damping ratio was formulated and compared with the similar empirical relationships proposed in the literature. The results provide useful guidelines for preliminary estimation of dynamic shear modulus and damping ratio values for fine grained soils based on laboratory tests. r 2007 Elsevier Ltd. All rights reserved. Keywords: Fine grained soils; Cyclic behavior; Dynamic shear modulus; Damping ratio; Modulus reduction
1. Introduction During seismic loading, soil layers are subjected to cyclic shear stresses with different amplitudes and frequencies that will induce transient and permanent deformations. The structures located on and the infrastructures located in these layers are going to be affected from these deformations and may be damaged. In addition, the change in the stress–strain-strength properties namely ‘‘dynamic properties’’ of soil layers during cyclic loading may have a significant influence on the stability of earth dams, embankments, retaining structures and natural slopes. Therefore, it is necessary to take into account the changes in dynamic characteristics of natural soil layers to decrease the vulnerability. The objective of this research is to gain an understanding of the possible mechanisms responsible for the dynamic Corresponding author.
E-mail address: [email protected] (A. Ansal). 0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.01.005
behavior of natural fine grained soils by performing cyclic triaxial tests and by comparing the results with the results reported in the literature. Based on the results of previous experimental investigations, it is well recognized that during cyclic loading with a stress or strain amplitude above a specific value, permanent loss of strength and stiffness is observed in fine grained soils with the increase in the number of cycles, because of the excess pore water pressure accumulation and particle structure breakdown [1–6]. Dynamic material properties are needed in order to evaluate response and the stability of the surface layers. The basic dynamic soil characteristics needed for dynamic analyses are the small strain shear modulus, shear modulus reduction and the damping ratio increase with shear strain amplitude. The small strain shear modulus, Gmax (the value of shear modulus at very small strains gp5 104%) is one of the important characteristic of soil deformability and plays a significant role in dynamic response analysis. Gmax is generally estimated by using laboratory or field
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tests related to seismic wave propagation and reduction curves are obtained based cyclic tests conducted in the laboratory. Widespread laboratory studies by using resonant-column, cyclic simple shear, cyclic torsional shear and cyclic triaxial devices have shown that the dynamic properties of fine grained soils are more difficult to understand than coarse grained soils and tests are time consuming as well. However, there are large number of studies on the dynamic shear modulus and damping factors of fine grained soils and several experimentally determined relationships for evaluating these parameters have been reported in the literature. The well-known study by Hardin and Black [7,8], and Hardin and Drnevich [9,10] presented empirical equations to estimate the maximum dynamic shear modulus and damping ratio for clays. Kim and Novak [11], Kagawa [12], Teachavorasinskun et al. [13] conducted different types of cyclic tests on many types of soil samples taken from different sites. Kokusho et al. [14], Vucetic and Dobry [15] suggested that the plasticity index (PI) is the primary parameter influencing the dynamic shear modulus and damping ratio in fine grained soils. The increase of the plasticity index causes an increase on the normalized dynamic shear modulus and a decrease on the damping ratio. By combining different sets of test results from the literature, Vucetic and Dobry [15] also plotted the curves showing the influence of the PI on the normalized dynamic modulus and damping ratio with respect to shear strain. 2. Basic relations The dynamic shear modulus varies with cyclic shear strain amplitude. At low strain levels, there is reduction in the dynamic shear modulus and the reduction accelerates as the strain amplitude increases. During the duration of the deformation process due to the cyclic loading, energy is dissipated through hysteretic damping. For small values of shear starin, (gp5 104%), the dynamic shear modulus Gsec remains practically constant and this is equal to its initial value, Gmax. In spite of number of cycles, no accumulation of plastic strains or excess pore pressure is observed during this range of cyclic strains. Significant numbers of experimental studies have shown that for fine grained soils, Gmax is mainly affected by void ratio, and effective confining stress [16,6]. The influence of the effective confining pressure, particularly is significant for soils of low plasticity [5]. The effects of confining pressures and plasticity index on shear modulus ratio were formulated by Ishibashi and Zhang [17] in the form s0m , g and PI. Theoretically, no dissipation of energy takes place at strains below the elastic threshold strain amplitude. On the other hand, a number of experimental studies indicated that some energy is dissipated even at very low strain levels [18], thus the damping ratio is never zero. Beyond the linear strain threshold, the increase in the size of hysterisis loops indicates that the damping ratio increases with the increasing cyclic strain amplitude.
