The effects of the electron energy distribution function (EEDF) on the structure of a dusty plasma sheath are investigated. Here, it is assumed that the electrons obey a Druyvesteyn-type distribution with a parameter
$x$
controlling the shape of the EEDF. The Druyvesteyn-like distribution tends to a Maxwellian distribution as
$x$
varies from 2 to 1. Using the orbital motion limited theory, the incident electron current on the dust is evaluated for a given
$x$
. The results of numerical simulations are compared with those of a Maxwellian distribution. It was found that the sheath dynamics depends strongly on the magnitude of
$x$
. The sheath thickness increases monotonically with increasing
$x$
. However, the absolute dust charge decreases and, as a result, the accelerating ion drag force is weakened and thus the dust number density is enhanced. For a plasma with a Druyvesteyn-like distribution, the Bohm speed is a function of
$x$
and increases with increasing
$x$
.