  A thin-walled tube, e.g., a drinking straw, manifests an instability when
bent by localizing the curvature change in a small region. This instability has
been extensively studied since the seminal work of Brazier nearly a century
ago. However, the scenario of pressurized tubes has received much less
attention. Motivated by rod-shaped bacteria such as E. coli, whose cell walls
are much thinner than their radius and are subject to a substantial internal
pressure, we study, theoretically, how this instability is affected by this
internal pressure. In the parameter range relevant to the bacteria, we find
that the internal pressure significantly postpones the onset of the
instability, while the bending stiffness of the cell wall has almost no
influence. This study suggests a new method to infer turgor pressure in
rod-shaped bacteria from bending experiments.
