6 Reliability Analysis 6.1 Introductory Remarks A very natural application of SFEM and the other probabilistic analytical and numerical methods [313]is the reliability assessment for both homogeneous [45,256,354]and heterogeneous structures [87,102,231,262].The starting point of the analysis is to assume the limit state function in terms of any structural state parameters -displacements,stresses,temperatures or strains (as well as some combination of them in the coupled problems).Then,starting from statistical input on the structural parameter,probabilistic structural analysis is carried out and, finally,starting from the limit state function,the reliability index is computed.The reliability index should have the same properties as the classical Kolmogoroff probability and,in the same time,the damage function. Following the stochastic structural analyses,First Order Reliability Method (FORM)and Second Order Reliability Method (SORM)are most frequently used [87,114,115,209].The methods do not provide satisfactory results for non- symmetric PDF of the input and output in the same time and that is why the higher order moments are proposed.Considering numerous applications of the Weibull PDF in the composite material area,the corresponding Second Order Third Moment (W-SOTM)approach proposed for homogeneous media is described below.To illustrate this approach,let us denote the limit state function as g.The expected values,variances and skewnesses of this function are calculated or computed first using up to the second orders of this function,the limit state function derivatives with respect to the input random variables vector b as well as using its probabilistic moments(o;as a standard deviation).There holds ELg]=g+ 0b2 (6.1) a'(g)-s 到成-a (6.2)
6 Reliability Analysis 6.1 Introductory Remarks A very natural application of SFEM and the other probabilistic analytical and numerical methods [313] is the reliability assessment for both homogeneous [45,256,354] and heterogeneous structures [87,102,231,262]. The starting point of the analysis is to assume the limit state function in terms of any structural state parameters - displacements, stresses, temperatures or strains (as well as some combination of them in the coupled problems). Then, starting from statistical input on the structural parameter, probabilistic structural analysis is carried out and, finally, starting from the limit state function, the reliability index is computed. The reliability index should have the same properties as the classical Kolmogoroff probability and, in the same time, the damage function. Following the stochastic structural analyses, First Order Reliability Method (FORM) and Second Order Reliability Method (SORM) are most frequently used [87,114,115,209]. The methods do not provide satisfactory results for nonsymmetric PDF of the input and output in the same time and that is why the higher order moments are proposed. Considering numerous applications of the Weibull PDF in the composite material area, the corresponding Second Order Third Moment (W-SOTM) approach proposed for homogeneous media is described below. To illustrate this approach, let us denote the limit state function as g. The expected values, variances and skewnesses of this function are calculated or computed first using up to the second orders of this function, the limit state function derivatives with respect to the input random variables vector b as well as using its probabilistic moments (σi as a standard deviation). There holds ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + n i i i b g E g g 1 2 2 2 2 1 [ ] σ ∂ ∂ (6.1) { } ∑ ∑ = = ⎥ − ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = + n i i i i i n i i i i S E g b g b g b g g b g g g 1 2 2 2 2 1 2 2 2 2 2 2 ( ) σ [ ] ∂ ∂ ∂ ∂ σ ∂ ∂ ∂ ∂ σ (6.2)
Reliability Analysis 297 2 S8)={ + 28 +8 (6.3) dg ag∂g +38 ab,ab, So-E'[g ]-3E[g lo(g) (g) These formulae can be derived using the classical perturbation approach described previously.Next,parameters B.A of the Weibull distribution [8]are obtained as a solution of the following system of equations: (6.4) 計r (6.5) 5()- +}2r小司 (6.6) where the Gamma function is defined as Jerldr (for x>0 (6.7) T(x)= 0 nn-1 lim (for any xe∈) nex(x+1)x+2)(x+n-1) Finally,the reliability index is obtained as ej (6.8) The application of this type of analysis to a simple two-component composite beam is shown in [179],for instance.From the computational point of view it should be underlined that the mathematical packages for symbolic computation are very useful in inversion of the Gamma function and in obtaining a direct numerical solution of the equations system presented above. The methodology shown above and applied for homogeneous media can be used for simulation of the composite materials as well.Having proposed a general algorithm for usage of the limit function g,the corresponding various limit
Reliability Analysis 297 { } ( g ) S E [ g ] E[ g ] ( g ) b g b g g b g b g g b g S( g ) g g n i i i i i i n i i i i 3 1 3 3 2 2 3 1 2 2 2 2 2 2 3 3 1 3 3 2 ⎪ σ ⎭ ⎪ ⎬ ⎫ σ − − σ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎪ ⎩ ⎪ ⎨ ⎧ σ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = (6.3) These formulae can be derived using the classical perturbation approach described previously. Next, parameters x , β, λ of the Weibull distribution [8] are obtained as a solution of the following system of equations: E g + x ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Γ + β λ 1 [ ] 1 (6.4) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Γ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Γ + β β σ λ 1 1 2 ( ) 1 2 2 g (6.5) ( ) 1 1 2 1 1 1 2 3 1 3 ( ) 1 2 3 3 g S g β β β β σ λ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + Γ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Γ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − Γ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Γ + (6.6) where the Gamma function is defined as ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ∈ℜ + + + − > Γ = − →∞ ∞ − − ∫ ( ) ( 1)( 2)...