  Hydrostatically pressurized circular rings confined to two dimensions (or
cylinders constrained to have only z-independent deformations) undergo Euler
type buckling when the outside pressure exceeds a critical value. We perform a
stability analysis of rings with arc-length dependent bending moduli and
determine how weakened bending modulus segments affect the buckling critical
pressure. Rings with a 4-fold symmetric modulation are particularly susceptible
to collapse. In addition we study the initial post-buckling stages of the
pressurized rings to determine possible ring folding patterns.
