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Lifetime prediction of fatigue sensitive structural elements
Structural Safety, 12 (1993) 105-111
105
Elsevier
Lifetime prediction of fatigue sensitive structural elements * F. Casciati, P. Colombi and L. Faravelli Department of Structural Mechanics, University of Pavia via Abbiategrasso 211, 127100 Pavia, Italy
Abstract. The propagation of uncertainty from the mechanical and geometrical properties of a structural element to fracture mechanics parameters, as the stress intensity factor and the J integral, is studied using a stochastic finite element method. The structural analysis code adopted for the computation of the fracture mechanics parameters is regarded as a "black box" by which numerical experiments can be performed. The results of these experiments are arranged into a response surface scheme. This means that no modifications are requested into the structural analysis algorithm. The method evaluates the cumulative distribution function of a single output variable and the conditional probability distribution function of a variable for given values of another. The numerical example deals with an infinite long pipeline with an axial semi-elliptical surface crack. The goal is to provide a complete characterization of the stochastic relation between the J integral and the crack depth when the geometrical and mechanical input variables are random.
Kay words: fracture mechanics; lifetime; response surface.
1. Introduction Fatigue crack propagation is a structural problem which is complicated by the sensitivity of the output to the material properties, to the structural geometry, to the external actions and to the environmental conditions. A pure deterministic approach is unable to model the influence of all these parameters and a probabilistic analysis is required. The main step in the evaluation of the fatigue lifetime is the calculation of the stress intensity factor or the J integral [1]. These are the parameters for two possible different ways of writing the differential equation which governs the problems. The computation of these parameters is affected by the uncertainties on the input quantities, the geometrical and mechanical properties and external loads. This makes the crack growth a p h e n o m e n o n of a stochastic nature. The aim of this paper is to characterize this stochastic dependency. For this purpose the input uncertainties are propagated into the evaluation of the J integral by a stochastic finite element method [2-5].
* Discussion is open until November 1993 (please submit your discussion paper to the Editor, Ross B. Corotis). 0167-4730/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
106 2. Mechanical model Different techniques are available for the evaluation of the stress intensity factor or the J integral. For the general situation of a 3D cracked body the so-called virtual crack extension method due to Parks [11] must be used. For a 2D body with surface crack a simplified analysis can be performed using the line spring methodology. The surface crack is replaced by a through the thickness crack. The crack front is discretized by couples of nodes: each node has the same coordinates but different degrees of freedom. The crack is sewn by truss elements whose stiffness matrix S depends on the local depth of the crack. This matrix is calculated by the technique described in [12] for the modes I, II and III of cracking. The finite element analysis of the structural component provides the relative displacements of the two sides of the crack and hence, using the matrix S the forces in the truss element. As these forces are evaluated, the stress intensity factor K~, K n and Kin are obtained as shown in [12]. The elastic part, JeJ, of the J integral is finally evaluated by the classical fracture mechanics relations:
l--u 2 l+v Je,- E (K? --~K?I ) + T K ? I I
(1)
where u is the Poisson coefficient and E is the Young modulus. The contribution, Jpt, to the J integral due to the plastic deformation, J = J¢~ + Jpl, is calculated using the procedure given in [12]. Briefly, the elasto-plastic stiffness matrix S ~p for the line spring elements is obtained defining a suitable yield surface [10,12]. The calculation starts with the evaluation of the plastic part of the crack tip opening displacement, 6tor, obtained by integration, on the load path, of the following relation [12]: g~, = go, + ( t / 2
- a)0 v'
(2)
In eqn. (2) 6 and 0 are the relative displacement and rotation, respectively, of the two sides of the crack, t is the thickness of the pipe and a the crack depth. Finally, J p~ is obtained using the following relation:
/pl = mO.y~pl
(3)
where o-y is the yield stress and m a coefficient dependent on the strain field geometry near the crack tip.
