  In this paper, we derive the first variation formulas for surfaces in
3-dimensional Euclidean space by using the ``strain-displacement relations''
known in thin shell theory. For applications to architectural surface design,
we focus on the objective function which has linear Weingarten surfaces as
stationary points. This article aims to provide an elementary and
cross-disciplinary exposition for applications without any tensor calculus, and
thus, we do not give any new mathematical results.
