The stochastic elastoplastic analysis is investigated in this study for structures under plane stress/strain conditions. The considered uncertain system parameters are including the Young's modulus, Poisson's ratio and hardening parameter. A novel stochastic analysis framework with the consideration of material nonlinearity is implemented basing on the typical finite element method (FEM). Within the new analysis system, the neural network (NN) approach is incorporated for the subsequent stochastic qualification. Both Gaussian and non-Gaussian distribution types can be handled. By adopting limited groups of response outputs as the training dataset, the statistical features (i.e., the estimated means, variance, probability density function (PDFs), cumulative distribution functions (CDFs)) for the concerned structural displacement and stress could be effectively obtained. Therefore, the nonlinear performance of the structure against both serviceability and strength limit states can be effectively investigated with the consideration of various system uncertainties. One numerical example is thoroughly implemented to illustrate the accuracy, applicability and effectiveness of the proposed approach.