Analytical Solution Of One-dimensional Consolidation Equation Of Saturated Soils Based On Reduced Order Method And Its Application

Terzaghi was committed to the construction of soil,through a series of assumptions,established one-dimensional consolidation theory.However,the soil consolidation law is quite complex,not only depending on the category and state of the soil but also varies from factors such as the initial conditions of the soil layer,the drainage conditions,and additional external loads.To be able to establish a mechanical model closer to the soil consolidation traits,domestic and foreign scholars constantly correct and improve the assumptions in the Terzaghi one-dimensional consolidation theory.However,the consolidated equation obtained when considering a variety of factors is generally a second-order nonlinear partial differential equation,not all second-order nonlinear partial differential equations can be solved analytically.At present,most scholars' consolidated analysis is numerical solutions,which also causes engineering applications in the consolidation theory to be greatly limited.To obtain analytical solutions,some scholars put forward new assumptions when solving second-order nonlinear partial differential consolidation equations,or use an approximation method to convert them into a linear equation.The approximate analytical solution obtained by this method is more complex,and the number of cases in which analytical solutions can be obtained is still rare.The difficulty of mathematical solutions limits the development of consolidation theory.Because of this,in this paper,a reduced-order method is proposed to solve the consolidation equation,which can avoid the difficult problem of solving the second-order nonlinear partial differential equation,and the simplified analytical solutions of three consolidation cases are given.The main contents and innovative achievements are as follows:1.The basic principle of solving the consolidation equation by reducing the order method is given.By comparing the simplified analytical solution of consolidation under the linear elasticity hypothesis with the Terzaghi one-dimensional consolidation theory,the rationality of the reduced-order method for solving the consolidation equation is verified,and the application of the simplified analytical solution for solving the consolidation coefficient is given.2.A method for solving the consolidation equation of saturated soil considering threshold gradient under instantaneous loading is presented,and a simplified analytical solution that can be expressed by elementary function is obtained.At the same time,this analytical solution is compared with analytic and numerical solutions of Pascal et al.,to verify the rationality and correctness of the simplified analytical solution presented in this paper.Using parameter R as the consolidation index considering the threshold gradient,the influences of the threshold gradient on the location of moving boundary,the distribution of excess pore water pressure,the average degree of consolidation of the soil layer,and the settlement of foundation are analyzed.3.A preliminary study on the self-weight consolidation is carried out,and a reduction method considering the initial hydraulic gradient on the self-weight consolidation is given in the case of small deformation,and a simplified solution is derived from the effective stress principle and boundary conditions,at the same time,it is proposed that the combined calculation of water and earth pressure is reasonable when considering the threshold gradient.4.A method for solving the nonlinear consolidation equation is presented by introducing an equivalent coefficient ?,which comprehensively reflects the nonlinearity of soil.The equivalent coefficient ? formula is determined by the numerical solution fitting,and a simplified analytical solution of consolidation considering the nonlinearity of soil is established.The rationality of the simplified analytical solution is verified from the perspective of engineering applications.Based on the analytical solution,it is easy to consider other factors affecting consolidation,which provides a new idea for solving the consolidation problem.