The investigation in this thesis is concerned with response statistics of shell structures with geometrical and material nonlinearities under stationary and non-stationary Gaussian random excitations.

A series of hybrid strain based three node flat triangular shell elements is developed for linear and nonlinear shell structural analyses. In the former the Hellinger Reissner variational principle is employed. The elements are obtained by combining a triangular bending element and a plane stress element, and incorporating the drilling degrees-of-freedom. For the nonlinear analysis the updated Lagrangian formulation and the incremental Hellinger-Reissner variational principle are applied, with incremental displacements and strains being the independently assumed fields. Accordingly, the incremental second Piola-Kirchhoff stress and the incremental Washizu strain are selected as the incremental stress and strain measures. Schemes to transform the second Piola-Kirchhoff stresses to Cauchy stresses are included. The director and simplified versions of the stiffness and consistent mass matrices are derived. Material nonlinearity is of elasto-plastic type with isotropic hardening which is formulated by the J₂ flow theory with Ilyushin's yield criterion. In all cases, explicit expressions for the stiffness and consistent mass matrices are obtained. Various shell structures studied in this thesis show that the element formulations are accurate, effective, flexible, and applicable to thin to moderately thick shells with geometrical and material nonlinearities.

In parallel, the stochastic central difference (SCD) method and its associated computational strategies are applied to determine response statistics of general structures. The strategies include the time co-ordinate transformation (TCT) and adaptive time schemes (ATS). The SCD method with the strategies has excellent accuracy and effectiveness, and does not cause computational instability.

The SCD method is subsequently extended to include a relatively general nonstationary random excitation that consists of a deterministic and stochastic components. In conjunction with the TCT and ATS, it is applied to compute the random responses of general nonlinear shell structures that are discretized by the derived shell elements. Numerical results employing the proposed methodologies are presented and the effectiveness of the methodologies addressed.

The thesis concludes with recommendations for further investigations.