<p>Thin elastic layers bonded on one or both surfaces to rigid substrates are ubiquitous in coatings, adhesives, and integrated circuits. Their response to applied surface tractions is commonly represented by reduced mattress (or Winkler) models and shear-lag models. Classical reduced relations typically ignore normal-tangential coupling and are often confined to compressible and isotropic solids. In this work, we develop a Hankel-transform formulation for axisymmetric loading of transversely isotropic layers and perform a systematic thin-layer (or long-wavelength) asymptotic expansion to obtain high-order surface traction-displacement relations. The resulting reduced relations provide extended Winkler and shear-lag-type models that retain the coupling between normal and shear responses and accommodate transverse isotropy. These asymptotic models provide a practical basis for predicting deformation fields and stress transfer in layered and bonded systems.</p>