Mathematical Study on Dynamic Shear Modulus and Damping Ratio of Seashore Soft Soil Chuang Yu College of Architecture and Civil Engineering, Wenzhou University, Wenzhou 325035, China
Wei Wang* Department of Architecture, Shaoxing University, Shaoxing 312000, China *corresponding author, email:
[email protected]
Jinjun Guo Department of Civil Engineering, Luoyang Institute of Science and Technology, Luoyang 471023, China
ABSTRACT In order to understand dynamic behavior of soft soil, mathematical study on dynamic shear modulus (DSM) and damping ratio is carried out. Based on mechanism of developing process, primary mathematical behavior which DSM model should fulfill is analyzed. What's more, conventional hyperbolic model for DSM is introduced and its shortcoming is signalized. According to the investigated data, a new exponential model for DSM is proposed and new damping ratio formula is deduced, which can well describe dynamic behavior of seashore soft soil. Mathematical relationship between hyperbolic model and exponential model for DSM is proved. Finally, a good accuracy of the new model is proved by two tested data. This research is useful for relative engineering numerical simulation.
KEYWORDS: Dynamic shear modulus, damping ratio, mathematical model, seashore soft soil
INTRODUCTION Seashore soft soil is widely distributed in China's Yangtze River Delta region and other seashore areas, which has high water content and low strength. Such kind of soft soil has obvious structure performance. Once subjected to vibration disturbance, its original structure will be damaged gradually, and its strength will decrease significantly. At present, with the rapid development of economic, highway, high-speed railway and magnetic levitation transportation systems are greatly increased in the Yangtze River Delta region and other seashore areas. Due to the special geological conditions, most of these projects are inevitably built on seashore soft soil ground. The impact of vibration on the soil ground should be considered in these projects design. The dynamic behavior of seashore soft soil is an important subject in civil engineering such as
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environment engineering, disaster prevention and reduction engineering, and so on. Earthquake, high-speed vehicle vibration and cycle wave load all belong to the scope of this subject. Dynamic shear modulus G and damping ratio λ are two important parameters of the soil dynamic properties. Furthermore, they are also the requisite dynamical parameters in the seismic safety evaluation of the project site and the soil seismic response analysis. Whether these parameters are reasonable or not will directly affect the safety and the economy of engineering structure (Lv, et al. 2003; Cai, et al. 2010; Chien and Oh 1998). At present, the curves of G and λ variation with the dynamic shear strain amplitude γ are applied in the site seismic response analysis, seismic stability evaluation of geotechnical structures, the dynamic soil-structure interaction analysis and the offshore site seismic stability evaluation (Yuan, et al. 2000; Zhang, et al. 2005; Shang, et al. 2006). Many scholars have done a lot of comprehensive study for G and λ from the laboratory investigation and the theoretical model, and have achieved fruitful results. Lin (2010) studied the shear modulus and damping ratio characteristics of gravelly deposits. Lv (2003) conducted the site test in Bohai seabed oil field, and given the recommended value of G and λ of various types soils. Li (2006) researched the dynamic characteristics of the soil in the Taiyuan area, and summed up its dynamic characteristics. Cai (2010) conducted the study of the typical soil’s G and λ of the Fuzhou region. Ma (2003) discussed the silty soil’s G and λ in the Yellow River Delta. Chen (2004; 2005) studied in detail the variation of the recently deposited soil’s G and λ in Nanjing and neighboring areas. Zhang (2004) studied the dynamic characteristics of soil in Tianjin seashore beaches. Hardin and Drnevich (2002) proposed the use hyperbolic model to fit the degradation law about G increased with γ. Zhu and Wu (1988) pointed out that the hyperbolic model had big errors in the low-amplitude vibration by experimental observation. Li (2006) pointed out that the hyperbolic model had underestimated damping ratio when the dynamic action level below a certain value, based on three types of cohesive soil’s G and λ test results. Therefore, it is necessary to analyze deeply the relationship of G and λ with γ based on test results statistics, and find a reasonable mathematical model to describe, and apply to engineering. This paper starts with the basic nature which the mathematical model should have, try to put forward a new G-γ model, determines λ of seashore soft soil, and demonstrate its relationship with the traditional model.