3. Stochastic model The stochastic finite element approach adopted in this paper is based on the evaluation of a response surface (response surface method). This surface describes the output of a mechanical system as a function of the input variables. This is done by regression analysis on the output of numerical experiments, appropriately planned [6]. The number of experiments to be performed could be rather high if several input random variables are considered. The response surface methodology adopted in this paper models the input random quantities as random variables, stochastic processes or random fields. The reader is referred to [6-8] for the details of the procedure which is just summarized in the following part of this section.
107
The input random variables are grouped in two classes as the result of a preliminary sensitivity analysis: primary random variables and secondary random variables. The primary random variables are the random variable whose randomness deeply influences the randomness of the structural response. The secondary random variables are the variables whose randomness is of marginal importance for the randomness of the structural response. Let x be the vector of the principal random variables and y the structural response of interest. Suitable transformations Y of y and Xj of xj are introduced in order to make the model more consistent. This relation between Y and Xi is described using a second order polynomial [6]. In matrix notation one has: y = a o + x T O t l q- xT(II2 X -}- ~:
(4)
The coefficients a0, a~ and ~2 are evaluated by a regression analysis over the results of the numerical experiments [6]. The term E on the right-hand side takes into account the model error and the effect of the secondary random variables. Let y~ and Y2 be two response variables and Y1, Y2 their appropriate transformations. When the stochastic finite element analysis illustrated above is solved, both the marginal probability distribution function of I11 and II2 are known. The variables II1 and Y2 are generally correlated and, hence the model given in eqn. (4) for Y~ can be updated in the form: Y1 = ~(~ -c xT°trl q- xTilcr2 X q'- c1Y2 q'- E~
(s)
a 0', till,' ot 2' and c~ are the coefficients of the new regression problem. Classical statistical theory [9] provide the way to evaluate the correlation coefficient Pr,.r2 between I11 and Y2. If more information is necessary one needs also the joint probability distribution of I11 and Y2. In terms of probability distribution function it is given by:
where
PYDY2(~I, ~'2) =
PY, iva({1 [ Kz)PY~({2)
(6)
The cumulative distribution function Py2((2) is obtained applying level-2 methods to eqn. (4) [6]. Moreover, the conditional cumulative distribution function PY, i r2((1 ] ~'2) is found starting from model (5) with II2 = {2.
4. Lifetime prediction Consider an infinite long pipeline with a surface crack in the axial direction. The radius R and the thickness t of the pipe are 268 mm and 10 mm, respectively. The crack is assumed to be semi-elliptic: the nominal value of the major semi-axes b is 29 mm. Three different values of the depth a of the crack are considered. They correspond to the following value of the ratio a/t: 0.2 (shallow crack deep), 0.46 (moderate deep crack) and 0.7 (deep crack). Due to the symmetry of the problem only one half of the pipe was discretized in finite elements by the finite element code A B A Q U S [10]. The mesh is given in Fig. 1 and was realized for a length equal to 15 times the semi-axis b of the crack. It was assumed that this length is sufficient to realize the condition of infinite long pipeline. The ratio radius/thickness allows a discretization by shell elements. This makes it possible to utilize line spring elements for the computation of the J integral: 12 line spring elements were used for crack discretiza-
108
Fig. 1. The finite element mesh used for the structural analysis.
tion. The line spring elements have different lengths as shown in Fig. 2. The material behaviour is assumed to be elasto-plastic with isotropic hardening. The material characteristics are: • Young modulus: E = 206.8 GPa; • Yield stress: ~ry = 482.5 MPa; • Poisson coefficient: u = 0.3. The finite element mesh was refined around the cracked zone using the option M P C given by A B A Q U S . By using it, it is in fact possible to force two nodes with the same coordinates to have the same degrees of f r e e d o m as well. This option makes it also possible to " s e w n " the crack, i.e. to modify the major semi-axis b. The external loads are given as an internal pressure of 12 MPa. Concentrated loads are present at the end of the piece of the pipeline considered in the analysis in order to reproduce the continuity condition. All these external actions are considered as deterministic quantities.
