The probabilistic solutions of the elastic stretched beam are studied under the excitation of Kanai-Tajimi ground motion. Finite difference scheme is adopted to formulate the nonlinear multi-degree-of-freedom system about the random vibration of the beam. The state-space-split is employed to make the high-dimensional Fokker-Planck-Kolmogorov equation reduced to 4-dimensional Fokker-Planck-Kolmogorov equations which are solved by the exponential polynomial closure method for the probabilistic solutions of the system responses. The rules for selecting the state variables are proposed in order to reduce the dimensionality of Fokker-Planck-Kolmogorov equation by the state-space-split method. The numerical results obtained by the state-space-split and exponential polynomial closure method, Monte Carlo simulation method, and equivalent linearization method are presented and compared to show the computational efficiency and numerical accuracy of the state-space-split and exponential polynomial closure method in analyzing the probabilistic solutions of the strongly nonlinear stretched beam systems formulated by finite difference scheme and excited by Kanai-Tajimi ground motion.