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

Abstract

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.

Keywords:

• precise integration algorithm
• dual vector forms
• transversely isotropic continuum
• uniform rectangular loads
• integral transform method

Acknowledgement

This research was supported by Grant 51409038, 51421064, 51138001 from the National Natural Science Foundation of China, Grant GZ1406 from the Open Foundation of State Key Laboratory of Structural Analysis for Industrial Equipment, Grants 2013M530919 and 2014T70251 form China Postdoctoral Science Foundation, Grant L2013016 from the Liaoning Province Department of Education research project, Grant 1202 from Open Foundation of State Key Laboratory of Ocean Engineering, and Grant DUT15RC(4)23 from the fundamental research funds for the central universities for which the authors are grateful for which the authors are grateful.