Elastic displacement fields of multi-layered transversely isotropic materials under rectangular loads

Based on the precise integration algorithm (PIA) and the technique of dual vector forms, analytical solutions of displacements for the complete elastic field induced by the vertical loadings uniformly distributed over a rectangular area are deduced. The rectangular area is assumed horizontally resting on the surface or embedded in the interior of a linearly elastic, homogeneous transversely isotropic continuum. The planes of transverse isotropy are chosen to be parallel to the horizontal boundary surface. Associated with the approach of dual vector, the classical integral transform method is adopted to convert the partial differential governing equations into the concise first-order ordinary differential matrix equation. As a highly accurate method to solve sets of the first-order ordinary differential equations, the procedure of PIA is introduced to evaluate the key matrix equation. As a result, any desired accuracy of the elastic solutions can be achieved. Additionally, dual vector forms of equations in the transformed domain make the assembly of the global stiffness matrix simple and convenient. Finally, some illustrative examples are analysed to verify the proposed solutions and elucidate the influences of the dimensions of the loading area, the type and degree of material anisotropy and the stratified characters on the load-displacement relationship in the transversely isotropic media.