( 1) ! ( 0) ( ) 1 0 1 lim for any x x x x x n n n e t dt for x x x n t x (6.7) Finally, the reliability index is obtained as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − β λ x R exp (6.8) The application of this type of analysis to a simple two-component composite beam is shown in [179], for instance. From the computational point of view it should be underlined that the mathematical packages for symbolic computation are very useful in inversion of the Gamma function and in obtaining a direct numerical solution of the equations system presented above. The methodology shown above and applied for homogeneous media can be used for simulation of the composite materials as well. Having proposed a general algorithm for usage of the limit function g, the corresponding various limit
298 Random Composites functions adequate to composite materials are summarised below.The most simplified and natural formulation of the limit function is a difference between allowable and computed values of the structural state function or functions. All limit state functions proposed and used for composites can be divided basically into three different groups.The most generalised functions,independent from the composite components type,and even from homogeneity or heterogeneity of a medium and fracture character as well as physical mechanisms of the whole process,can be classified into the first group.The functions included in the second one obey a precise definition of material fracture mechanism in terms of elastoplastic behaviour,crack formation and its propagation into the composite during the whole process.The last group is characterised by the presence of the failure function in the limit function and is therefore usually oriented to the specific groups and types of composite materials. The most general relations are maximum stress and strain laws formulated in terms of longitudinal and transverse stresses and strain for both compression and tension as follows: maximum stress law: 8x(K)= o-oxox≥0) OLe+Gx(Gx <0) (6.9) 8,(K)= 0-0,6,20) OLe +o,(,<0) 8,(X)=O-s maximum strain law: 8x(X)= 「e-exex≥0) ELe+Ex(Ex<0) (6.10) 8,(X)= e-e,e,≥0 ELe+E,(E,<0) 8,(X)=EL-Esl As can be seen,the limit functions are independent from of composite material type (fibre-reinforced or laminated)as well as from the character of its components (polymer-based,metal matrix,etc.).They originate from the mechanics of homogeneous media.However,brittle or ductile character of material damage is not taken into account in the analysis as well as the possibility of crack formation during the fatigue process.That is why more sophisticated criteria are proposed as,for instance,the one formulated as
298 Random Composites functions adequate to composite materials are summarised below. The most simplified and natural formulation of the limit function is a difference between allowable and computed values of the structural state function or functions. All limit state functions proposed and used for composites can be divided basically into three different groups. The most generalised functions, independent from the composite components type, and even from homogeneity or heterogeneity of a medium and fracture character as well as physical mechanisms of the whole process, can be classified into the first group. The functions included in the second one obey a precise definition of material fracture mechanism in terms of elastoplastic behaviour, crack formation and its propagation into the composite during the whole process. The last group is characterised by the presence of the failure function in the limit function and is therefore usually oriented to the specific groups and types of composite materials. The most general relations are maximum stress and strain laws formulated in terms of longitudinal and transverse stresses and strain for both compression and tension as follows: - maximum stress law: ( ) ( ) ⎩ ⎨ ⎧ + < − ≥ = 0 0 ( ) , , L c X X L t X X g X X σ σ σ σ σ σ ( ) ( ) ⎩ ⎨ ⎧ + < − ≥ = 0 0 ( ) , , L c y y L t y y g y X σ σ σ σ σ σ g y X =σ LT − σ S ( ) (6.9) - maximum strain law: ( ) ( ) ⎩ ⎨ ⎧ + < − ≥ = 0 0 ( ) , , L c X X L t X X g X X ε ε ε ε ε ε ( ) ( ) ⎩ ⎨ ⎧ + < − ≥ = 0 0 ( ) , , L c y y L t y y g y X ε ε ε ε ε ε g y X LT S ( ) = ε − ε (6.10) As can be seen, the limit functions are independent from of composite material type (fibre-reinforced or laminated) as well as from the character of its components (polymer-based, metal matrix, etc.). They originate from the mechanics of homogeneous media. However, brittle or ductile character of material damage is not taken into account in the analysis as well as the possibility of crack formation during the fatigue process. That is why more sophisticated criteria are proposed as, for instance, the one formulated as