The dissipated energy in each cycle is commonly represented by the area of the hysteresis loop for that cycle. The mathematical models defining the shape of the stress–strain loops may be considered as bi-linear, hyperbolic or Ramberg–Osgood type [19]. Just as dynamic shear modulus is influenced by soil plasticity, so is damping ratio [14,20]. Damping is also influenced by effective confining pressure particularly for low plastic soils as shown by Ishibashi and Zhang [17] who also developed empirical relationships for damping ratio for plastic and nonplastic soils. 3. Testing apparatus The cyclic loading of soil samples was carried out by using a servo-controlled pneumatic triaxial apparatus similar to the one described by Kokusho [5]. The apparatus is capable of carrying out stress controlled cyclic and strain-controlled static loading tests. The maximum cell pressure, the maximum axial load, the maximum axial displacement are 10 kgf/cm2, 200 kgf and 20 mm, respectively. The frequency can be changed between 0.001 and 10 Hz. To have reliable small strains (103%) measurements, two non contact displacement transducers were attached on the top cap of the sample. Additional displacement transducers were also attached outside of the cell in order to measure large axial deformations. The load cell for monitoring the axial load is encased in the triaxial cell to eliminate the piston friction. The support columns are inside the cell to assure vertical alignment and to minimize the sample disturbance during sample preparation. 4. Sample and test procedure The soil samples used in this research were taken from various sites in Turkey after the 1999 Kocaeli Earthquake as a part of post earthquake investigations. The physical properties of the samples are summarized in Table 1. A total of 98 samples taken from 56 different boreholes were used in this investigation. The soil samples were taken with thin walled shelby tubes of 75 mm in diameter and were extruded by cutting the tubes longitudinal in order to minimize the disturbance. The samples were prepared according to the specifications of the Japanese Standard [21]. The trimmed solid samples had a diameter of 50 mm and a height of 105. Eight strips of filter paper were placed around the sample in order to maintain the quick completion of primary consolidation and to balance of pore water pressure throughout the sample during the test. The samples were isotropically consolidated in the triaxial cell with a given effective pressure and a back pressure. The effective cell pressures were approximately 50% higher than the preconsolidation pressures determined separately by conventional odometer tests to minimize the effect of sample disturbance as suggested by Ladd and Foott [22]. The cyclic axial load was
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Table 1 Summary of the properties of tested samples Depth range of samples (m)
Water content, wo (%)
Void ratio, eo
Liquid limit, wL (%)
Plasticity index, PI (%)
Soil type
2.50–23.55
25–52
0.68–1.40
38–70
9–40
ML, MH, CL, CH
0.04 S1
Cyclic stress ratio, σd / 2σc
0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
0
5
10
15
20 25 30 Number of cycles, N
35
40
45
50
Fig. 1. Actual loading sequence used during first group of tests.
then applied to the sample under undrained conditions with frequency of 0.5 Hz. The investigation program consisted of two sets of tests with different loading stages. The first set was composed of multi stage tests that were used where the number of available test samples is limited. In these tests, samples were first consolidated and later were subjected five cycles of cyclic axial stresses starting from quite low stress amplitudes under undrained conditions. The samples were cyclically loaded step by step with gradually increasing stress amplitudes covering a wide strain range of 103–101% under a constant confining pressure. For evaluation of the stress–strain properties, the 3rd cycle was selected as the representative cycle for that strain range. The second test type was again a multi stage but this time the sample was cyclically loaded to some level of virgin strain and then cyclically loaded at a considerably smaller level of strain. The application of small cyclic strains with increasing and decreasing amplitudes gives no prestraining effects on the dynamic properties for a range of strain as explained in the following paragraphs. The dynamic shear modulus and the damping ratio were again calculated from the stress vs. strain hysterisis loop for the 3rd cycle of every five cycles. On the both group of tests, the failure was defined corresponding to 10% double strain amplitude. The actual
test records of loading sequences for the first group tests are shown in Fig 1. 5. Test results For the first group of tests conducted on normally consolidated samples, an empirical expression for the small strain dynamic shear modulus, Gmax, suggested by Ishihara [23] Gmax ¼ A
ðB eÞ2 0 C ðs Þ ð1 þ eÞ c
(1)
was adopted, where Gmax is in MPa, e ¼ void ratio, and s0c ¼ effective confining pressure in kPa. The constants in Eq. (1) are calculated using a nonlinear curve fitting algorithm based on laboratory measured values as A ¼ 466, B ¼ 3.4, and C ¼ 0.66. In Fig. 2, small strain dynamic shear modulus normalized by the void ratio function, F(e) are plotted against the confining stress, s0c , where F ðeÞ ¼
ð3:4 eÞ2 ð1 þ eÞ
(2)
As can be observed from Fig. 2, the correlation coefficient for the Gmax/F(e) vs. s0c relationship is relatively high indicating well established correlation and validating the adopted Gmax formulation.
