Study on Deformation Law of Circular Foundation Under Combined Loading

ZHANG Qiyi, and SHI Hongda

1) Department of Ocean Engineering, College of Engineering, Ocean University of China, Qingdao 266100, P. R. China

2) Key Laboratory of Ocean Engineering of Shandong Province, Qingdao 266100, P. R. China

Study on Deformation Law of Circular Foundation Under Combined Loading

ZHANG Qiyi1),2),*, and SHI Hongda1),2)

1) Department of Ocean Engineering, College of Engineering, Ocean University of China, Qingdao 266100, P. R. China

2) Key Laboratory of Ocean Engineering of Shandong Province, Qingdao 266100, P. R. China

The foundations of some ocean engineering structures are built to withstand not only the vertical gravity loadV, but also the horizontal loadHinduced by sea waves and current. The horizontal load includes the concentrated force load, the moment loadM, and the torque loadTtermed also as combined loading. It is of academic and engineering significance to study the deformation law of submarine seabed due to combined loading. On the basis of the three-dimensional elastic mechanics solution of circular foundation, numerical methods are used to analyze the deformation law of submarine soil under circular foundation with six degrees of freedom. The finite element analysis results give the elastic deformation law of soil in three dimensional spaces, modify the theoretical elasticity solution, and presents nonlinear soil deformation mechanism under the circular foundation with six degrees of freedom.

degrees of freedom; circular foundation; deformation law; combined loading; elastic mechanics solution

1 Introduction

Although a variety of sophisticated constitutive laws have been developed to describe elastic-plastic soil behavior during the last decades (Bransby and Randolph, 1999; Cassidyet al., 2002, 2004; Gottardi and Butterfield, 1993, 1999), solutions based on linear elastic theory are still a convenient and rational tool in geotechnical practice provided that the level of induced stresses in the soil is considerably lower than its ultimate strength.

Traditional foundation deformation law is based on the theory of elastic mechanics (Endleyet al., 1981; Al- Bermani and Kitipornchai, 1990; Gazetaset al., 1985; Georgiadis and Butterfield, 1988; Nova and Montrasio, 1991; Poulos and Davis, 1974; Selvadurai, 2002). It is aimed at continuum in the half-space, mainly applied to monotonic loading and capable of solving linear elasticity deformation law on a single load, such as the concentrated load, the moment load or the torque load. When the above mentioned classical elasticity theory is applied to soil, the soil behaves elasticity-plasticity at the beginning of the loading (Martin, 1994) because it is granular material. Therefore, the deformation law should be modified for soil application.

In addition, foundations of some ocean structures are influenced not only by huge gravity load, but also by horizontal and torque loads produced by environmental loading such as wind, waves, and currents (Zhang, 2008). The horizontal load is also referred to as the combined loading. There is a strong coupling between soil deformation and load component under the combined loading (Barari and Ibsen, 2012) which is particularly important to study the soil deformation law under this loading.

This study focuses on the deformation law of a circular foundation on seabed, and the vertical, horizontal, moment and torque loads acting on the foundation are investigated. Based on theoretical results (Poulos and Davis, 1974; Butterfieldet al., 1997), numerical methods are adopted to study the coupling relationship between the deformation law of seabed soil and the load components acting on the footing, and the effect of the Poisson’s ratio on the soil deformation law.

Fig.1 shows the numerical model used in this study, whereVis the vertical load,HxandHyare thex- andy-component of the horizontal load, respectively,MxandMyare thex- andy-component of the moment load, respectively, andTis the torque load.

Fig.1 Circular foundation model with six degrees of freedom.

2 Theoretical Analysis and Numerical Calculations

This paper gives the theoretical derivation of soil deformation laws under monotonic loading using elastic theory, and at the same time, the soil deformation law under combined loading mode is studied in detail using the Galerkin finite element method.

2.1 Theoretical Analysis

Poulos and Davis (1974) studied the soil deformation law in elastic semi-infinite space with the foundation subjected to the vertical, horizontal, moment, and torque load. The method used to solve the elasticity problem was developed on the basis of the integral transform technique and the dual integral equation technique. The solutions are given as follows:

whereEandμare the elastic modulus and Possion’s ratio of soil,Ris the radius of circular foundation. The load components and the corresponding deformation relationships are shown in Figs.1 and 2.

Fig.2 The relationships between load components and soil deformation.

Because undrained saturated soft clay is incompressible, the Poisson’s ratio of soil is taken as 0.5 and the above theoretical results are accurate. According to the research by Poulos, full roughness should be considered for the interface between a circular foundation and marine sediment. With Poisson’s ratioμ<0.5, Bell (1991) showed that none of Eqs. (1)–(6) gives precise results for rough foundations on seabed. Therefore, the above theoretical solutions should be further modified for the compressible soil with a Poisson’s ratio not equal to 0.5.

2.2 Numerical Calculations

In order to study the coupling relationship between load components and soil deformation under the combined loading, the calculations follow two steps: 1) calculate the effect of Poisson’s ratio on the foundation deformation to obtain the monotonic loading mode; 2) calculate the coupling relationship between seabed soil deformation law and load components acting on the foundation for the combined loading mode.

2.2.1 Numerical model

The three-dimensional finite element mesh is used to simulate the circular foundation with the radiusRas shown in Fig.3. The grid domain of the circular foundation has a radius of 10R. Zero-displacement boundary conditions prevent out-of-plane displacements of the vertical boundaries, and the base of the grid is fixed in all the three coordinate directions.

