Reliability indicator for layered composites with strongly non-linear behaviour

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Abstract

This paper set out a method to determine safety factors when designing composite laminate with strongly non-linear behaviour. The failure probability is used to assess the laminate reliability.

The second moment-FORM (first order reliability method) method is chosen and the Nataf probability transformation is used in order to take account for a possible correlation between the different variables. The conception point is found from the Rackwitz–Fiessler method. Space conception parameters variability is then introduced in the reliability evaluation.

Finally, the method is applied to a composite plate, to show up the conception and the space material parameters variability importance.

Introduction

Due to the lack of knowledge of the long-term composite materials behaviour, the design of sufficiently reliable composite structures leads the designers to use very high safety coefficients. As a result, the composite solution, in leading industries such as aeronautics, looses a large part of its interest because of oversizing. This paper tries to present an objective reliability index for damaged composites with strongly non-linear behaviour. To calculate this index, a reliability computation algorithm is coupled with a deterministic finite element code, in which, the non-linear behaviour of each ply is assumed to be damaged, visco-elastic [1], [2]. The great lines of the proposed modelling are presented in Table 1. Although the damage is anisotropic, it is modelled by a single damage variable D, within the context of continuum mechanics. This variable affects the elastic modules by means of a damage tensor H̲̲(D). The viscoelastic model is a spectral type one where the release times and weighting functions are presented in Fig. 1. The viscoelastic strains are decomposed in elementary viscoelastic mechanisms, whose evolutions are given in Table 1.

Generally, design and correct dimensioning leads to check an inequality of the type G(s,r)>0wheres is the mechanical response provided by the behaviour model, r resistances (threshold of damage, failure…) and G a failure function. When the inequality is verified for the nominal values of the different variables, which confidence can be granted to the design or the dimensioning according to the uncertainties of the variables? A reliability index is computed, through a probabilistic approach, taking into account failure scenarios and nominal parameters uncertainties.

As an example, this approach is applied to a laminated plate [0, 90]s simply supported at two edges and loaded by a uniform pressure and a single force acting in its centre. The geometric, loading and material data are firstly defined as being uniform uncorrelated random variables. Secondly, the impact of the space variability of the longitudinal Young modulus is studied.

Section snippets

Probabilistic approach

This approach let the statistical data available on the dispersions of the mechanical model parameters to be accounted for. The evaluation of the failure probability Pf:Pf=G(X)0fX(X)dXis required to determine the laminate reliability index.

Indeed, the simplicity of this expression hides some major operational difficulties: on one hand, the density of joint probability fX(X) of the basic variables X=(X1,,Xn) is generally unknown and only incomplete information limited to the marginal laws and

Application to layered composite plate

The previous method described is applied to the case of a laminated plate [0/90]S (L = 600 mm, l = 60 mm, h = 16 mm) simply supported at two edges and subjected to the combined action of a uniform pressure and a concentrated force in its centre (Fig. 3).

The failure function is defined by:G(x)=30%-D(%)where D is a damage variable depending on a set of nominal parameters.

Firstly, the reliability index as well as the parameters sensitivities, have been obtained considering the geometry (geom.) and loading

Conclusion

Within the context of the building of a reliable design tool for the design and the dimensioning of structures made of materials composite with organic matrix, a second moment reliability method has been coupled with a finite element code in which the element formulation as well as the strongly non-linear behaviour modelling are perfectly controlled [10]. This method allows the evaluation of the failure probability by taking account for dispersions on nominal parameters (geometry, loading,

Acknowledgements

This work was carried out under the AMERICO project (Multiscale Analysis Innovating Research for CFRP) directed by ONERA (French Aeronautics and Space Research Center) and funded by the DGA/STTC (French Ministry of Defense), which is gratefully acknowledged.

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    To overcome this problem, reliability analysis should be implemented to reflect the uncertainties in design variables through evaluating the probability of failure (Pf). There are many available reference work dealing with reliability study of laminated composites (e.g. [1–20]) and as the recent review on this subject by Chiachio et al. [21] reveals, most of these researches reflect distinct goals and conclusions, because a wide range of failure criteria, performance indices, design variables and reliability methods could be adopted. As Chiachio et al. conclude in their study, introduction of computational efficient approaches for reliability analysis of laminated composites is still required [21].

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