Ali Asghar Bagheri - Department of Civil Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
Alireza Mortezaei - Department of Civil Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
Mohammad Ali Sayarinejad - Department of Civil Engineering, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

Many abilities of Monte Carlo simulation methods have led to their increasing use in solving various reliability problems of structures. These methods are based on generating random samples in order to simulate events and estimate their results. Achieving certain accuracy requires a significant number of simulation operations. Acceptable accuracies can be achieved with a smaller number of samples by adopting different approaches. In this article, for the first time, symmetric structures are analyzed using the theory of graphs and canonical forms, the frequency of the structures is calculated, and their reliability is also checked using these theories. Also, the study and investigation of the calculation of frequencies and eigenvectors corresponding to symmetric structural models from the point of view of the decomposition of the stiffness matrix in probabilistic reliability analysis, which we will achieve this goal by using the proposed theories. In this article, in addition to obtaining all canonical forms, the relationship between all canonical forms is obtained. Finally, a new Rayleigh-based theory called the proposed improved Rayleigh theory is presented, which is used to extract the natural frequencies of structures. Also, in this research, several samples of different frames and structures were presented using the proposed method and finally, numerical results were presented and the solution of the three-story frame with ۲۴ degrees of freedom was presented. It can be seen from the results that the proposed method has a much higher speed and accuracy than the Monte Carlo method.

canonical forms, improved Rayleigh-Ritz method, Graph, symmetrical structure analysis