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
Reliability Analysis 301 Let us observe that the contact problem is axisymmetric with respect to the vertical axis introduced at the centre of the spherical particle and at the bottom of this sphere (Figure 6.2).It is assumed that there is no pressure between the composite constituents and therefore the contact appears at the point C only.The distance between the points on the contacting surfaces and the plane transverse to the vertical axis of both surfaces is assumed to be small and can be described as 2(R-R2） 22-31= 2R R, (6.14) where r denotes the distance between the points M,N and the symmetry axis introduced at C.If the composite is loaded by the vertical force P acting along the vertical axis at the point C,then some local strains are induced in the neighbourhood of this point.They are a result of a contact phenomenon on a small circular surface (contact area).Assuming that the composite constituents radii R and R2 are sufficiently greater than the radius of the contact area,then the results of the Bussinesq problem of the linear elastic half-space loaded by the concentrated force can be adopted here.For this purpose let us denote by wi the vertical displacement induced by the local vertical strain of the point M belonging to the matrix;w2 is the corresponding displacement of the point N in a vertical direction. Finally,assuming that the tangential plane in point C remains unmovable during a local compression,the close-up of the two points M and N can be expressed by some real n as [344] 刀=a-6m+w,)=R-R) (6.15) 2RR2 If M and N belong to the contact area,their displacements w for i=1,2 can be written as =1x小gkd πE (6.16) which follows the symmetry of the pressure intensity q and the corresponding local strains with respect to the vertical axis at the point C.Integration in this formula is carried out over the entire contact surface.Therefore k+k川gddp=a-民-R) (6.17) Now,we are looking for such an expression for g to fulfil the above equation.It can be obtained for the pressure distribution on the contact surface represented by
Reliability Analysis 301 Let us observe that the contact problem is axisymmetric with respect to the vertical axis introduced at the centre of the spherical particle and at the bottom of this sphere (Figure 6.2). It is assumed that there is no pressure between the composite constituents and therefore the contact appears at the point C only. The distance between the points on the contacting surfaces and the plane transverse to the vertical axis of both surfaces is assumed to be small and can be described as ( ) 1 2 1 2 2 2 1 2R R r R R z z − − = (6.14) where r denotes the distance between the points M, N and the symmetry axis introduced at C. If the composite is loaded by the vertical force P acting along the vertical axis at the point C, then some local strains are induced in the neighbourhood of this point. They are a result of a contact phenomenon on a small circular surface (contact area). Assuming that the composite constituents radii R1 and R2 are sufficiently greater than the radius of the contact area, then the results of the Bussinesq problem of the linear elastic half-space loaded by the concentrated force can be adopted here. For this purpose let us denote by w1 the vertical displacement induced by the local vertical strain of the point M belonging to the matrix; w2 is the corresponding displacement of the point N in a vertical direction. Finally, assuming that the tangential plane in point C remains unmovable during a local compression, the close-up of the two points M and N can be expressed by some real η as [344] ( ) ( ) 1 2 1 2 2 1 2 2R R r R R w w − η = α − + = (6.15) If M and N belong to the contact area, their displacements wi for i=1,2 can be written as ∫∫ − = ϕ π ν qdsd E w i i i 2 1 (6.16) which follows the symmetry of the pressure intensity q and the corresponding local strains with respect to the vertical axis at the point C. Integration in this formula is carried out over the entire contact surface. Therefore ( ) ( ) 1 2 1 2 2 1 2 2R R r R R k k qdsd − + ∫∫ ϕ = α − (6.17) Now, we are looking for such an expression for q to fulfil the above equation. It can be obtained for the pressure distribution on the contact surface represented by
302 Random Composites the coordinates of the hemisphere with the radius a constructed on a contact surface.If go is taken as the pressure at the point C,then one can show that qds=10A a (6.18) where A--sin) (6.19) which gives k+血6：-r产n'pp=a-图-） (6.20) a 2RR2 Finally,the parameters a and a can be determined for this problem as 3 P(k +k2 RR2 a=3 (6.21) 4 R-R2 9n2 P2(k+k2)(R-R2) 16 RR which gives maximal pressure on the contact surface equal to 3P (6.22) 9= 2a2 Then,the normal stress can be defined as o:=-J3grdrz (6.23) -1*石4同 Let us note that the shear stresses are equal to 0,which result from the spherical symmetry of the reinforcing particle.However in the case of ellipsoidal reinforcement the shear stresses differ from 0. The main purpose of further analysis is to determine the probabilistic characteristics of maximal contact stresses as well as contact surface geometrical parameters.Since the spherical particle surrounding the matrix is considered,let us assume that the difference R-R2=e is smaller than R2.This parameter is treated as
