In order to take into account the uncertainties linked to the variables in the evaluation of the statistical properties of structural response, a reliability approach with probabilistic aspect was considered. This is called the Probabilistic Transformation Method (PTM). This method is readily applicable when the function between the input and the output of the system is explicit. However, the situation is much more involved when it is necessary to perform the evaluation of implicit function between the input and the output of the system through numerical models. In this work, we propose a technique that combines Finite Element Analysis (FEA) and Probabilistic Transformation Method (PTM) to evaluate the Probability Density Function (PDF) of response where the function between the input and the output of the system is implicit. This technique is based on the numerical simulations of the Finite Element Analysis (FEA) and the Probabilistic Transformation Method (PTM) using an interface between Finite Element software and Matlab. Some problems of structures are treated in order to prove the applicability of the proposed technique. Moreover, the obtained results are compared to those obtained by the reference method of Monte Carlo. A second aim of this work is to develop an algorithm of global optimization using the local method SQP, because of its effectiveness and its rapidity of convergence. For this reason, we have combined the method SQP with the Multi start method. This developed algorithm is tested on test functions comparing with other methods such as the method of Particle Swarm Optimization (PSO). In order to test the applicability of the proposed approach, a structure is optimized under reliability constraints.
Cite this paper
S. Ouhimmou, A. Hami, R. Ellaia and M. Tkiouat, "Contribution to Development of Reliability and Optimization Methods Applied to Mechanical Structures," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 19-24. doi: 10.4236/am.2013.41005.
 M. Lemaire, “Evaluation of Reliability Index Associated to Structural Mechanical Models,” French Journal of Mechanic, Vol. 2, 1992, pp. 145-154.
 B. Sudreta and A. Der Kiureghian, “Comparison of Finite Element Reliability Methods,” Probabilistic Engineering Mechanics, Vol. 17, No. 4, 2002, pp. 337-348.
 A. Der Kiureghian and J.-B. Ke, “The Stochastic Finite Element Method in Structural Reliability,” Probabilistic Engineering Mechanics, Vol. 3, No. 2, 1988, pp. 83-91.
 A. M. Hasofer and N. C. Lind, “Exact and Invariant Second-Moment Code Format,” Journal of the Engineering Mechanics Division, Vol. 100, No. 1, 1974, pp. 111-121.
 H. O. Madsen, S. Kenk and N. C. Lind, “Methods of Structural Safety,” Prentice-Hall Inc., Upper Saddle River, 1986.
 R. Rackwitz, “Reliability Analysis: A Review and Some Perspectives,” Structural Safety, Vol. 23, No. 4, 2001, pp. 365-395. doi:10.1016/S0167-4730(02)00009-7
 M. Lemaire, A. Mohamed and O. Flores-Macias, “The Use of Finite Element Codes for the Reliability of Structural Systems,” 1997.
 A. Mohamed and M. Lemaire, “Linearzed Mechanical Model to Evaluate Reliability off Shore Structures,” Structural Safety, Vol. 17, No. 3, 1995, pp. 167-193.
 S. Kadry, “A Proposed Technique to Evaluate the Stochastic Mechanical Response Based on Transformation with Finite Element Method,” International Journal of Applied Mathematics and Mechanics, Vol. 2, No. 2, 2006, pp. 94-108.
 G. Muscolino, G. Ricciardi and N. Impollonia, “Improved Dynamic Analysis of Structures with Mechanical Uncertainties under Deterministic Input,” Structural Safety, Vol. 15, No. 2, 2000, pp. 199-212.
 European Committee for Standardization, “Eurocode 3: Design of Steel Structures,” Eyrolles, Paris, 1992.
 P. Siarry, J. Dréo, A. Pétrowski and E. Taillard, “Metaheuristics for Hard Optimization,” Eyrolles, Paris, 2003.
 R. Fletcher, “Practical Methods of Optimization,” 2nd Edition, John Wiley and Sons, Hoboken, 2000.
 Z. Xinchao, “A Perturbed Particle Swarm Algorithm for Numerical Optimization,” Applied Soft Computing, Vol. 10, No. 1, 2010, pp. 119-124.
 Y. Cooren, “Development of an Adaptive Algorithm of Particulate Swarm Optimization. Applications in Medical Engineering and Electronics,” Ph.D. Thesis, 12 Val de Marne University, Paris, 2008.
 A. Mohamed and M. Lemaire, “Linearized Mechanical Model to Evaluate Reliability of Offshore Structures,” Structural Safety, Vol. 17, 1995, pp. 167-193.