Abstract
Abstract This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing amplitudes of the blast. Both perfect shells and shells with axisymmetric geometric imperfections are analyzed. The impact energy threshold causing buckling is identified and compared with the energy barrier that exists between the buckled and unbuckled static equilibrium states of the energy landscape associated with the pre-loaded pressure. The extent to which the impact energy of the threshold blast exceeds the energy barrier depends on the details of its shape and width. Targeted blasts that approximately replicate the size and shape of the energy barrier buckling mode defined in the paper have an energy threshold that is only modestly larger than the energy barrier. An extensive study is carried out for more realistic Gaussian-shaped blasts revealing that the buckling threshold energy for these blasts is typically in the range of at least 10–40% above the energy barrier, depending on the pressure pre-load and the blast width. The energy discrepancy between the buckling threshold and energy barrier is due to elastic waves spreading outward from the impact and dissipation associated with the numerical integration scheme. Buckling is confined to the vicinity of the pole such that, if the shell is not shallow, the buckling thresholds are not strongly dependent on the location of the clamping boundary, as illustrated for a shell clamped halfway between the pole and the equator.
Highlights
The design of imperfection-sensitive shell structures such as spherical shells under external pressure makes heavy use of a knockdown factor which accounts for structural imperfections by reducing the expected buckling load below the prediction for the perfect version of the structure
The literature on shell buckling is replete with theoretical and experimental papers addressing the evaluation of knockdown factors
For a shell with unstable post-buckling behavior, the energy barrier to buckling at a given static load is the difference between energy of the shell/load system in the quasi-static buckled state from that in the unbuckled state
Summary
The design of imperfection-sensitive shell structures such as spherical shells under external pressure makes heavy use of a knockdown factor which accounts for structural imperfections by reducing the expected buckling load below the prediction for the perfect version of the structure. The axisymmetric quasi-static pre-buckling and post-buckling normal deflections of the perfect shell clamped at the equator are presented in Fig. 2 for the full range of pressures and paired with the plot of pole deflection. The energy barrier plays a central role in providing an understanding of the robustness of the pre-loaded shell against disturbances such as local impacts, and it enables a quantitative means of rationalizing the energies of disturbances required to buckle the shell To back up this statement, an extensive study has been conducted in this paper of perfect and imperfect elastic spherical shells that are clamped at the equator and subject to blast-like impacts of various amplitudes and shapes in the vicinity of the pole. − (SE + PE ) n with the subscripts s denoting saddle and n denoting node
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