  Curved thin sheets are ubiquitously found in nature and manmade structures.
Within the framework of classical thin plate theory, the stiffness of thin
sheets is independent of its bending state. This assumption, however, goes
against intuition. Simple experiments with a cantilever sheet made of paper
show that the cantilever stiffness largely increases with the transversal
curvature. We here demonstrate by using simple geometric arguments that thin
sheets subject to two-dimensional bending necessarily develop internal
stresses. The coupling between the internal stresses and the bending moments
can increase the stiffness of the plate by several times. We develop a theory
that describes the stiffness of curved thin sheets with simple equations in
terms of the longitudinal and transversal curvatures. The theory perfectly fits
experimental results with a macroscopic cantilever sheet as well as numerical
simulations by the finite element method. The results shed new light on plant
and insect wing biomechanics and provide an easy route to engineer micro- and
nanomechanical structures based on ultrathin materials with extraordinary
stiffness tunability.