302 Random Composites the coordinates of the hemisphere with the radius a constructed on a contact surface. If q0 is taken as the pressure at the point C, then one can show that A a q qds 0 ∫ = (6.18) where ( ) ϕ π 2 2 2 sin 2 A = a − r (6.19) which gives ( ) ( ) ( ) 1 2 1 2 2 0 1 2 0 2 2 2 2 sin 2 R R r R R a r d a k k q − − = − + ∫ ϕ ϕ α π π (6.20) Finally, the parameters a and α can be determined for this problem as ( ) ( )( ) ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ + − = − + = 3 1 2 1 2 2 1 2 2 2 3 1 2 1 2 1 2 16 9 4 3 R R P k k R R R R P k k R R a π α π (6.21) which gives maximal pressure on the contact surface equal to 2 2 3 a P q π = (6.22) Then, the normal stress can be defined as ( ) ( ) ( )⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = + = − + = − + − − ∫ 2 2 3 0 3 / 2 3 2 2 0 5 / 2 3 2 2 1 3 a z z qz r z q qrdrz r z a a σ z (6.23) Let us note that the shear stresses are equal to 0, which result from the spherical symmetry of the reinforcing particle. However in the case of ellipsoidal reinforcement the shear stresses differ from 0. The main purpose of further analysis is to determine the probabilistic characteristics of maximal contact stresses as well as contact surface geometrical parameters. Since the spherical particle surrounding the matrix is considered, let us assume that the difference R1-R2=ε is smaller than R2. This parameter is treated as
Reliability Analysis 303 input design parameter in further sensitivity analysis.The characteristics mentioned above are necessary in the final stochastic reliability computations and, considering the complexity of the equations describing reliability parameters,the stochastic second order perturbation method is proposed.The second order perturbation follows a traditional approach in this area (and the lack of convergence studies with respect to the Taylor expansion order).The third probabilistic moment method reflects the need of unsymmetric random variables modelling.Adopting the same notation as before (see Chapter 1)the skewness parameter S(u)is calculated by se,卢aae-kYn.h (6.24) In further applications,the Weibull distribution is used with the probability density function defined as (6.25) where B is the Weibull shape parameter,A denotes the scale parameter and is the location parameter,which indicates the smallest value of the random variable x for which the probability density function is positive.Considering this definition,the Weibull PDF is used for general mechanical applications,where many random variables must be nonnegative (Young modulus and some geometrical parameters, for instance)and especially in composite failure and fatigue modelling.Let us note that if discrete representation of a random variable b(x;0)is used,then statistical estimators may be applied to approximate any order probabilistic moments of this variable. Starting from probabilistic moments and the stochastic perturbation methodology presented above,we compute the first three probabilistic orders of the vertical stresses Eo.(x;0),Var(o.(x;@))and S(o.(x;@))as (6.26) (6.27)
Reliability Analysis 303 input design parameter in further sensitivity analysis. The characteristics mentioned above are necessary in the final stochastic reliability computations and, considering the complexity of the equations describing reliability parameters, the stochastic second order perturbation method is proposed. The second order perturbation follows a traditional approach in this area (and the lack of convergence studies with respect to the Taylor expansion order). The third probabilistic moment method reflects the need of unsymmetric random variables modelling. Adopting the same notation as before (see Chapter 1) the skewness parameter S(ui) is calculated by ( ) ( ) ( ) [ ] ∫ +∞ −∞ = u − E u p b db u S u i i R i i ( ) 1 3 3 σ (6.24) In further applications, the Weibull distribution is used with the probability density function defined as ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ − = − x x x x x x x x pR 0, exp ; β 1 β λ λ λ β (6.25) where β is the Weibull shape parameter, λ denotes the scale parameter and x is the location parameter, which indicates the smallest value of the random variable x for which the probability density function is positive. Considering this definition, the Weibull PDF is used for general mechanical applications, where many random variables must be nonnegative (Young modulus and some geometrical parameters, for instance) and especially in composite failure and fatigue modelling. Let us note that if discrete representation of a random variable b(x;θ) is used, then statistical estimators may be applied to approximate any order probabilistic moments of this variable. Starting from probabilistic moments and the stochastic perturbation methodology presented above, we compute the first three probabilistic orders of the vertical stresses E[ ] σ (x;ω) z , Var( ) σ (x;ω) z and S( ) σ (x;ω) z as [ ] ∑ ( ) = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + n i i i z z z b b E 1 2 2 2 2 0 1 σ σ σ σ (6.26) ( ) ( ) () () [ ]z n i i i i z i z n i i i z z i z z z S b b E b b b b b σ σ σ σ σ σ σ σ σ σ σ 2 1 3 2 2 1 2 2 2 2 2 2 − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = (6.27)
