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R. Heuer, H. Irschik, F. Ziegler:
"Nonlinear sure and random response of shallow shells";
Sadhana, 20 (1995), 427 - 439.

Deterministic and random vibrations are considered for the case of shear-deformable shallow shells composed of multiple perfectly bonded layers. The nontrivial generalization of the flat plate vibrations is expressed by the fact of "small amplitude" vibrations to exist about the curved equilibrium position together with the snap-through and snap-buckling type large amplitude vibrations about the flat position. The geometrically nonlinear vibrations are treated by applying Berger´s approximation to the generalized von Karman-type plate equations considering hard hinged supports of the straight boundary segments of skew or even more generally shaped polygonal shells. Shear deformation is considered by means of Mindlin´s kinematic hypothesis and a distributed lateral force loading is applied. Application of a multi-mode expan¬sion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordi¬nates with cubic and quadratic nonlinearities. For reasons of convergence, a light viscous modal damping is added. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the "perturbation method of multiple scales". By means of a nondimensional formulation and introducing the eigen-time of the basic mode of the associated linearized problem renders a unifying result with respect to the planform of the shell. Within the scope of random vibrations, it is assumed that the effective forces can be modeled by uncorrelated, zero-mean wide-band noise processes. Considering the set of modal equations to be finite, the Fokker-Planck-Kolmogorov (F.P.K.) equation for the transition probability density of the generalized coordinates and velocities is derived. Its stationary solution gives the probability of eventual snapping after a long time has elapsed. However, the probability of first occurrence follows from the (approximate) integration of the nonstationary F.P.K.-equation. The probability of first dynamic snap-through is derived for a single mode approximation with the influence of higher modes taken into account. Using the two-mode expansion, also the probability distribution of the asymmetric snap-buckling is evaluated.