  Harmonic mappings have long intrigued researchers due to their intrinsic
connection with minimal surfaces. In this paper, we investigate shearing of two
distinct classes of univalent conformal mappings which are convex in horizontal
direction with appropriate dilatations. Subsequently, we present a family of
minimal surfaces constructed by lifting the harmonic mappings obtained through
shear construction method given by Clunie-Sheil. Furthermore, we contribute to
addressing an open problem partially, proposed by Boyd and Dorff, by
identifying the resulting minimal surfaces for certain values of the parameters
in one of the classes of mappings. Notably, this family of minimal surfaces
transforms from the well-established Enneper's surface to a Helicoid.