FUNDAMENTAL THEORIES Dynamic Shear Modulus G Degradation of the dynamic shear modulus G is usually expressed as G-γ relationship. According to a great quantity of different soil tests, G-γ variation is shown in Fig. 1. From Fig. 1, it can be seen that a reasonable G-γ mathematical model must have the following basic properties: (1) When γ is zero, the corresponding G is the maximum Gmax; (2) When γ tends to infinity, the corresponding G goes to zero; (3) The first derivative of model is less than zero to ensure that G is monotonically decreasing with γ; (4) The second derivative of model is greater than zero to ensure that G is the concave function of γ.
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G
Gmax
0
γ Figure 1: Investigated G-γ curves
Damping Ratio λ Damping ratio λ of soil means the viscous nature of soil, usually using hysteresis hoop to show the impact of the viscosity of the soil on stress-strain relationship. The size of this impact can be measured from the shape of the hysteresis loop, as shown in Fig. 2.
dynamic stress
σd
A
E
C D
o
γ
dynamic strain
B
Figure 2: Dynamic hysteresis loop and damping ratio of soil Damping ratio λ is the ratio of actual damping factor and critical damping factor, and can be calculated from the relationship between the energy loss coefficient (Chen, et al. 2007):
λ=
A1 4π A2
where, A1 is the area enclosed by hysteresis hoop ACBDA, A2 is the area of △AOE.
(1)
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TRADITIONAL HYPERBOLIC MODELS The Model Expression According to the variation regulation of dynamic shear modulus with the dynamic strain in Fig. 1, 1/G and γ are linear relationship basically:
1 = 1 + aγ G
(2)
where a is the positive undetermined parameter. Based on the statistics above, Hardin and Drnevich (2002) gave the empirical formula of G/Gmax and γ, named as the hyperbolic model:
f =
G 1 = Gmax 1 + aγ
(3)
According to the knowledge of advanced mathematics, it is easy to get that the formula 3 has the following properties (Wang, 2012):
f (0) = 1, f (∞) = 0; df df = −a (1 + aγ ) −2 < 0, dγ dγ d2 f 2 = 2a 2 (1 + aγ ) −3 > 0 dγ
γ =0
= − a;
(4)
Equation (4) shows that the hyperbolic model can satisfy the basic mathematical properties required by the reasonable dynamic shear modulus model. The parameter a is the opposite number of the initial slope of G-γ. Using λ max as the initial damping ratio, the damping ratio λ based on the hyperbolic model can be expressed as:
λ = λmax (1 −
G aγ )= Gmax 1 + aγ
(5)
Limitations of the Model Traditional hyperbolic model has been widely used due to its simple expression and other advantages. With the gradual deepening of the study, it was discovered that traditional hyperbolic model had larger fitting error. In fact, 1/G-γ is not a simple straight line but approximately a broken line, and it has line points, especially the broken line of loose specimens in small confining pressure is more obvious, shown as in Fig. 3. Using the linear equation based hyperbolic model, usually lead to the phenomenon that forepart shear modulus is smaller and the second half is larger (Zhu and Wu 1988; Kallioglou et al., 2009). For the loose soil specimens in low confining pressure, this bias can be very large. These biases are usually beyond those permitted in engineering, so it is necessary to conduct in-depth study to establish a more appropriate mathematical model.
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1/G
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0
γ Figure 3: Investigated 1/G -γ curves
A NEW MATHEMATICAL MODEL Expression of Dynamic Shear Modulus Based on 1/G-γ diagram in Fig. 3, we propose the exponential function to simulate it, namely:
1 = exp(kγ ) G
(6)
According to the nature of exponential function, Equation 6 is a typical concave function. 1/G-γ curve is composed by the approximate two broken line can be effectively simulated according to the change of the parameter k. 1/G-γ curve simulated by hyperbolic and exponential function is shown in Fig. 4. From Fig. 4, the fitting effect of the exponential function is significantly better than the hyperbolic model. By Equation 6, the exponential model of the G-γ decay curve of seashore soft soil can be expressed as:
F=
G = e − kγ Gmax
(7)
where k is a positive undetermined parameter. From the mathematical analysis of the equation (7), we can obtain the mathematical nature as follows:
(8)
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1/G
Line
Exponential model
0 γ Figure 4: Fitted results for 1/G-γ of hyperbolic and exponential model Equation 8 describes that the exponential model of the G-γ can fully meet the basic mathematical properties proposed in Section 2. Meanwhile we can get that the parameter k is the opposite number of the initial slope of G-γ, and it has a clear physical meaning. The value is mainly related to the confining pressure.
