  Nanomechanical resonators based on strained silicon nitride (Si$_3$N$_4$)
have received a large amount of attention in fields such as sensing and quantum
optomechanics due to their exceptionally high quality factors ($Q$s).
Room-temperature $Q$s approaching 1 billion are now in reach by means of
phononic crystals (soft-clamping) and strain engineering. Despite great
progress in enhancing $Q$s, difficulties in fabrication of soft-clamped samples
limits their implementation into actual devices. An alternative means of
achieving ultra-high $Q$s was shown using trampoline resonators with engineered
clamps, which serves to localize the stress to the center of the resonator,
while minimizing stress at the clamping. The effectiveness of this approach has
since come into question from recent studies employing string resonators with
clamp-tapering. Here, we investigate this idea using nanomechanical string
resonators with engineered clampings similar to those presented for
trampolines. Importantly, the effect of orienting the strings diagonally or
perpendicularly with respect to the silicon frame is investigated. It is found
that increasing the clamp width for diagonal strings slightly increases the
$Q$s of the fundamental out-of-plane mode at small radii, while perpendicular
strings only deteriorate with increasing clamp width. Measured $Q$s agree well
with finite element method simulations even for higher-order resonances. The
small increase cannot account for previously reported $Q$s of trampoline
resonators. Instead, we propose the effect to be intrinsic and related to
surface and radiation losses.
