Maxwell model geotextile encased stone column in soft soil improvement

Use your smartphone to scan this QR code and download this article ABSTRACT In the field of geotechnical – soft soil improvement, themathematical model or mechanical model is one of the important input parameters for the design calculations or studies. The determination of the appropriateness of the models has a great influence on the accuracy results of design and calculation as well as the sustainable stability of soft ground after improvement. On the contrary, the selection of inadequate calculationmodelswill lead to increased costs of soft soil improvement, possibly even leading to the destabilization of the work and causing immense loss of people and property. Recently, many projects major highway after construction design in use has not meet the requirements of the standard, leading to wasted money and time of individuals, organizations, and the state of post-treatment. Therefore, the research and application of usingmathematical or mechanical models in accordance with the new soft soil improvement method will greatly help as well as add additional options for soft soil improvement in Vietnam. The soft soil deformation is not only related to load but also to load time. The change in stress and deformation of weak soil over time is called rheology, and in this study is the viscoelastic behavior. From the above reasons, we try to apply a generalized Maxwell model to explain the viscoelastic behavior of a soft soil. In particular, the time-dependent behavior of a viscoelastic soft soil was represented by using the Maxwell rheological model. The Matlab programming code helps to solve numerically all the equation of themathematical exhibition of the generalizedMaxwellmodel results. We acknowledge that the generalized Maxwell model is superior in demonstrating the time-dependent behavior of soft soil. The results probably show that this is one of the effective models to predict the behavior of soft soils in ground improvement with GEC.


INTRODUCTION
The nature of soft soil such as low bearing capacity, high compressibility and low permeability cause difficulties for designing/constructing geotechnical engineering structures such as embankments, petroleum refinery, thermal power plants and infrastructure on them. Encased column is a successful well-known method that reduce the soft soil problem and can be improve ground and bearing system in very soft soils with undrained shear strength c u < 5 kN/m 2 . The load and the loading time exacerbate the soft soil deformation. Rheological properties describe the variety of soft soil stress and strains during the time such as creep, stress relaxation, long-term strength, and after effect elasticity 1 . Next the theory of the rheological model will be described and then a generalized Maxwell model is to constitute the mechanical rheological properties of soft soil. An ordinary differential equation is generated from the designed model then numerically solved by a Matlab programming code. The relaxation modulus E (t) of the generalized Maxwell model and its application are discussed. Thereafter, this study includes a manual application of the generalized Maxwell model to the time-dependent settlements of GEC for improvement of soft soil.

METHODOLOGY Theory of Maxwell Model and Generalized Maxwell Model
We begin with examining all the characteristics of rheological viscoelastic behavior. Then we can reconstruct the behavior of the soft soil system with changeable elements to account for the elasticity and viscosity. To identify the stress inside the soft soil, we need to measure the force of a soft soil.
Where σ represents the stress, F the force and A the surface of the cross section of soft soil perpendicular to the direction of the force.
We also assume that the behavior of soft soil as a linear-elastic soil as written in Hooke's Law (equation (2)).
Where σ describes the stress of the spring in MPa, E the elastic modulus in MPa and ε the applied strain. A simple model that describes such behavior would be an ordinary spring to which a strain is applied. A simple model could only illustrate the linear increase of the stress in the time, but after this time the relaxation does not appear at all. So, the process of relaxation must be considered into the model. In rheology, this phenomenon is called viscosity, and in a model, it is represented by a dashpot in series with a spring 2 . This system is called a Maxwell element (Figure 1). To model viscous behavior of a soft soil, a dashpot with a known stress-strain relation is used. Stress of dashpot elements is related with strain differentiated by time t, and the constitutive relation is time- With ε is the strain, σ d is the stress of the dashpot, σ s is the stress of the spring and η is the viscosity of the soft soil.
To describe the appropriate behavior of this system a constitutive equation that accounts for both the strain of the dashpot as well as the strain of the spring is needed. The constitutive equations of the series model of spring and dashpot are as follows 2 : Figure 2 shows the relaxation of stress response σ (t) under strain input ε 0 applied at t = 0 and kept constant 2 . Thus, the Maxwell model is characterized by the following simple relaxation function.

Model Description
The behavior of geotextile encased stone column (GEC) on soft foundation soil has been idealized by the proposed foundation model ( Figure 4) and assuming soft ground was supposed as a viscoelastic material. The paper focused on finding the suitabil-

Estimating the viscosity and strain rate
The viscosity and strain rate of the soft clay will be taken from the studies of Anders Augustesen, Morten Liingaard and Poul V. Lade (2004) 6

Varying the relaxation times and elasticity modulus
The relaxation time is dependent on the value of young's modulus E and the viscosities η. In this paper α = 0.5 will assumed for all the analysis 2 .