TABLE 1 Probabilistic definition of the input quantities assumed as random variables in the analysis Physical quantity
Distribution
Mean value
Standard deviation
Thickness t Radius R Major semi-axis b Young modulus Yield stress
Gaussian Lognormal Lognormal Gaussian Gaussian
10 mm 268 mm 28.93 mm 206.8 GPa 482.5 MPa
0.67 2.68 2.97 6.67 14.2
mm mm mm GPa MPa
109
/
S"
I / I I
/
/
j
Fig. 2. Detail of the finite element mesh around the crack tip.
1
I
0.9
I
I
a/t-0.2
I
a/t-0.46
!
a/t-0.7
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 I
0 0
10
20
30 J Integral
!
I
40
50
(N/mm)
Fig. 3. Cumulative distribution ~nction ofthe Jintegral ~ r t h r e e d i f f e r e n t v a l u e s o f t h e r a t i o a/t.
60
110 1
,
,
O
.,~ 0
e~ o .,,t
Ps(.,)lj(.,)(J(a)lY(ad=
o. 9
3.718 N / m m )
........
0.8 0.7
" "~
PJc.,wc°,~CS(.)lS(.~) = 4.063 N/ram)
0.6
0.5
O
0.4 0
~
ps(,,)lS(°t)(y(a)JJ(al) = 4.520 N / m m )
0.3
0.2 ~
0.1 A
o
"J' 5
/ 10
I
,
I
I
I
15
20
25
30
35
J Integral
4/)
(N/mm)
Fig. 4. Conditional distribution functions of the J integral for a / t = 0.46 for given values of the same J integral computed for a / t = 0.2. The dashed line indicates the marginal distribution function.
The input variables assumed random in the analysis are: the elastic modulus E, the yield stress %, the radius R, the thickness t and the semi-axis b of the crack. Their probabilistic definition is given in Table 1. Only the semiaxis b, the thickness t and the radius R of the pipe are regarded as primary random variables. The mechanical properties of the material (the Young modulus and the yield stress) are regarded as secondary random variables, since it is assumed that the material is well controlled so that the randomness of the material parameters is small. It is worth noting that the major semi-axis b of the crack can only assume discrete values. Eigth classes were used in the analysis to describe the variability of this parameter. First the cumulative distribution function of the J integral is evaluated for the considered crack depths (a/t equal to 0.2, 0.46 and 0.7, respectively). These results are shown in Fig. 3. Finally, the conditional probability distribution functions of the values of the J integral for a/t = 0.46, conditioned to give values of the J integral for a/t = 0.2 were calculated. The last result is shown in Fig. 4. In this figure the conditional probability distribution functions are plotted as a solid line, while the marginal probability distribution function is plotted as a dashed line. The conditional distribution functions of the values of the J integral for a/t = 0.7, conditioned to a given value of the J integral for a/t = 0.2 and a/t = 0.46 were also evaluated.
5.
Conclusions
This paper illustrates a technique for the propagation of the uncertainty present in the evaluation of the fracture mechanics parameters. The proposed approach can be coupled with any sort of computer code of structural analysis since no modification is required inside the
111
structural analysis algorithm. The proposed numerical example illustrates in detail the adopted procedure for the evaluation of the distribution function of the J integral for a given crack depth, conditioned to given values of the J integral for lower crack depths. This completely characterizes the relation between the J integral and the crack size from a stochastic point of view. Open problems for further research developments are the investigation of the influence of the randomness of the external loads on the randomness of the J integral and the determination of the most unfavourable combination of external actions. Also the modeling of the mechanical parameters as random fields would make the symmetry used in the analysis unrealistic.
Acknowledgement This research has been supported by funds from both the Italian Ministry of University and Research (MURST) and the Italian Space Agency (ASI).
