Semi-analytical Solutions On Tilted Transversely Isotropic Multi-layered Foundation

It is a significant question for discussion about multi-layered elastic structure of soilmechanics and composite material mechanics. For more than a century, with the developmentof mathematics and computational mechanics, elastic material model is improved fromcompletely isotropic model to anisotropic one and researchers have done a lot of researchwork in this field. Considering the natural soil is mostly formed by sedimentation, thetransversely isotropic model has been widely applied in the settlement of subgrade ofgeotechnical engineering.In the past, the transversely isotropic model with perpendicular symmetry axis in thismulti-layered system is often considered by researchers. However, the natural subsoil andsome other composite materials always do not be composed with this kind of standardmaterial structure. The structure and distribution of rock and soil layers always be changed bygeological processes, consequently a transversely isotropic model with tilted symmetry axisis seemed to be closer to simulate these material structures in this case. In order to solve thismore general problem, starting from the essential equations of elasticity, the transverselyisotropic model with tilted symmetry axis is derived by coordinate transformation from theone with perpendicular symmetry axis. Then, the state space equations of the new model areestablished by using the state space theory and integral transformation method fromdifferentiation and elimination of fundamental equations. A six order exponential matrix(transfer matrix) of one-layered elastic structure can be derived from the state space equations.In this paper, a discussion and analysis of calculation methods of derivation of six ordermatrix exponential has been done. Combining the transfer matrix of each layer with theboundary conditions, the new multi-layered system is solved by using the transition matrixmethod or other effective methods. Two methods（the transition matrix method and the layerelement analytic method）are used for solving the layered system. In addition, someexpressions of initial boundary conditions of several typical loads are also discussed. It isconvenient for engineering application of the solving methods of layered system this paperadvanced.