Coupled Shaft–Plate Load-Transfer Solution for Helical Piles in Depth-Dependent Elastic Media

Helical piles differ from conventional straight piles because vertical load transfer (LT) is governed by the coupled action of distributed shaft shear and localized bearing resistance mobilized by the helix plate. Existing LT models and continuum solutions for axially loaded piles generally do not explicitly account for this discrete bearing surface and the associated load redistribution mechanism. Building on established continuum pile and embedded plate solutions, this study develops a coupled analytical formulation for the vertical response of a single helix pile embedded in a depth-dependent elastic medium. The pile is modeled as a compressible shaft coupled with an embedded circular bearing surface representing the helix plate. The shaft and soil responses are linked through displacement compatibility at the helix level, allowing the helix load to be determined as part of the solution. Integral-form analytical solutions are derived for the pile displacement, axial force, and soil stress fields using Hankel transforms and Green’s function techniques. The formulation is verified by degeneration to a benchmark embedded circular plate problem and comparison with finite-element results. These results show that the helix induces a distinct axial force discontinuity and localized stress redistribution, marking a transition from shaft-controlled transfer to bearing-controlled load redistribution. Increasing the helix embedment depth, helix diameter, or soil stiffness gradient enhances load sharing by the helix, but the response becomes progressively less sensitive to further increases in these parameters over parts of the investigated ranges. These trends arise from displacement compatibility between shaft resistance and helix bearing. The proposed formulation provides a mechanics-based reference for interpreting load redistribution in helical piles and for assessing simplified and numerical models.