Analytical Solution Of Multi-layered Elastic Subgrade Under Arbitrary Loading In Rectangular Coordinates

It is a significant question for discussion in Soil Mechanics to study of multi-layered elastic subgrade. Natural subgrade is popularly layered distributing. For the sake of solving calculational problem of multi-layered subgrade, many scholars bent themselves to the research. Moreover, Boussinesq solution based on semi-infinite elastic space model is still applied to calculate additional stress and settlement at subgrade. The solution of additional stress and settlement about multi-layered half-plane and half-space system is researched on based this in the paper. Starting from the essential equations of elastic Mechanics in rectangular coordinates system, the analytical solution is given for multi-layered elastic subgrade under arbitrary loading by using theory of the state space and transferring matrix method in the paper.Starting from the essential equations of elastic mechanics about multi-layered half-plane question, the state controlling equation of one-layered elastic subgrade is derived by using Fourier transform. The answer of one-layered elastic subgrade is given with the theory of the state space. Then the analytical solution of stress and strain is given for multi-layered elastic foundation under arbitrary loading by means of transferring matrix method.To multi-layered half-space subgrade, the problem of axisymmetric was often researched by predecessor in the cylindrical coordinate system; It is very less for researching of non-axisymmetric problem and progressional method was used for non-axisymmetric problem. The method is very complex and sometimes is not convergent or very slow. So the answer for one-layer subgrade is given with the theory of the state space by using Laplace transform in rectangular coordinate system.Then the analytical solution is given for multi-layered elastic subgrade under arbitrary loading by means of transferring matrix method in rectangular coordinate system. The key to solving multi-layered subgrade is changing the traditional orientation of rectangular coordinates system, origin of coordinates is enough distant from accessional loading. Consequently, equations of elastic theories are triumphantly transformed equations of the state controlling equations.Obviously, the new method presented in the paper has discovered a simple, effective and expedient tool, for solving the problem pertinent to half-space isotropic multi-layered elastic subgrade.