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846
30
Gmax/F(e) (MPa)
20
10 9 8 7
Correlation Coefficient = 0.9099
6 5
60
70
80 90 100
200
300
400
500
Effective Confining Pressure, σc′
Fig. 2. Relationship between measured values of Gmax/F(e) and s0c . 400 300
Ha rdin and Black. 1968. Kagaw a. 1992 Kallioglou et all. 1999. Equation (1) Laboratory measured
Dynamic shear modulus, Gmax
200
100 70 60 50 40 30 20
0.55
0.57
0.64
0.66
0.76
0.79
0.81
0.84
0.89
0.97
1.02
1.06
1.07
1.08
1.11
1.17
1.27
1.27
1.40
1.50
10
Void ratio of the samples tested, e
Fig. 3. Comparison of small strain dynamic shear modulus calculated by Eq (1) with the previously suggested relationships and laboratory measured values for different samples.
The small strain dynamic shear moduli determined for the samples tested in the cyclic triaxial system were also calculated using Eq. (1), and relationships proposed by Hardin and Black [7,12] and Kallioglou et al. [24] as shown in Fig. 3 with respect to each other and in comparison to laboratory measured values. The formulation suggested by Hardin and Black [7] is generally valid for stiff kaolinite clay and was based on resonant column tests that yield higher Gmax values when compared to dynamic triaxial test as observed in Fig. 3. The differences between the relationship proposed by Kagawa [12] and this study are limited and most likely due to the testing scheme adopted by [12] where the cyclic tests were conducted right after the completion of the primary consolidation. A very good agreement is observed with the relationship proposed by Kallioglou et al. [24]. The expression proposed by
Kallioglou et al. [24] is for undisturbed clay samples which have similar physical properties used in this investigation. As shown in Fig. 3, the agreement among the Gmax calculated from Eq. (1) and laboratory measured values are relatively good. However, it is interesting to note that plasticity index appears to have no significant effect on the small strain dynamic shear moduli obtained experimentally and modeled relatively well using Eq. (1). Okur and Ansal [25] presented results of the multi-stage loading program to show the trends between plasticity index and the shear modulus ratio and the damping ratio curves based on several additional tests conducted. Fig. 4 shows the dynamic shear modulus reduction and damping ratio increase with increasing shear strain amplitude. The results confirm that especially plasticity index is the governing factor both in dynamic shear modulus reduction
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1.2 C16, PI=40
Dynamic shear modulus ratio, G/Gmax
S3, PI=35
1
C15, PI=27 C12, PI=25 C11, PI=20
0.8
C13, PI=18 S2, PI=15 C10, PI=12
0.6
C9, PI=10 C5, PI=9
0.4
0.2
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) 25 C16, PI=40 C15, PI=27
20
C12, PI=25 C11, PI=20
Damping ratio, D (%)
C13, PI=18 C10, PI=12
15
C9, PI=10 C5, PI=9
10
5
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%)
Fig. 4. Variation of dynamic shear modulus and damping ratio with shear strain amplitude for samples with different plasticity index.
and damping ratio increase as stated in [14,26]. The effect of void ratio is much less significant than plasticity as shown in Fig. 5, most likely because it has much less effect on inter particle forces that govern the fine grained soil behavior.
