The Reflection-Transmission Matrix Method For The Consolidation Of The Multilayered Saturated Soil And Its Application

The consolidation of the layered saturated soil is an important issue in civil engineering and a better method for treating this issue has been pursued for a long time. To this end, in this study, the reflection–transmission matrix(RTM) method for treating the layered saturated soil under consolidation is developed based on the Biot’s theory, and the consolidation of the layered saturated soil is analyzed by the developed RTM method. Combining the fundamental solution due to the RTM method with the Fredholm integral equation method, the statically loaded single pile in viscoelastic layered saturated soil is addressed in this study. The major studies of this paper consist of the following parts:(1) The RTM method for treating the layered saturated soil under axisymmetric consolidation or non-axisymmetric consolidation is developed to obtain the fundamental solutions for both the saturated soil subjected to an axisymmetric load and a horizontal load.First, the Mc Namee displacement functions are introduced to represent the displacements, stresses and pore pressure of the saturated soil under axisymmetric consolidation. The governing equations of the Biot’s theory are decoupled using the aforementioned expressions, and then the partial differential equations for the McNamee displacement functions are obtained. The ordinary differential equations corresponding to the above-mentioned partial differential equations are obtained using the Laplace and Hankel transforms. The expressions for the displacements, stresses and pore pressure in the Laplace-Hankel transform domain are obtained using the general solutions of the above-mentioned ordinary differential equations. Then the state vector and static wave vector of the saturated soil as well as the transform matrix relating the aforementioned two vectors are defined in the global coordinate system, and the expressions of the RTMs for the static wave vector of the saturated soil are presented.For the layered saturated soil under non-axisymmetric consolidation, the governing equations of the non-axisymmetric Biot’s theory are decoupled by introducing the Schiffman displacement functions. As with the axisymmetric case, the general solutions for the layered saturated soil are obtained using the Laplace-Hankel transform method. The RTMs for the layered saturated soil under non-axisymmetric consolidation can also be obtained employing the general solutions. As the state vector, static wave vector, and the transform matrix relating the two vectors are all defined in the global coordinate system, the RTMs obtained in this study thus have a reasonable physical meaning.With the RTMs for the layered saturated soil, the solutions for the layered saturated soil subjected to an axisymmetric source or a horizontal source are derived respectively. Comparison of results obtained from the proposed RTM method with some existing results and those obtained by the transfer matrix(TM) method validates the developed RTM method. Some numerical results are obtained using the proposed RTM method for the layered saturated soil.(2) The vertically and horizontally loaded single piles in viscoelastic layered saturated soil are studied based on the fundamental solutions obtained via the developed RTM method and the Fredholm integral equation method.By using the Muki’s method and the axisymmetric fundamental solutions for the saturated soil, the second kind of the Fredholm integral equation for the axial force of the fictitious pile is obtained by assuming that the normal strain of the fictitious pile and that of the saturated soil at the corresponding position are equal. The integral equation is simplified by means of the Laplace transforms method. Then the approximate Fredholm integral equation of the second kind in the time domain is obtained by the Schapery method. The solution of the pile in the viscoelastic layered saturated soil at an arbitrary time can now be obtained via numerical solution of the approximate Fredholm integral equation.For the horizontally loaded single pile in the viscoelastic layered saturated soil, again using the Muki’s method and the non-axisymmetric fundamental solutions, the second kind of the Fredholm integral equation for the shear force of the fictitious pile is obtained by equalizing the rotation angle of the axis of the fictitious pile with respect to the vertical axis and that of the layered saturated soil at the corresponding position. The integral equation is simplified with the Laplace transform, and similarly, the approximate Fredholm integral equation of the second kind for the horizontally loaded single pile in the time domain is obtained by the Schapery method. The approximate Fredholm integral equation can be solved numerically, yielding the solution of the horizontally loaded single pile in the viscoelastic layered saturated soil at an arbitrary time.