Two separate problems are discussed: axisymmetric equilibriumconfigurations of a circular membrane under pressure andsubject to thrust along its edge, and the buckling of a circularcylindrical shell.
An ordinary differential equation governing the circular membrane is imbedded in a family of n-dimensional nonlinear equations.Phase plane methods are used to examine the number of solutionscorresponding to a parameter which generalizes the thrust, as well asother parameters determining the shape of the nonlinearity and theundeformed shape of the membrane. It is found that in any number ofdimensions there exists a value of the generalized thrust for which acountable infinity of solutions exist if some of the remaining parametersare made sufficiently large. Criteria describing the number ofsolutions in other cases are also given.
Donnell-type equations are used to model a circular cylindricalshell. The static problem of bifurcation of buckled modes fromPoisson expansion is analyzed using an iteration scheme and pertubationmethods. Analysis shows that although buckling loads are usuallysimple eigenvalues, they may have arbitrarily large but finite multiplicitywhen the ratio of the shell's length and circumference is rational.A numerical study of the critical buckling load for simple eigenvaluesindicates that the number of waves along the axis of the deformed shellis roughly proportional to the length of the shell, suggesting the possibilityof a "characteristic length." Further numerical work indicatesthat initial post-buckling curves are typically steep, although the loadmay increase or decrease. It is shown that either a sheet of solutionsor two distinct branches bifurcate from a double eigenvalue. Furthermore,a shell may be subject to a uniform torque, even though one isnot prescribed at the ends of the shell, through the interaction of twomodes with the same number of circumferential waves. Finally,multiple time scale techniques are used to study the dynamic bucklingof a rectangular plate as well as a circular cylindrical shell; transitionto a new steady state amplitude determined by the nonlinearity isshown. The importance of damping in determining equilibrium configurationsindependent of initial conditions is illustrated.