A simple relationship is adapted to model dynamic shear modulus reduction with cyclic shear strain amplitude using a hyperbolic model
6. Dynamic shear modulus and damping ratio
where ga is cyclic shear strain amplitude, gr is reference strain defined as a function of plasticity index, PI. The empirical relationship for the reference strain, gr was estimated as an exponential function with respect to PI and the constants in this formulation was calculated using
Dynamic shear modulus and damping ratio are the two important parameters in the evaluation of cyclic behavior of soil elements as well as in site response analysis.
G 1 ¼ Gmax 1 þ ga =gr
(3)
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848 1.1
Dynamic shear modulus ratio, G/Gmax
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
IT, PI=12, e=0.83, σ 'c=150 kPa C10, PI=12, e=0.96, σ 'c=200 kPa C1, PI=12, e=1.42, σ'c=300 kPa
0.1 0 0.0001 0.0002
0.0005 0.001
0.002
0.005
0.01
0.02 0.03 0.05
0.1
0.2 0.3
0.5 0.7 1
2
3 4 5
Cyclic shear strain amplitude, γa (%) 24 22 20
IT, PI=12, e=0.83, σ'c=150 kPa C10, PI=12, e=0.96, σ'c=200 kPa C1, PI=12, e=1.42, σ'c=300 kPa
Damping ratio, D (%)
18 16 14 12 10 8 6 4 2 0 0.0001 0.0002
0.0005 0.001
0.002
0.005
0.01
0.02 0.03 0.05
0.1
0.2 0.3
0.5 0.7 1
2
3 4 5
Cyclic shear strain amplitude, γa (%)
Fig. 5. Effects of void ratio on dynamic shear modulus reduction and damping ratio increase with cyclic shear strain amplitude.
a nonlinear regression analysis as gr ¼
1 21 1 expð106 PI1:585 Þ
(4)
A similar empirical expression was also used to model the increase in damping ratio with cyclic shear strain amplitude based on plasticity index: D ¼ C0 þ C1 C2PI C3 expðC4 ga C5PI Þ (5) where C0–C5 are empirical constants that were determined using a nonlinear curve fitting algorithm based on the cyclic triaxial test results as C0 ¼ 6.3, C1 ¼ 19.2, C2 ¼ 0.9976, C3 ¼ 1.54, C4 ¼ 8.73, and C5 ¼ 0.99.
The modeling achieved by Eqs (4) and (5) are presented with respect to test data in Fig. 6 and as can be observed, the proposed empirical models for the modulus reduction and damping ratio increase with respect to cyclic shear strain amplitude are yielding relatively very good fits for the given test results. Hardin and Drnevich [10,27] have studied the relationships between dynamic shear modulus and damping ratio and suggested to adopt G D¼f (6) Gmax for sandy soils. Based on the results obtained in this study as shown in Fig. 7, it was possible to establish by simple
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a
849
1.2
Dynamic shear modulus ratio, G/Gmax
C16, PI=40 1
S3, PI=35 C12, PI=25 C11, PI=20
0.8
S2, PI=15 C5, PI=9
0.6
0.4
0.2
0 0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%)
b
25 C16, PI=40
Damping ratio, D (%)
20
C15, PI=27 C10, PI=12
15
10
5
0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) Fig. 6. Comparison between modeled and measured: (a) Dynamic shear modulus reduction and (b) damping ratio increase with cyclic shear strain amplitude.
regression analyses the following relationship: G D ¼ 18:114 þ 20:033 G max
(7)
The purpose is to develop a simple approach to estimate the damping ratio and damping ratio increase with cyclic shear strain amplitude as a function of to dynamic shear modulus ratio. Since it is easier and less time consuming and less susceptible to possible errors, one may need to
calculate damping ratio based on dynamic shear modulus. The correlation coefficient (R ¼ 0.9799) being relatively high would justify such a simplification. 7. Dynamic behavior The amplitude of cyclic shear strain at which a clear degradation in stiffness begins to occur has been known to depend on several factors such as effective confining stress
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850
25
D = -18.114(G/Gmax) + 20.033
Damping ratio (%)
20
R2 = 0.9602
15
10
5
0
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Dynamic shear modulus Ratio, G/Gmax
0.8
0.9
1
Fig. 7. Relationship between dynamic shear modulus ratio and damping ratio.
