Abstract
We investigate stabilizing and eschewing factors on bistability in polar-orthotropic shells in order to enhance morphing structures. The material law causes stress singularities when the circumferential stiffness is smaller than the radial stiffness (β < 1), requiring a careful choice of the trial functions in our Ritz approach, which employs a higher-order geometrically nonlinear analytical model. Bistability is found to strongly depend on the orthotropic ratio, β, and the in-plane support conditions. An investigation of their interaction offers a new perspective on the effect of the hoop stiffness on bistability: while usually perceived as promoting, it is shown to be only stabilizing insofar as it prevents radial expansions; however, if in-plane supports are present, it becomes a redundant feature. Closed-form approximations of the bistable threshold are then provided by single-curvature-term approaches. For significantly stiffer values of the radial stiffness, a strong coupling of the orthotropic ratio and the support conditions is revealed: while roller-supported shells are monostable, fixed-pinned ones are most disposed to stable inversions; insight is given by comparing to a simplified beam model. Eventually, we show that cutting a central hole is a suitable method to deal with stress singularities: while fixed-pinned shells are barely affected by a hole, the presence of a hole strongly favours bistable inversions in roller-supported shells.
Highlights
Shells with more than one stable equilibrium state have enjoyed considerable interest in the engineering community, since their multistability enables them to adapt to changing loading conditions in beneficial ways
We have presented a higher-order, geometrically nonlinear analytical model with up to three degrees of freedom, in order to study bistable behaviour of polar-orthotropic shells
A Ritz approach that relates the deflection field to in-plane stresses via Gauss’s Theorema Egregium has been used to find alternative stable configurations; these were identified by solving a nonlinear eigenvalue problem whose solution unveiled minima in the strain energy functional
Summary
Shells with more than one stable equilibrium state have enjoyed considerable interest in the engineering community, since their multistability enables them to adapt to changing loading conditions in beneficial ways. Polar orthotropy allows us to vary the internal directional stiffness of shells, which sheds light into the statically indeterminate interplay between radial and hoop stiffness, and points towards optimized values that stabilize (or diminish) bistable inversions This knowledge enables us to make bistable shells more efficient, to save material, and to make such structures more versatile by allowing the tailoring of the shape of alternative equilibrium configurations; it enables us to judge where the unique features of shells are required and where simpler beam structures suffice. The trial functions employed—which are of real but, in general, not integer order—incorporate a novel extension for higher-order approaches that is based on the geometrically linear solution of a bent plate, in order to address the stress singularities arising in shells with an increased radial stiffness (β < 1). We do not expect to observe additional stable configurations stemming from the material law
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