References [1] D. Broek, Elementary Engineering Fracture Mechanics, Martinus Nijhoff, Dordrecht, 1982. [2] F. Casciati, P. Colombi and L. Faravelli, Filter technique for stochastic crack growth, in Computational Stochastic Mechanics, eds. P.D. Spanos and C.A. Brebbia, Elsevier, Amsterdam, 1991, pp. 485-496. [3] F. Casciati, P. Colombi and L. Faravelli, Fatigue lifetime evaluation via response surface methodology, in Proc. European Safety and Reliability Conf. 92, Elsevier, Amsterdam, 1992, pp. 550-561. [4] F. Casciati, P. Colombi and L. Faravelli, Stochastic crack growth by filter technique, in Proc. Sixth Int. Conf. on Applications of Statistics and Probability m Civil Engineering, Eds. L. Esteva and S.R. Ruiz, Mexico City, Mexico, 1991, Vol. 1, pp. 74-81. [5] F. Casciati, P. Colombi and L. Faravelli, Fatigue crack size probability distribution via filter technique, Fatigue Fract. Eng. Mater. Struct., 15 (5) (1992) 463-475. [6] F. Casciati and L. Faraveili, Fragility Analysis of Complex Structural Systems, Research Studies Press Ltd., Taunton, 1991. [7] L. Faravelli, Response surface approach for reliability analysis, J. Eng. Mech., ASCE, 115 (12) (1989) 2763-2780. [8] L. Faravelli, Finite element analysis of stochastic nonlinear continua, in Computational Mech. of Probability and Reliab. Analysis, eds. Liu W.K. and Belytschko T., Elmepress, Lausanne, 1989, pp. 263-280. [9] L. Faravelli, Response variable correlation in stochastic finite elements analysis, Meccanica, 22 (2) (1988) 102-106. [10] Hibbit, Karlsson & Sorensen Inc., ABAQUS Manual: Vol. 1, User's Manual; Vol. 2, Theory Manual; Vol. 3., Example Problems Manual. Providence, RI, USA, 1982. [11] D.M. Parks, The virtual crack extension method for nonlinear material behaviour, Comp. Meth. in Appl. Mech. Eng., 12 (1977) 353-364. [12] D.M. Parks, The inelastic line spring: estimates of elastic-plastic fracture mechanics parameters for surfacecracked plates and shells, J. Pressure Vessel Tech., 103 (1981) 246-254.
105
Elsevier
Lifetime prediction of fatigue sensitive structural elements * F. Casciati, P. Colombi and L. Faravelli Department of Structural Mechanics, University of Pavia via Abbiategrasso 211, 127100 Pavia, Italy
Abstract. The propagation of uncertainty from the mechanical and geometrical properties of a structural element to fracture mechanics parameters, as the stress intensity factor and the J integral, is studied using a stochastic finite element method. The structural analysis code adopted for the computation of the fracture mechanics parameters is regarded as a "black box" by which numerical experiments can be performed. The results of these experiments are arranged into a response surface scheme. This means that no modifications are requested into the structural analysis algorithm. The method evaluates the cumulative distribution function of a single output variable and the conditional probability distribution function of a variable for given values of another. The numerical example deals with an infinite long pipeline with an axial semi-elliptical surface crack. The goal is to provide a complete characterization of the stochastic relation between the J integral and the crack depth when the geometrical and mechanical input variables are random.
Kay words: fracture mechanics; lifetime; response surface.
1. Introduction Fatigue crack propagation is a structural problem which is complicated by the sensitivity of the output to the material properties, to the structural geometry, to the external actions and to the environmental conditions. A pure deterministic approach is unable to model the influence of all these parameters and a probabilistic analysis is required. The main step in the evaluation of the fatigue lifetime is the calculation of the stress intensity factor or the J integral [1]. These are the parameters for two possible different ways of writing the differential equation which governs the problems. The computation of these parameters is affected by the uncertainties on the input quantities, the geometrical and mechanical properties and external loads. This makes the crack growth a p h e n o m e n o n of a stochastic nature. The aim of this paper is to characterize this stochastic dependency. For this purpose the input uncertainties are propagated into the evaluation of the J integral by a stochastic finite element method [2-5].