Cyclic stress ratio amplitude, σd/2σc
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1
0
20
40
60
80
100
120
Number of cycles, N Fig. 8. The loading scheme for the second group of tests.
and number of cycles. If the effective confining stress is small, dynamic shear modulus ratio, G/Gmax and damping ratio, D start to change at lower levels of cyclic shear strain amplitudes for saturated fine grained soils [26,28–31]. It was observed that there is a lower limit below which cyclic degradation will be negligible and that this lower limit threshold would be dependent on the pore water pressure buildup. The studies on the influence of plasticity index also showed that the value of threshold generally increases with the plasticity index. The second group of dynamic triaxial tests was conducted to determine the threshold between nonlinear
elastic and elasto-plastic behavior and the undrained stress–strain behavior of fine grained soils under irregular cyclic loading. The samples were consolidated to effective stresses equal to the calculated in situ mean effective stress from consolidation tests. At the initial stage, cyclic loads were applied from small stress levels and increased after every five cycles. When the pore water pressure began to build up, the stress level once again decreased to the initial value, then increased again as shown in Fig. 8. Loading and unloading cycles showed that the nonlinear elastic threshold is approximately equal to strain levels corresponding to the 90% of G/Gmax (Fig. 9) that appears
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1.1
A
Dynamic shear modulus ratio,G/GMAX
1
B Sample Y8, PI=28
C
0.9
E 0.8 0.7 0.6
G D
0.5 0.4 0.3
H
F
0.2 0.1 0 0.002 0.003
0.005
0.01
0.02 0.03
0.05 0.07 0.1
0.2
0.3 0.4 0.5 0.7
1
2
3
4 5
Cyclic shear strain amplitude, γa (%)
Fig. 9. Variation of shear modulus ratio in the second group of tests.
1.2
25
1
N=15
N=150 20
N=75
0.8
N=115
15
0.6 10
0.4
5
0.2 0 0.0001
Damping ratio, D (%)
Shear modulus ratio, G/Gmax
SB1
0.001
0.01 0.1 Cyclic shear strain amplitude, γa (%)
1
10
0
Fig. 10. Variation of shear modulus and damping ratio for different cycles.
to be in the range of 0.01–0.02%. If the stress amplitude is reduced prior to point B (Fig. 9), dynamic shear modulus returns back to its initial value, but when the stress amplitude level exceeds the limit defined by point B, particle structure deformations starts to take place and upon unloading a reduction in the dynamic shear modulus is observed depending on the order exceedance as shown by point C and later by point E that was unloaded from point D (Fig. 9). The modulus reduction observed is due to the rapidly increasing levels of cyclic strains leading to particle structure breakdown and related softening. However, below this threshold point B, engineering properties of soil layers remain practically unaffected. Point F represents approximately the flow threshold, the unloading before this
threshold would lead to recovery of the approximately 60% of the initial Gmax (point G), however once this threshold is exceeded, the recovery is only in the order of 20% as shown by point H in Fig. 9. As pointed out before, the cyclic stress ratio were increased after every five cycles in the first group of tests. Fig. 10 shows the variation of dynamic shear modulus ratio and damping ratio against cyclic shear strain amplitude for different cycles for test SB1. The hysterisis loops corresponding to different stages of loading are given in Fig. 11. As expected, the shape of the hysterisis loops for the cycles 10–15 have the same area for every cycle that indicates nonlinear elastic response of the soil element. The dynamic shear modulus ratio is at its 100% value. The nonlinear
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0.06
SB1 Cyclic stress ratio amplitude σd/2σc
Cyclic stress ratio amplitude σd /2σc
0.01 0.008 0.006 0.004 0.002 0 -0.002
N=11-15
-0.004
Gmax=34 MPa
-0.006
SB1
0.04
0.02
0
-0.02
N=71-75 G=26.8 MPa
-0.04
G/Gmaks=76 %
G/Gmax=98 %
-0.008 -0.002 -0.0015 -0.001 -0.0005
D=5.6 %
D=3.6 %
0
-0.06 -0.02
0.0005 0.001 0.0015 0.002 0.0025 0.003
-0.015 -0.01
Cyclic shear strain amplitude, γa (%) 0.5
SB1
Cyclic stress ratio amplitude σd /2σc
Cyclic stress ratio amplitude σd /2σc
0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1
N=116-120 G=12.9 MPa G/Gmaks=36.6% D=12.6%
-0.15 -0.2 -0.25 -0.15
-0.005
0
0.005
0.01
0.015
0.02
0.025
Cyclic shear strain amplitude, γ (%) a
SB1
0.4 0.3 0.2 0.1 0 -0.1 -0.2
N=146-150 G=1.5 MPa G/Gmax=4% D=20.3%
-0.3 -0.4 -0.5
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-2
0.25
Cyclic shear strain amplitude, γa (%)
-1
0
1
2
3
4
Cyclic shear strain amplitude, γa (%)
Fig. 11. Shapes of hysterisis curves for different cyclic shear stress amplitudes and cycles.