Settlement calculation of GEC and the timedependent behavior of soft clay soils
The ground improvement works are to provide a steady platform to support the operation of SL6000 (Kobelco) and the CC2800 (Terex Demag) cranes on the designated routes, which can support up to 500 kPa of transient loads and long-term primary settlement should be less than 250 mm. The soft clay is reinforced by the GEC with a diameter of 0.8 m, depth of treatment from 12 m and the tensile stiffness of the geotextile encasement J = 3000 kN/m. The GEC are arranged in square grids with spacing 2.3 m.
The settlement was estimated based on Raithel and Kempfert's analytical calculation 10 . This calculation is conducted using data obtained from soil parameters in Table 1. ε To study the suitability of the generalized Maxwell model, the viscosity η is set to 0.1, 0.5, 1, 5 and 10 MPa·d and strain rate . ε is checked with 1%, 5%, 10% and 20%. How to determine the Maxwell model parameters is a major concern, in this section some issues are considered to evaluate the response of the generalized Maxwell model described above. Several studies are performed and relaxation time is considered, which is usually used to assess the time-dependent behavior of viscoelastic soft soil.

Constant strain rate with different viscosities
In this case the influence of viscosity on the generalized Maxwell model is studied. The numerical simulations are carried out with different number of Maxwell elements. It can be seen that when the viscosity or the number of Maxwell elements increases greater values of stress are  obtained. This is highlighted in Figure 5, in which the effect of numbers of Maxwell elements on the maximum stress σ reached it can be observed.
The general observation is that the higher the viscosity, the higher the stresses for a certain strain rate. For example, Figure 5 (a and b) show the comparison of the stress versus time by viscosity 5 Mpa . d and 10 MPa . d with same constant strain rate 5% per day for a period of 500 days, and it is observed that the stress reached a maximum level 1,428 MPa and 2,365 MPa respectively after a period of time of 20 days for case 5 Maxwell elements. The evolution of stress with time for different number of Maxwell elements is shown in Figure 5. As can be seen the stress response is nonlinear although the strain rate has a linear variation with time.

Constant Maxwell elements with different strain rates and viscosities
In this case two different viscosities with five different strain rates are considered. From a point of view of the sensitivity to constitutive parameters it can be said that when viscosity decreases, the stress is lower for the same time.
It is seen that the maximum stress moves to the right faster for higher strain rate. In Figure 6, the results of the generalized Maxwell model are shown, and it seems that the bigger the strain rate, the stiffer the soil. Figure 6 also shows the results obtained by changing strain rate from 1% to 20% while keeping the other parameters constant. It can be seen that the smaller the value of strain rate, the longer the time to reach the maximum stress. An example, Figure 6a shows the result of strain rate 1% per day, during the first 100 days an increasing strain is applied, the strain rate is constant and equals 1% per day, during these days the stress appears to increase non-linearly with the strain. After 100 days the strain remains constant at maximum strain, and the stress decreases exponentially to a stress-relaxation limit. Figure 7 presents the effects of strain rate on the stress of the generalized Maxwell model for soft soil. For the short period of time (1 to 5 days), the stress is found to increase linearly with strain rate before the required strain value is reached. The stress developed almost linearly with low strain rate (1% to 10% per day) until the maximum strain is reached and then decreases with high strain rate for a period of 10 to 20 days. Finally, for the larger period of more than 50 days, the stress is gradually reduced from very low strain rate to very high strain rate.

Settlement calculation of GEC
The settlement of a GEC is estimated using the elastic modulus E and relaxation modulus E (t).
It is shown that the viscosity mainly affects the overall relaxing rate of the foundation soil. For the same time and same strain rate, the settlements increase with the decrease of viscosity. The increase of Maxwell elements causes smaller values of settlement, when the viscosity and strain rate are same. Fig. 8 shows the results at different moduli of relaxation modulus E (t) under constant loading with the viscosity η = 5 Mpa . d and 10 Mpa . d. It is easy to make the conclusion that the smaller the modulus of relaxation E (t) is, the bigger the settlement is, and the smaller the strain rate to reach the

LIMITATION OF THE PROPOSED APPROACH
The proposed analysis of using relaxation modulus for estimating the settlement of GEC is primarily intended for understanding the behavior of viscoelastic soils. It has been mentioned in the study that the generalized Maxwell model is capable of representing the behavior of a viscoelastic soil when considered in terms of the short-term or long-term settlement under loading followed by a relaxation phase under sustained loading. The used data surely restricts the approach of estimating the generalized Maxwell model parameters being valid only for the loaded viscoelastic soft soils. The verifiable relationship between the generalized Maxwell model parameters and soft soil properties is an important problem to be solved, which needs the support of sufficient measurement data from laboratory and field experiments.

CONCLUSIONS
Based on the above studies, the following conclusions can be made: • Based on the theory of linear viscoelasticity, a generalized Maxwell viscoelastic model is developed to account for the time-dependent behaviour of soil foundations improvement with GEC under concentrated line load.
• Analytical solution of settlement in the foundation was estimated based on Raithel and Kempfert's analytical calculation model with Relaxation Modulus.
• The viscoelastic theory shows its potential in modeling the long-term foundation deformation.

CONFLICT OF INTEREST
The authors pledge that there are no conflicts of interest in the publication of the paper.