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Mathematical Study on Dynamic Shear Modulus and Damping Ratio of Seashore Soft Soil

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Mathematical Study on Dynamic Shear Modulus and Damping Ratio of Seashore Soft Soil 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 - 2509 - Vol. 17[2012], Bund. Q 2510 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. Vol. 17[2012], Bund. Q 2511 Gmax G 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 C E o D 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): A λ = 1 (1) π 4 A2 where, A1 is the area enclosed by hysteresis hoop ACBDA, A2 is the area of △AOE. Vol. 17[2012], Bund. Q 2512 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 (2) G 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: G 1 f == (3) + γ Gamax 1 According to the knowledge of advanced mathematics, it is easy to get that the formula 3 has the following properties (Wang, 2012): =∞= ff(0) 1, ( ) 0; ddff− =−aa(1 +γ )2 < 0, =− a ; (4) γγγ =0 dd 2 d f =+>23γ − 2 2(1)aa 0 dγ 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: Gaγ λλ=−=(1 ) (5) max + γ Gamax 1 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. Vol. 17[2012], Bund. Q 2513 /G 1 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 ) (6) G 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: G −kγ Fe== (7) Gmax where k is a positive undetermined parameter. From the mathematical analysis of the equation (7), we can obtain the mathematical nature as follows: (8) Vol. 17[2012], Bund. Q 2514 Line /G 1 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.
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