* Discussion is open until November 1993 (please submit your discussion paper to the Editor, Ross B. Corotis). 0167-4730/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
106 2. Mechanical model Different techniques are available for the evaluation of the stress intensity factor or the J integral. For the general situation of a 3D cracked body the so-called virtual crack extension method due to Parks [11] must be used. For a 2D body with surface crack a simplified analysis can be performed using the line spring methodology. The surface crack is replaced by a through the thickness crack. The crack front is discretized by couples of nodes: each node has the same coordinates but different degrees of freedom. The crack is sewn by truss elements whose stiffness matrix S depends on the local depth of the crack. This matrix is calculated by the technique described in [12] for the modes I, II and III of cracking. The finite element analysis of the structural component provides the relative displacements of the two sides of the crack and hence, using the matrix S the forces in the truss element. As these forces are evaluated, the stress intensity factor K~, K n and Kin are obtained as shown in [12]. The elastic part, JeJ, of the J integral is finally evaluated by the classical fracture mechanics relations:
l--u 2 l+v Je,- E (K? --~K?I ) + T K ? I I
(1)
where u is the Poisson coefficient and E is the Young modulus. The contribution, Jpt, to the J integral due to the plastic deformation, J = J¢~ + Jpl, is calculated using the procedure given in [12]. Briefly, the elasto-plastic stiffness matrix S ~p for the line spring elements is obtained defining a suitable yield surface [10,12]. The calculation starts with the evaluation of the plastic part of the crack tip opening displacement, 6tor, obtained by integration, on the load path, of the following relation [12]: g~, = go, + ( t / 2
- a)0 v'
(2)
In eqn. (2) 6 and 0 are the relative displacement and rotation, respectively, of the two sides of the crack, t is the thickness of the pipe and a the crack depth. Finally, J p~ is obtained using the following relation:
/pl = mO.y~pl
(3)
where o-y is the yield stress and m a coefficient dependent on the strain field geometry near the crack tip.
3. Stochastic model The stochastic finite element approach adopted in this paper is based on the evaluation of a response surface (response surface method). This surface describes the output of a mechanical system as a function of the input variables. This is done by regression analysis on the output of numerical experiments, appropriately planned [6]. The number of experiments to be performed could be rather high if several input random variables are considered. The response surface methodology adopted in this paper models the input random quantities as random variables, stochastic processes or random fields. The reader is referred to [6-8] for the details of the procedure which is just summarized in the following part of this section.
107
The input random variables are grouped in two classes as the result of a preliminary sensitivity analysis: primary random variables and secondary random variables. The primary random variables are the random variable whose randomness deeply influences the randomness of the structural response. The secondary random variables are the variables whose randomness is of marginal importance for the randomness of the structural response. Let x be the vector of the principal random variables and y the structural response of interest. Suitable transformations Y of y and Xj of xj are introduced in order to make the model more consistent. This relation between Y and Xi is described using a second order polynomial [6]. In matrix notation one has: y = a o + x T O t l q- xT(II2 X -}- ~:
(4)
The coefficients a0, a~ and ~2 are evaluated by a regression analysis over the results of the numerical experiments [6]. The term E on the right-hand side takes into account the model error and the effect of the secondary random variables. Let y~ and Y2 be two response variables and Y1, Y2 their appropriate transformations. When the stochastic finite element analysis illustrated above is solved, both the marginal probability distribution function of I11 and II2 are known. The variables II1 and Y2 are generally correlated and, hence the model given in eqn. (4) for Y~ can be updated in the form: Y1 = ~(~ -c xT°trl q- xTilcr2 X q'- c1Y2 q'- E~
(s)
a 0', till,' ot 2' and c~ are the coefficients of the new regression problem. Classical statistical theory [9] provide the way to evaluate the correlation coefficient Pr,.r2 between I11 and Y2. If more information is necessary one needs also the joint probability distribution of I11 and Y2. In terms of probability distribution function it is given by:
where
PYDY2(~I, ~'2) =
PY, iva({1 [ Kz)PY~({2)
(6)
The cumulative distribution function Py2((2) is obtained applying level-2 methods to eqn. (4) [6]. Moreover, the conditional cumulative distribution function PY, i r2((1 ] ~'2) is found starting from model (5) with II2 = {2.
