Compu~rs & Srrurrures Vol. 53. No. 6. pp. 1407-1418. 1994 Elsevier Science Ltd Pergamon 00457949(94)EO240-3 Printed in Great Britain 004s7949/94 $7.00 + 0.00 RELIABILITY BASED STRUCTURAL OPTIMIZATION: A SIMPLIFIED SAFETY INDEX APPROACH M. V. Reddy,? R. V. Grandhit and D. A. Hopkins TDepartment of Mechanical and Materials Engineering, Wright State University, Dayton. OH 45435, U.S.A. SNASA Lewis Research Center, Cleveland, OH 44135, U.S.A. (Received 12 May 1993) Abstract-A probabilistic optimal design methodology for complex structures modelled with finite element methods is presented. The main emphasis is on developing probabilistic analysis tools suitable for optimization. An advanced second-moment method is employed to evaluate the failure probability of the performance function. The safety indices are interpolated using the information at the mean and most probable failure point. The minimum weight design with an improved safety index limit is achieved by using the extended interior penalty method of optimization. Numerical examples covering beam and plate structures are presented to illustrate the design approach. The results obtained by using the proposed approach are compared with those obtained by using the existing probabilistic optimization techniques. A a, b CDF cov C.Y 6 MPP PI s X* Z Zl =E B n 4 c 5 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA NOMENCLATURE transformation matrix gradient of performance function with respect to ith random variable design variable vector cumulative distribution function coefficient of variation covariance matrix jth constraint function most probable failure point specified probability mean value vector of random variables most probable point in x space performance function linearized performance function specified performance limit reliability or safety index failure surface standard normal function standard deviation correction factor INTRODUCTION In the last two decades, deterministic optimization techniques have been successfully applied to a large number of structural optimization problems. How- ever, it is generally recognized that there is always some uncertainty involved in any structural system due to the variations in material properties, improper definition of loading environment and the manufac- turing tolerances. For instance, in the design of an aircraft wing, the strength of the wing is a random variable since it varies considerably from sample to sample. In the design of mechanical system, the actual dimensions of any machined part are random since the dimensions may lie anywhere within specified tolerance bands. Similarly, in the design of aircraft and rockets the actual loads acting on the structure depend on the atmospheric conditions prevailing at the time of the flight, which may not be precisely predicted in advance. Hence, the loads are random variables in the design of such flight vehicles. Despite the generally recognized nondeterministic character of the parameters defining the engineering structures, reliability methods were applied in structural design on a relatively limited scale. The main difficulties in dealing with nondeterministic problems are the lack of information about the variability of the system parameters and the high cost of calculating their statistics. The aforementioned difficulties were circumvented with the introduction of a second moment reliability method, which is a procedure for a probabilistic design where the random parameters influencing the design appear only through their means and covari- antes. Thus, recent research efforts are being directed towards the development of a probabilistic approach for the optimum design problem involving random parameters. The reason for the development of re- liability based methods for structural design is also due to the necessity of performing accurate safety and reliability analyses for special-purpose engineering structures like airframes, turbine engines and space structures. Incorporation of probability concepts in the structural design requires the development of a suitable optimization formulation that enables a sol- ution of the design problem. A formulation was suggested by Charnes and Cooper [l] in which the stochastic optimization problem is converted into an equivalent deterministic one by a chance constraint 1407