elastic response begins to change at cycle 65–70 corresponding to point B in Fig. 9. It must be underlined that there is no exact transformation point between nonlinear elastic and elasto-plastic response of cohesive soils. But it can be seen from Fig. 9 that after decreasing of stress from point B back to the initial value, the curve returns to point C with a reduction of the dynamic shear modulus ratio. The area of the loops begin to change at the cycles between 115 and 120 despite the same cyclic stress ratio which means the plastic alteration began in the particle structure of the sample. Between cycles 145 and 150, the change is very evident. This transformation point is defined the flow threshold, gP, where the sample is behaving in a viscoplastic mode since it rigidity is approximately 10% of it is initial
value and it has reached the steady state phase where the residual strength is mobilized. Even though it is very empirical, one conventional procedure that can be adopted is to draw three tangent straight lines to three parts of the modulus reduction curve as shown in Fig. 12 and estimate the elastic and flow threshold strain levels from the intersection of these three straight lines. Similar response patterns were observed for all samples tested and correlations were established for the elastic and flow strain thresholds in terms of plasticity index, where gE ¼
0:035 1 þ 11:92 expð0:1PIÞ
(8)
ARTICLE IN PRESS Dynamic shear modulus ratio, G/Gmax
D.V. Okur, A. Ansal / Soil Dynamics and Earthquake Engineering 27 (2007) 843–854
853
1.2 1 0.8 0.6 γE
0.4
γP
0.2 0 0.0001
0.001
0.01
0.1
1
10
Cyclic shear strain amplitude, γa (%) Fig. 12. Tangents for nonlinear elastic, elasto-plastic and viscoplastic phases of the dynamic modulus reduction response and estimation of the elastic and flow thresholds.
gP ¼
1 1:39 0:33PI 0:28
(9)
thresholds were determined using an empirical approach and simple approximate correlations were proposed based on the plasticity index.
8. Conclusion References Cyclic behavior of saturated fine grained soils were studied in detail based on a set of undrained stress controlled dynamic triaxial tests conducted on low to medium plastic fine grained soil samples obtained from various sites in Turkey. The idea was to confirm the effecting soil parameters and to develop semi-empirical correlations to evaluate the dynamic properties of cohesive soil layers. The results confirm the strong influence of both plasticity and the effective confining stress on the behavior of the low-medium fine grained soils. An empirical expression was developed to calculate the maximum dynamic shear modulus, Gmax based on void ratio and confining pressure. The strain-dependent change of dynamic shear modulus ratio, G/Gmax and damping ratio, D, of fine grained soils is highly related with the plasticity index. Since the plasticity index is easily determined parameter in geotechnical engineering, empirical relationships were formulated for fine grained soil modulus reduction and damping increase curves. The curves determined for the shear modulus ratio agrees very well with the test data and the literature. One of the important features of fine grained soil layers subjected to cyclic excitations is the reduction of stiffness. The results obtained from a series of tests with changing stress amplitudes on undisturbed soil samples have indicated that the reduction of stiffness starts to become effective once a critical cyclic shear strain level defined as elastic threshold is exceeded. It was also possible to define a second critical strain level defined as flow threshold where the soil sample has reached the steady state conditions and starts to behave as viscoplastic material. These two critical
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