Fig.3 The finite element model grid.

The finite element model shown in Fig.3 includes two parts: the circular foundation and the submarine seabed.

The seabed comprises 26800 second order reduced integrated hexahedral hybrid elements (C3D20RH), and the circular foundation is represented using a discrete rigid body (R3D4). In the numerical analysis, the interface between the seabed and the circular foundation is assumed to be fully rough. The hybrid element formulation uses a mixture of displacement and stress variables (as opposed to a sole displacement variable) to approximate the equilibrium equations and compatibility conditions. Hybird elements are usually recommended for modeling the response of nearly incompressible materials, which is appropriate for undrained saturated soil conditions.

2.2.2 Numerical loading mode

In order to obtain the deformation curve of the foundation due to the combined loading, the loading control and displacement control methods are adopted in the finite element numerical calculations (Butterfieldet al., 1997). Compared to the loading control method, the displacement control method can give the displacement-loading curves accurately by determining the reaction force from the bottom of the foundation (Tan, 1990; Yun and Bransby, 2007).

The displacement control mode is used to study the deformation law of soil. In order to investigate the relationship between the deformation law and the load components, the displacement control procedure is conducted in ten steps in the numerical analysis.

3 Monotonic Loading

In the case of the monotonic loading, the relationships between the soil deformation law and the vertical loadV, horizontal loadsHxandHy, moment loadsMxandMy, and torque loadTare calculated, and the results of deformation with varying Poisson’s ratio are shown in Table 1, wherezis the vertical displacement caused by vertical concentrated loadV; the horizontal displacementshxandhyare caused by horizontal loadsHxandHy, respectively;ωxandωyare the overturning displacement caused by moment loadsMxandMy, respectively; andθis the torsion displacement caused by torque load. The calculated results of these variables are shown in Figs.4, 5, 6, and 7.

The numerical results show that the variation of Poisson’s ratio has no effect on the horizontal load, but has a great influence on the vertical, moment, and torque loads. Some correction factors for the theoretical formulas are given as follows:

With the correction factors the deformation law of clay under monotonic load is given as:

Table 1 The relationships between Poisson’s ratio and load components

Fig.4 Relationship between vertical load and vertical displacement.

Fig.5 Relationship between horizontal load and horizontal displacement.

Fig.6 Relationship between moment load and overturning displacement.

Fig.7 Relationship between torque load and torsion displacement.

4 Combined Loading with 3 Degrees of Freedom

On the basis of numerical calculations, the deformation law of combined loading with 3 DOF is studied. Considering vertical loadV, horizontal loadHand moment loadMacting on a circular foundation with the radiusR, the relationship between load components and deformation can be expressed as follows.

4.1 Coupling Coefficient Between Vertical and Horizontal Loads

By adjusting the Poisson’s ratio, the relationships among vertical, horizontal loads and deformation of soil are investigated. And the results show zero coupling coefficients between vertical loadVand horizontal displacementUhx, nor between horizontal loadHxand vertical displacementUz, which can be expressed as

4.2 Coupling Coefficient Between Moment and Horizontal Loads

Table 2 shows the calculated coupling coefficient between moment load and horizontal load. When Poisson’s ratioμ=0.5, the coupling coefficients indicate the occurrence of the elastic deformation of soil.

On the basis of numerical analysis results, the coupling coefficient can be expressed as follows

Table 3 shows the coefficients between moment loadMand horizontal loadHwith different Poisson’s ratios of soil.

Table 2 The coupling coefficient (μ=0.5)

Table 3 The relationship between coupling coefficient and Poisson’s ratio

4.3 Coupling Coefficient Between Vertical and Moment Loads

With different Poisson’s ratios, zero coupling coefficients between vertical loadVand moment loadMare obtained and expressed as

Based on the above results, the final deformation matrix of the circular foundation with 3 degrees of freedom is obtained:

5 Combined Loading with 6 Degrees of Freedom

According to the numerical analysis, the deformation law of circular foundation with 6 DOF is also investigated. Considering vertical loadV, horizontal loadsHxandHy, moment loadsMxandMy, and torque loadTacting on the circular foundation with the radiusR, the deformation matrix is

6 Conclusions

Based on the elastic-plastic finite element numerical analysis, the deformation law for a circular foundation on seabed with six degrees of freedom is analyzed. The conclusions are as follows:

1) For the monotonic loading mode, including vertical loadV, horizontal loadH, moment loadM, and torque loadT, the deformation law of soil varies greatly with the Poisson’s ratioμexcept for horizontal loadH.

2) In the case of the combined loading mode, the deformation matrix is more complex. And there is a strong coupling effect between horizontal loadHand moment loadMon the deformation law.

Acknowledgements

The authors wish to express their gratitude to Prof.Maotian Luan of Dalian University of Technology. The financial support for this study is through the grants 50909050 from the National Natural Science Foundation of China, and ZR2009FQ004 from the Natural Science Foundation of Shandong Province.

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(Edited by Xie Jun)

* Corresponding author. Tel: 0086-532-88527758

E-mail: zhangqiyi@163.com

(Received February 25, 2012; revised June 14, 2012; accepted May 7, 2013)

? Ocean University of China, Science Press and Springer-Verlag Berlin Heidelberg 2014