NONLINEAR DETERMINISTIC AND RANDOM RESPONSE OF SHALLOW SHELLS

Deterministic and random vibrations are considered for the case of she ar deformable shallow shells composed of multiple perfectly bonded lay ers. The nontrivial generalization of the flat plate vibrations is exp ressed by the fact of''small amplitude'' vibrations existing about the curved equilibrium position together with the snap-through and snap-b uckling type large amplitude vibrations about the flat position. The g eometrically nonlinear vibrations are treated by applying Berger's app roximation to the generalized von Karman-type plate equations consider ing hard hinged supports of the straight boundary segments of skew or even more generally shaped polygonal shells. Shear deformation is cons idered by means of Mindlin's kinematic hypothesis and a distributed la teral force loading is applied. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, wher e the eigenfunctions of the corresponding linear plate problem are use d as space variables, renders a coupled set of ordinary time different ial equations for the generalized coordinates with cubic and quadratic nonlinearities. For reasons of convergence, a light viscous modal dam ping is added. The nonlinear steady-state response of shallow shells s ubjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the ''perturba tion method of multiple scales''. The use of a nondimensional formulat ion and introduction of the eigen-time of the basic mode of the associ ated linearized problem provides a unifying result with respect to the planform of the shell. Within the scope of random vibrations, it is a ssumed that the effective forces can be modelled by uncorrelated, zero -mean wide-band noise-processes. Considering the set of modal equation s to be finite, the Fokker-Planck-Kolmogorov (FPK) equation for the tr ansition probability density of the generalized coordinates and veloci ties is derived. Its stationary solution gives the probability of even tual snapping after a long time has elapsed. However, the probability of first occurrence follows from the (approximate) integration of the nonstationary FPK equation. The probability of first dynamic snap-thro ugh is derived for a single mode approximation with the influence of h igher modes taken into account. Using the two-mode expansion, the prob ability distribution of the asymmetric snap-buckling is also evaluated .