4. Lifetime prediction Consider an infinite long pipeline with a surface crack in the axial direction. The radius R and the thickness t of the pipe are 268 mm and 10 mm, respectively. The crack is assumed to be semi-elliptic: the nominal value of the major semi-axes b is 29 mm. Three different values of the depth a of the crack are considered. They correspond to the following value of the ratio a/t: 0.2 (shallow crack deep), 0.46 (moderate deep crack) and 0.7 (deep crack). Due to the symmetry of the problem only one half of the pipe was discretized in finite elements by the finite element code A B A Q U S [10]. The mesh is given in Fig. 1 and was realized for a length equal to 15 times the semi-axis b of the crack. It was assumed that this length is sufficient to realize the condition of infinite long pipeline. The ratio radius/thickness allows a discretization by shell elements. This makes it possible to utilize line spring elements for the computation of the J integral: 12 line spring elements were used for crack discretiza-
108
Fig. 1. The finite element mesh used for the structural analysis.
tion. The line spring elements have different lengths as shown in Fig. 2. The material behaviour is assumed to be elasto-plastic with isotropic hardening. The material characteristics are: • Young modulus: E = 206.8 GPa; • Yield stress: ~ry = 482.5 MPa; • Poisson coefficient: u = 0.3. The finite element mesh was refined around the cracked zone using the option M P C given by A B A Q U S . By using it, it is in fact possible to force two nodes with the same coordinates to have the same degrees of f r e e d o m as well. This option makes it also possible to " s e w n " the crack, i.e. to modify the major semi-axis b. The external loads are given as an internal pressure of 12 MPa. Concentrated loads are present at the end of the piece of the pipeline considered in the analysis in order to reproduce the continuity condition. All these external actions are considered as deterministic quantities.
TABLE 1 Probabilistic definition of the input quantities assumed as random variables in the analysis Physical quantity
Distribution
Mean value
Standard deviation
Thickness t Radius R Major semi-axis b Young modulus Yield stress
Gaussian Lognormal Lognormal Gaussian Gaussian
10 mm 268 mm 28.93 mm 206.8 GPa 482.5 MPa
0.67 2.68 2.97 6.67 14.2
mm mm mm GPa MPa
109
/
S"
I / I I
/
/
j
Fig. 2. Detail of the finite element mesh around the crack tip.
1
I
0.9
I
I
a/t-0.2
I
a/t-0.46
!
a/t-0.7
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 I
0 0
10
20
30 J Integral
!
I
40
50
(N/mm)
Fig. 3. Cumulative distribution ~nction ofthe Jintegral ~ r t h r e e d i f f e r e n t v a l u e s o f t h e r a t i o a/t.
60
110 1
,
,
O
.,~ 0
e~ o .,,t
Ps(.,)lj(.,)(J(a)lY(ad=
o. 9
3.718 N / m m )
........
0.8 0.7
" "~
PJc.,wc°,~CS(.)lS(.~) = 4.063 N/ram)
0.6
0.5
O
0.4 0
~
ps(,,)lS(°t)(y(a)JJ(al) = 4.520 N / m m )
0.3
0.2 ~
0.1 A
o
"J' 5
/ 10
I
,
I
I
I
15
20
25
30
35
J Integral
4/)
(N/mm)
Fig. 4. Conditional distribution functions of the J integral for a / t = 0.46 for given values of the same J integral computed for a / t = 0.2. The dashed line indicates the marginal distribution function.