Reliability Analysis 299 8(X)= 2X XivX 8 π。 2X In sec X2+X, Keπ2 2X2+X, (6.11) where X,is loading stress,X2 yield strength,X3 tensile stress,X,fracture toughness,Xs initial crack length,X crack length and calculation of Kie is presented by [91].This limit state function allows us to combine brittle and ductile fracture type of the analysed material specimen,even in the elastoplastic range. However,as in previous formula,it is quite non-sensitive to the composite material type.Considering that,the limit state functions are combined with the failure stress or strain functions in the form of so-called quadratic polynomial failure criteria, for instance.The limit state functions proposed using such a criterion can be used for the unidirectional composite laminate in both stress and strain formulations: -Hill-Chamis: 8(X)=1-OxFAxOx,8(X)=1-EKGAxEx (6.12) Hoffman and Tsai-Wu [352]: g(X)=1-axFAxOx-Fix,8(X)=1-EKGAxEx-GBx (6.13) Starting from the equations describing the limit function g,its probabilistic moments are calculated using the formula proposed above,but in such a case the knowledge of failure function probabilistic moments is necessary.In this context, analogous to the previous considerations,the second order perturbation method can be applied to randomise any of the reliability criteria i.e.Tsai-Hill failure criterion. 6.2 Perturbation-based Reliability for Contact Problem To illustrate the reliability analysis implementation,the stochastic perturbation reliability analysis of the linear elastic contact analysis is carried out for a composite reinforced with spherical particles.Since the solution for the deterministic problem is known and has been worked out analytically,the probabilistic analysis is made using the package MAPLE.The reliability limit function and probabilistic moments of the contact stress computations as well as some sensitivity numerical studies are carried out by the use of this program together with the visualisation of all computed functions.This methodology can be successfully applied for randomisation of all contact problem reliability studies, where contact stresses are described by the closed form equations.Otherwise, Stochastic Finite [88,162]or Boundary Element Method [46,51,185]
Reliability Analysis 299 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = 2 3 1 1 2 1 6 2 3 1 2 2 lnsec 2 8 ( ) X X X K X X X X X g X c π π π (6.11) where X1 is loading stress, X2 yield strength, X3 tensile stress, X4 fracture toughness, X5 initial crack length, X6 crack length and calculation of K1c is presented by [91]. This limit state function allows us to combine brittle and ductile fracture type of the analysed material specimen, even in the elastoplastic range. However, as in previous formula, it is quite non-sensitive to the composite material type. Considering that, the limit state functions are combined with the failure stress or strain functions in the form of so-called quadratic polynomial failure criteria, for instance. The limit state functions proposed using such a criterion can be used for the unidirectional composite laminate in both stress and strain formulations: - Hill-Chamis: A X X T g(X) =1 −σ X F , σ , A X X T g X ε XG ε 1 , ( ) = − (6.12) - Hoffman and Tsai-Wu [352]: T A X X B X T g X 1 X F , F , ( ) = −σ σ − , T A X X B X T g X 1 X G , G , ( ) = − ε ε − (6.13) Starting from the equations describing the limit function g, its probabilistic moments are calculated using the formula proposed above, but in such a case the knowledge of failure function probabilistic moments is necessary. In this context, analogous to the previous considerations, the second order perturbation method can be applied to randomise any of the reliability criteria i.e. Tsai-Hill failure criterion. 6.2 Perturbation-based Reliability for Contact Problem To illustrate the reliability analysis implementation, the stochastic perturbation reliability analysis of the linear elastic contact analysis is carried out for a composite reinforced with spherical particles. Since the solution for the deterministic problem is known and has been worked out analytically, the probabilistic analysis is made using the package MAPLE. The reliability limit function and probabilistic moments of the contact stress computations as well as some sensitivity numerical studies are carried out by the use of this program together with the visualisation of all computed functions. This methodology can be successfully applied for randomisation of all contact problem reliability studies, where contact stresses are described by the closed form equations. Otherwise, Stochastic Finite [88,162] or Boundary Element Method [46,51,185]
300 Random Composites computational implementations are to be made in order to get general approximate probabilistic solutions for the composite contact problems.Furthermore,the numerical approach to stochastic reliability,stochastic contact modelling and the relevant analytical computation aspects can be applied and explored in various areas of modern engineering,especially in the field of composite materials. Let us consider the contact phenomenon between two linear elastic isotropic regions characterised by the Young moduli (e,e2)and Poisson ratios (v,v2).Let us assume that the regions have spherical shapes with radii R and R2,respectively, and that the contact is considered in a point denoted by C,as it is shown in Figure 6.1 below.The 3D view of the particle-reinforced composite plane cross-section is shown in Figure 6.2. R Z17 Figure 6.1.Contact surface geometry Particle Matrix Figure 6.2.3D view of the particle-reinforced composite plane cross-section
300 Random Composites computational implementations are to be made in order to get general approximate probabilistic solutions for the composite contact problems. Furthermore, the numerical approach to stochastic reliability, stochastic contact modelling and the relevant analytical computation aspects can be applied and explored in various areas of modern engineering, especially in the field of composite materials. Let us consider the contact phenomenon between two linear elastic isotropic regions characterised by the Young moduli ( ) 1 2 e , e and Poisson ratios ( ) 1 2 ν ,ν . Let us assume that the regions have spherical shapes with radii R1 and R2, respectively, and that the contact is considered in a point denoted by C, as it is shown in Figure 6.1 below. The 3D view of the particle-reinforced composite plane cross-section is shown in Figure 6.2. R1 r R2 P M N z1 z2 C Figure 6.1. Contact surface geometry Figure 6.2. 3D view of the particle-reinforced composite plane cross-section Particle Matrix