Expression of Damping Ratio According to the definition, the formula based on damping ratio of the new model as follow:
λ = λmax + b(1 −
G ) Gmax
= λmax + b(1 − e − kγ )
(9)
= λmax + b − be − kγ where λ max has the same meaning as mentioned before, b is the undetermined parameter, which are all related to the basic nature of the soil.
Comparison with the traditional model Both of the exponential model which is proposed by the author and the traditional hyperbolic model can meet the basic mathematical properties of the G-γ curve in Section 2. In order to analyze their advantages and disadvantages, we need to further discuss the mathematical relationship between them. Make the two models have the same initial slope so as to compare expediently, that is, take a=k, then:
F − f = e − kγ −
1 1 + kγ
(10)
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Taking MacLaurin expansion for the exponential function in the above, it gives:
e − kγ =
1 e kγ 1
= 1 + kγ + ≤
k γ k γ3 k mγ m + + + + m! 2! 3! 2
2
3
(11)
1 1 + kγ
where m is the natural number. Substituting Equation 11 into Equation 10, we can get:
F≤ f
(12)
G/Gmax
1 f (γ) 0
1
F (γ)
k γ
Figure 5: Relation between hyperbolic and exponential models The above equation shows that, the hyperbolic model is always on the upward side of the exponential model when they have the same initial slopes, as shown in Fig. 5. This means that when the initial decay rates are the same, the G/Gmax decay rate simulated by the hyperbolic model is smaller than the corresponding exponential model. This may be the reason that the new model of this paper is more applicable in seashore soft soil.
TEST DATA VALIDATION Dynamic Shear Modulus Fitting The author has conducted a preliminary experimental study on the dynamic shear modulus of seashore soft soil in the seashore areas of Zhejiang, and selected two groups measured G/Gmax-γ curves for the new model fitting. The result is shown in Fig. 6. The points in the figure indicate the measured data and broken lines indicate the fitting results.
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1.2
max
0.8
G /G
1.0
0.6 0.4 0.2 0.0 0
5
10
γ / 10
15
20
-4
Figure 6: Tested and fitted G/Gmax-γ of seashore soft soil
Damping Ratio Fitting Zhang (2004) has done a detailed research for the damping ratio of the soil in Tianjin seashore tidal flats. Use the equation (9) of the new model to fit the damping ratio of C5-01 soil specimen, and take λ = 0.035 measured when γ is 2*10-6 as the initial value. The results are listed in Table 1. Table 1: Tested and fitted damping ratio of seashore soft soil γ/10
-4
0.05
0.1
0.3
0.5
1
3
5
10
Tested λ/10-2
3.58
3.72
4.25
4.72
5.73
8.10
9.30
10.20
Fitted λ/10-2
3.63
3.76
4.24
4.69
5.67
8.15
9.31
10.18
Relative error /%
1.37
0.95
- 0.28
- 0.74
- 1.14
0.65
0.14
- 0.22
The comparisons between the tested and fitted damping ratio show that the exponential model has a higher fitting accuracy, especially has the highest fitting accuracy for the damping ratio. This has certain promotional value.
CONCLUSIONS (1) The G-γ mathematical model of seashore soft soil must have the following basic properties: When γ is zero, the corresponding G is the maximum Gmax; when γ tends to infinity, the corresponding G is zero; the first derivative is less than zero, and the second derivative is greater than zero. (2) The 1/G-γ curve of seashore soft soil is usually not a straight line, but a curve with obvious turning points, resulting that the traditional hyperbolic model has a larger fitting error, while the exponential model can better fit the 1/G-γ curve.
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(3) When the initial slopes are the same, the hyperbolic model of the dynamic shear modulus is always above the exponential model, which may be the reason that the new model in this paper is more suitable for the seashore soft soil. (4) The exponential model proposed in this paper can effectively fit the G/Gmax-γ curve and λγ curve of the seashore soft soil. It is worth emphasizing that the present model is only validated by several typical soil test data, and it remains to do further discussion on the applicability to other soil types. In addition, the statistical rules and determination methods for the change of the model parameters with confining pressure and the soil types need to do further study.
ACKNOWLEDGMENTS The authors thank the reviewers who gave a through and careful reading to the original manuscript. This work is supported in part by the National Natural Science Foundation of China (NO. 41202222, NO. 41002091), the Key Project of Chinese Ministry of Education (NO. 211068), and the Nature Science Foundation of Zhejiang Province (NO. Y1080839).
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