Application of graph theory and canonical forms in reliability analysis of symmetric structures

Document Type : Research Paper


1 Department of Civil Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran

2 Seismic Geotechnical and High Performance Concrete Research Centre

3 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 24 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.


[1] G.K. Bhattacharyya and R.A. Johnson, Estimation of reliability in a multicomponent stress-strength model, J. Amer. Statist. Assoc. 69 (2012), no. 348, 966–970.
[2] O. Burukhina, A. Bushinskaya, I. Maltceva and S. Timashev, Mechanical system reliability analysis using reliability matrix method, IOP Conf. Ser.: Mater. Sci. Eng. IOP Pub. 365 (2018), no. 4, 042067.
[3] Y. Chen and J. Feng, Group-theoretic exploitations of symmetry in novel prestressed structures, Symmetry 10 (2018), no. 6, 229.
[4] M. Fragiadakis and S.E. Christodoulou, Seismic reliability assessment of urban water networks, Earthquake Engin. Struct. Dyn. 43 (2013), no. 3, 357–374.
[5] O. Goldschmidt, P. Jaillet and R. Lasota, On reliability of graphs with node failures, Networks 24 (1994), no. 4, 251–259.
[6] R.A. Johnson, 3 Stress-Strength Models for Reliability, Handbook of Statistics, 1988.
[7] Z. Kala, Reliability analysis of the lateral torsional buckling resistance and the ultimate limit state of steel beams with random imperfections, J. Civil Eng. Manag. 21 (2015), no. 7, 902–911.
[8] A. Kaveh and M.A. Sayarinezhad, Eigenvalues of matrices with special patterns using symmetry of graph, Sci. Iran. 10 (2003), no. 2, 220–226.
[9] A. Kaveh and M.A. Sayarinejad, Eigensolutions for matrices of special structures, Commun. Numer. Meth. Engin. 19 (2003), no. 2, 125–136.
[10] A. Kaveh and M.A. Sayarinejad, Graph symmetry and dynamic systems, Comput. Struct. 82 (2004), no. 23-26, 2229–2240.
[11] A. Kaveh and M.A. Sayarinezhad, Eigensolution of special compound matrices and applications, Asian J. Civil Eng. (Building and Housing) 6 (2005), no. 6, 495–509.
[12] A. Kaveh and M.A. Sayarinejad, Eigensolution of specially structured matrices with hyper-symmetry, Int. J. Numer. Meth. Eng. 67 (2006), no. 7, 1012–1043.
[13] D. Lehky, O. Slowik and D. Novak, Reliability-based design: Artificial neural networks and double-loop reliability-based optimization approaches, Adv. Eng. Software 117 (2018), 123–135.
[14] J. Li and J. Chen, Probability density evolution method for dynamic response analysis of structures with uncertain parameters, Comput. Mech. 34 (2004), no. 5, 400–409.
[15] J. Li and J. Chen, Stochastic Dynamics of Structures, John Wiley & Sons, 2009.
[16] M.A. Nabian and H. Meidani, Deep learning for accelerated seismic reliability analysis of transportation networks, Comput. Aided Civil Infrastruct. Engin. 33 (2018), no. 6, 443–458.
[17] H. Shariatmadar and G. Behnam Rad, Evaluation of active controlled structures using subset simulation method, Nationalpark-Forschung Schweiz 102 (2013), no. 3, 136–157.
[18] J.D. Sorensen, Notes in structural reliability theory and risk analysis, Aalborg University, 2004.
[19] P.D. Spanos and I.A. Kougioumtzoglou, Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation, Appl. Mech. 81 (2014), no. 5, 05101–05116.
Volume 14, Issue 11
November 2023
Pages 241-249
  • Receive Date: 08 November 2022
  • Revise Date: 15 January 2023
  • Accept Date: 18 February 2023