The optimal functioning of the long span thin walled elliptical cross section shells used as vacuum chambers for fast-cycling synchrotrons is provided by their buckling capacity. Also it is often necessary to design inter-stiffener panels of elliptical shells used as vacuum chambers to resist any tendency towards pressure induced buckling due to some combination of excessive out-gassing, fragility, radiation damage, magnetic field distortion,. The analysis for design is complicated because elliptical shell chambers display a complex form of nonlinear snap buckling behavior under the external pressure. Buckling analysis for shells is further complicated by the observation that geometric imperfections have an important influence on the buckling mode as well as on the buckling load-carrying capacity. Buckling loads are, in general, considerably lower than the lowest critical loads predicted from the idealized classical linear modeling of shell behavior. Consequently, there has been a tendency to abandon classical linear buckling analysis in favor of large-scale, nonlinear, finite element analyses of buckling. This has resulted in a loss of simplicity and explicitness in the design analysis of these and other shells. The buckling of elastic-plastic elliptical cross section shells subject to external pressure loading is investigated. Testing of the corrugated elliptical hallow shells are performed by the pressure prescribed conditions. A numerical method is then employed to investigate the elastic-plastic buckling of these elliptical shells in a variety of modes. A comparison is made between the numerical results and the experimental results.