304 Random Composites and 00:0: sS6)σ36，) (6.28) -Ejo:l-3Eo.b"o.)a@.) Having computed the first three probabilistic moments of contact stresses (expected values,standard deviations and skewness coefficients),the random field of the limit function g(z;)is to be proposed.Usually,it can be introduced as a difference between allowable and actual stresses o.(z:)induced in the composite as g(zo)=ou(@-o,(2:w) (6.29) Let us underline that allowable stresses are most frequently analysed as random variables in the interior of statistically homogeneous materials,whereas actual stresses are random fields.That is why the computational analysis presented later is carried out for the specific value of the vertical coordinate z.The random variable of allowable stresses o)is specified by the use of the first three probabilistic moments El@),Var((@)and S((@)).Then,the corresponding probabilistic characteristics of the limit function are calculated as 26 (6.30) (6.31) as well as
304 Random Composites and ( ) ( ) () () ( ) [] [] ( )} ( )z z z z z n i i i i z i z z i z n i i i z z i z z z z E E S b b b b b b b b S σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ 3 3 2 3 1 3 2 2 3 1 2 2 2 2 2 3 1 3 1 3 2 2 3 − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = (6.28) Having computed the first three probabilistic moments of contact stresses (expected values, standard deviations and skewness coefficients), the random field of the limit function g(z;ω) is to be proposed. Usually, it can be introduced as a difference between allowable and actual stresses ) σ z (z;ω induced in the composite as g( ) () ( ) z;ω = σ all ω −σ z z;ω (6.29) Let us underline that allowable stresses are most frequently analysed as random variables in the interior of statistically homogeneous materials, whereas actual stresses are random fields. That is why the computational analysis presented later is carried out for the specific value of the vertical coordinate z. The random variable of allowable stresses σ all ( ) ω is specified by the use of the first three probabilistic moments E[ ] σ all( ) ω , Var( ) σ all( ) ω and S( ) σ all ( ) ω . Then, the corresponding probabilistic characteristics of the limit function are calculated as ∑ ( ) = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + n i i i b b g E g g 1 2 2 2 2 0 1 [ ] σ (6.30) ( ) ( ) ( ) () () [ ] 2 1 3 2 2 1 2 2 2 0 2 2 2 0 S b b E g b g b g b b g g b g g g n i i i i i n i i i i − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = σ σ σ as well as (6.31)
Reliability Analysis 305 26） s6,)o36,) (6.32) ∂bab 3(g) -ELg1-3Ag1o(g川。g Inserting the limit state function g from(6.29)into (6.30)-(6.32)and assuming that the random variable of allowable stresses and the random field of actual stresses are uncorrelated it is obtained that (6.33) e-6-+06-腰 (6.34) S6)o6,）-E[o-o】 and,finally sg)=aa-o月 oi") (6.35) s6,)6) a-o:loo-o) Comparing the second order second moment(SOSM)approach with the second order third moment (SOTM)approach,it is seen that the expected values are described by exactly the same equation,while standard deviations(or variances) have some extra components connected with the skewness of analysed PDF;the third order parameter of the output PDF is taken into account in the SOTM-based analysis [282]
Reliability Analysis 305 ( ) ( ) ( ) ( ) () () } ( ) 1 [ ] 3 [ ] ( ) ( ) 1 3 2 2 3 3 3 2 3 1 3 2 2 0 3 1 2 2 2 2 0 2 0 3 0 g E g E g g g S b b b g b g g b g b b g g b g S g g g n i i i i i i n i i i i σ σ σ σ σ − − ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = + ∑ ∑ = = (6.32) Inserting the limit state function g from (6.29) into (6.30)-(6.32) and assuming that the random variable of allowable stresses and the random field of actual stresses are uncorrelated it is obtained that ∑ ( ) = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = − − n i i i z all z b b E g 1 2 2 2 2 0 0 1 [ ] σ σ σ σ (6.33) ( ) ( ) ( ) ( ) () () [ ] all z n i i i i z i z n i i i z all z i z all z S b b E b b b b b g σ σ σ σ σ σ σ σ σ σ σ σ σ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ = − + ∑ ∑ = = 2 1 3 2 2 1 2 2 2 0 0 2 2 2 0 0 (6.34) and, finally ( ) ( ) ∑ ( ) ( ) ( ) = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ + − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + − ⎩ ⎨ ⎧ = − n i i i z all z i z all z all z b b b S g 1 2 2 2 2 0 0 2 0 0 3 0 0 2 2 3 σ σ σ σ σ σ σ σ σ ∑ () () = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + n i i i i i i S b b b g b g g b g 1 3 2 2 0 3 3 σ [ ][ ] ( )} ( ) 3 0 0 3 0 0 0 0 2 0 0 1 3 all z E all z E all z all z σ σ σ σ σ σ σ σ σ σ − − − − − − (6.35) Comparing the second order second moment (SOSM) approach with the second order third moment (SOTM) approach, it is seen that the expected values are described by exactly the same equation, while standard deviations (or variances) have some extra components connected with the skewness of analysed PDF; the third order parameter of the output PDF is taken into account in the SOTM-based analysis [282]