The circumferential wave number of Fourier no., n, is different for each shell configuration and boundary condition. That is why several computer programs have been written to assist the analyst. For simply supported shells, the lowest harmonic pattern usually follows a sin function.
The general solution for cylinders the modal pattern is expressed as e-lamdax * (Sin n theta * Cos m pi x/2L + Cos n theta * Sin m pi x/2L). Thus a minimum value of m=1 corresponds to a minimum half wave. But, there are a number of factors that can alter these modal patterns. Foresberg in a 1964(?) AIAAJ paper provides a number of cases for cylinders giving approximate eigenvalues. In a more general presentation, Liessa, presents a number of formulae for a variety of shells of revolution in NASA SP-288 for the vibration of shells.
Buckling eigenvalues are close to the vibration eigenvalue set, but are different. There is an old adage that cylinders prefer diamonds and spheres prefer hexagons. It is best to start with an assumed Fourier wave number and search for the minimum using a range of at least a decade above or below. For each eigensolution, there will be an waveform that corresponds to the minumum. It may not necessarily be m=1.
Bifurcation also requires an understanding of modal coupling with the applied force. Energy can be transferred between adjacent modes that will produce a minimum buckling load differing from either applied and principle responding modes.
Suggest that you examine the literature very carefully before attempting to use a general FEM package without an experience factor assisting your results. The publications and computer programs reported at
may be of some benefit.
