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In the fields of engineering, nanoscience, and biomechanics, thin structural members, such as beams, plates, and shells, that are supported by an elastic medium are used in several applications. There is a possibility that these thin structures might buckle under severe loading conditions; higher-order, complicated elastic buckling modes can be found owing to the balance of rigidities between the thin members and elastic supports. In this study, we have shown a new and simple ‘power law’ relation between the critical buckling strain (or loads) and rigidity parameters in structural members supported by an elastic medium, which can be modelled as a Winkler foundation. The following structural members have been considered in this paper: i) a slender beam held by an outer elastic support under axial loading, ii) cylindrical shells supported by an inner elastic core under hydrostatic pressure (plane strain condition), and iii) complete spherical shells that are filled with an inner elastic medium.

The goal of this study was to investigate analytically the buckling phenomena of slender beam structures embedded in an elastic medium. In such cases, there is a possibility that spiral-shaped buckling modes can occur due to the restraining effect of the surrounding medium. As a result of analytical investigations, helical buckling occurs radically. Furthermore the pitch of the helix does not fluctuate as the radius of the helix increases. The interesting mechanical implications of the embedding effect are demonstrated by the theoretical approach established in this study.