A reliability method for optimization of +w; -w n fiber reinforced composite pipes F. Richard, D. Perreux * Laboratoire de Me ´canique Applique ´e R. Chale ´at, UMR 6604 CNRS 24, rue de l’Epitaphe, 25000 Besanc ¸on, France Received 30 March 1999; accepted 22 December 1999 Abstract This paper presents damage strength optimization of laminates using a reliability-based method. Two complementary analyses are developed: the first is a mechanical analysis of damage behavior using a thermodynamics framework that determines the failure criterion; the second is a reliability method which takes not only randomness, but also statistical uncertainties into account for failure probability calculation. This method is based on a first order reliability method that requires the resolution of a constrained problem of optimization. To solve this problem, a hybrid algorithm, combining evolutionary computation techniques with deterministic procedure is presented. Filament- wound pipe optimization under pressure is discussed as a working example.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Composite laminates; Optimization; Damage; Probabilistic method; Design; Evolutionary algorithm; Statistical uncertainties 1. Introduction Mechanical models are generally used for designing laminated composites. Scattering and subjective uncertain- ties such as neglect, mistakes, incorrect modeling and manu- facturing errors must be taken into consideration when designing for materials, stacking sequence, dimensions, etc. These problems are particularly significant for compo- site materials. All laminate composites must perform their expected functions with a high level of reliability during the prescribed service time. Particularly, when failure may cause loss of human lives, structural reliability is indispen- sable. This reliability assurance is also crucially important from an economical point of view. This is why reliability- based design methodology plays a role of vital importance in rational design [1,2]. Traditionally, structural design relies on deterministic analysis. We introduce an empirical safety factor that takes account of uncertainties in material properties, geome- try and loads. In probabilistic analysis, for each failure mode, a desirable or acceptable value of reliability must be available to the designer. This reliability target is chosen using personal judgment or rational analysis. Subjective uncertainties in parameters of the laws of probability or in the type of laws are often excluded, and the reliability is generally evaluated using the first order reliability index b [3]. The resulting reliability may thus be biased. Our analy- sis based on the principle of maximizing the probability of failure takes statistical uncertainties into account and provides a compensating alternative to this shortcoming. Unlike the Tsai-Wu [4] criterion often used for multi- axial laminates optimization based on a reliability method [5,6], a first damage criterion is defined as the limit state function of unidirectional ply. 2. First damage criterion In general, an N-layer laminate implies N possible layer failures, which can act as a series system or a parallel system. A series system is the lowest acceptable level for reliability; therefore when one layer fails the laminate fails. The first damage criterion is expressed in each layer. To define the damage threshold of the virgin layer, a variable definition of damage and the concept of effective stress [7] ~ s k applied to the kth layer expressed in the fiber system (Fig. 1) is used. If the damage is the microcracking of the matrix, two representations of the damaged lamina can be proposed depending on the orientation for the microcracks (Fig. 2). Three different types of damage can be defined D k I  - DE k 2 E k 2 ; D k II  - DG k 12 G k 12 ; D k III  - DG k 23 G k 23 1 Here E k 2 is the Young’s modulus in the direction-2, and G k 12 Reliability Engineering and System Safety 68 (2000) 53–59 0951-8320/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S0951-8320(00)00002-8 www.elsevier.com/locate/ress * Corresponding author. Tel.: + 33-3-8166-6012; fax: + 33-3-8140- 2901. E-mail address: dominique.perreux@univ-fcomte.fr (D. Perreux).