The input variables assumed random in the analysis are: the elastic modulus E, the yield stress %, the radius R, the thickness t and the semi-axis b of the crack. Their probabilistic definition is given in Table 1. Only the semiaxis b, the thickness t and the radius R of the pipe are regarded as primary random variables. The mechanical properties of the material (the Young modulus and the yield stress) are regarded as secondary random variables, since it is assumed that the material is well controlled so that the randomness of the material parameters is small. It is worth noting that the major semi-axis b of the crack can only assume discrete values. Eigth classes were used in the analysis to describe the variability of this parameter. First the cumulative distribution function of the J integral is evaluated for the considered crack depths (a/t equal to 0.2, 0.46 and 0.7, respectively). These results are shown in Fig. 3. Finally, the conditional probability distribution functions of the values of the J integral for a/t = 0.46, conditioned to give values of the J integral for a/t = 0.2 were calculated. The last result is shown in Fig. 4. In this figure the conditional probability distribution functions are plotted as a solid line, while the marginal probability distribution function is plotted as a dashed line. The conditional distribution functions of the values of the J integral for a/t = 0.7, conditioned to a given value of the J integral for a/t = 0.2 and a/t = 0.46 were also evaluated.
5.
Conclusions
This paper illustrates a technique for the propagation of the uncertainty present in the evaluation of the fracture mechanics parameters. The proposed approach can be coupled with any sort of computer code of structural analysis since no modification is required inside the
111
structural analysis algorithm. The proposed numerical example illustrates in detail the adopted procedure for the evaluation of the distribution function of the J integral for a given crack depth, conditioned to given values of the J integral for lower crack depths. This completely characterizes the relation between the J integral and the crack size from a stochastic point of view. Open problems for further research developments are the investigation of the influence of the randomness of the external loads on the randomness of the J integral and the determination of the most unfavourable combination of external actions. Also the modeling of the mechanical parameters as random fields would make the symmetry used in the analysis unrealistic.
Acknowledgement This research has been supported by funds from both the Italian Ministry of University and Research (MURST) and the Italian Space Agency (ASI).
References [1] D. Broek, Elementary Engineering Fracture Mechanics, Martinus Nijhoff, Dordrecht, 1982. [2] F. Casciati, P. Colombi and L. Faravelli, Filter technique for stochastic crack growth, in Computational Stochastic Mechanics, eds. P.D. Spanos and C.A. Brebbia, Elsevier, Amsterdam, 1991, pp. 485-496. [3] F. Casciati, P. Colombi and L. Faravelli, Fatigue lifetime evaluation via response surface methodology, in Proc. European Safety and Reliability Conf. 92, Elsevier, Amsterdam, 1992, pp. 550-561. [4] F. Casciati, P. Colombi and L. Faravelli, Stochastic crack growth by filter technique, in Proc. Sixth Int. Conf. on Applications of Statistics and Probability m Civil Engineering, Eds. L. Esteva and S.R. Ruiz, Mexico City, Mexico, 1991, Vol. 1, pp. 74-81. [5] F. Casciati, P. Colombi and L. Faravelli, Fatigue crack size probability distribution via filter technique, Fatigue Fract. Eng. Mater. Struct., 15 (5) (1992) 463-475. [6] F. Casciati and L. Faraveili, Fragility Analysis of Complex Structural Systems, Research Studies Press Ltd., Taunton, 1991. [7] L. Faravelli, Response surface approach for reliability analysis, J. Eng. Mech., ASCE, 115 (12) (1989) 2763-2780. [8] L. Faravelli, Finite element analysis of stochastic nonlinear continua, in Computational Mech. of Probability and Reliab. Analysis, eds. Liu W.K. and Belytschko T., Elmepress, Lausanne, 1989, pp. 263-280. [9] L. Faravelli, Response variable correlation in stochastic finite elements analysis, Meccanica, 22 (2) (1988) 102-106. [10] Hibbit, Karlsson & Sorensen Inc., ABAQUS Manual: Vol. 1, User's Manual; Vol. 2, Theory Manual; Vol. 3., Example Problems Manual. Providence, RI, USA, 1982. [11] D.M. Parks, The virtual crack extension method for nonlinear material behaviour, Comp. Meth. in Appl. Mech. Eng., 12 (1977) 353-364. [12] D.M. Parks, The inelastic line spring: estimates of elastic-plastic fracture mechanics parameters for surfacecracked plates and shells, J. Pressure Vessel Tech., 103 (1981) 246-254.