RELIABILITY ANALYSIS IN STRUCTURES SUBMITTED TO FATIGUE USING THE BOUNDARY ELEMENT METHOD

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1 RELIABILITY ANALYSIS IN STRUCTURES SUBMITTE TO FATIGUE USING THE BOUNARY ELEMENT METHO Edson enner Leonel, André Teófilo Beck, Wilson Sérgio Venturini, epartment of Structural Engineering, São Carlos School of Engineering, University of São Paulo São Carlos, SP, Brazil Abstract: In this paper, the boundary element method and reliability analysis are used to model and solve structural problems involving uncertainty, stress concentrations singularities and crack propagation. Particularly, the dual boundary element formulation is adopted for the analysis of solids with cracks. This procedure is shown to be efficient and sufficiently accurate. Both crack faces are described by the same geometry and no singularity appears. Since crack propagation parameters show large variability, it is particularly important to include uncertainty in the analysis. This is done by coupling the boundary element crack propagation model with structural reliability models. Two ways of performing this coupling are considered in this work. In the response surface method, the numerical (mechanic) limit state surface is appoximated by an analytical, surrogate model, as the search for the design point is performed. The second scheme is referred herein as numerical gradients. In this method, gradients of the limit state function are computed directly, based on the numerical response of the mechanical model. The limit state function remains implicit. The two schemes are applied to some example crack propagation problems. It is shown that the numerical gradients technique leads to much faster convergence, in comparison to the traditional response surface method. Keywords: Boundary Element Method, Structural Reliability, Fatigue, Crack Propagation, Uncertainty. 1. INTROUCTION This paper presents a coupling between Boundary Element (BEM) and structural reliability models. A mechanical model is constructed, based on BEM, in order to deal with arbitrary crack growth in complex geometries. The probability of collapse (fracture) is formulated in terms of structural reliability models, with the collapse limit state function being computed via boundary element model. Two alternative procedures to perform the coupling between mechanical (BEM) and structural reliability models are considered in this study. The first is the well known Response Surface Method (RSM), based on the construction of local polynomial approximations to the numerical, mechanical response. The second scheme, referred herein as Numerical Gradients (NG), gradients of the limit state function are computed directly, based on the numerical response of the mechanical model. The limit state equation remains implicit. Both schemes are applied in the solution of some representative crack propagation problems. It is shown in the paper that the numerical gradients technique leads to much faster convergence, in comparison to the traditional response surface method. 2. CRACK GROWTH LIFE PREICTION In this paper, the life prediction criterion of Paris and Erdogan (1963) is considered: da C K dn = n (1) where C and n are materials constants, a represents the crack length, N is the number of loading cycles and K is the range of stress intensity factors. To evaluate life under crack propagation, it is necessary to calculate stress intensity fields ahead of the crack. Stress intensity factors are functions of component and crack geometry, as well as far stress fields. For most practical cases, stress intensity factors have to be evaluated numerically. In this paper, BEM is used to model complex component and crack geometries, and stress intensity factors are obtained using the displacement correlation technique. This technique relates displacements at the crack tip with stress intensity factors, as expressed in Eq. (2), for 2 problems: 2 π µ 2 π µ KI = CO K = CS r II ( κ+ 1) r ( κ+ 1) (2) where CO is the Crack Opening isplacement, CS is the Crack Sliding isplacement and r is the distance between

2 the crack tip and the points where CO and CS are calculated. To calculate the equivalent stress intensity factor and crack propagation angle the maximum circumferential stress theory is adopted: 2 3 θp 2 θp θp θp 1 K I K I KEq = KI cos 3 KII cos sin and tan = ± KII K II (3) 3. BOUNARY ELEMENT FORMULATION For 2 elastic domains, the boundary element formulation can be obtained considering a bi-dimensional homogeneous domain Ω, with boundary Γ. The equilibrium expression is written, in terms of displacements, by: u 1 bi + u + = 0 1 2υ µ i, jj j, ji (4) In this equation, µ represents the shear modulus, υ is the Poisson s coefficient, u i are the components of displacement and b i the body forces. Using Betti s theorem, one can obtain the representation of the singular integral for displacements, without the action of body forces, given by: c f cu f + P f cu c dγ= P cu f c dγ * * (, ) ( ) (, ) ( ) ( ) (, ) il l il l l il Γ Γ (5) where P l and u l are the tractions and displacements on the boundary, is the principal value integral of Cauchy and term c il is equal δ il /2 for smooth contours, and * * Pil and u are the fundamental solutions, for tractions and displacements, il respectively. For the elastic problem, it can be written: * 1 1 ulk ( f, c) = (3 4 ) ln +,, 8 (1 ) υ δlk rr l k πµ υ r * 1 P (, ), (1 2 ) 2,, (1 2 )(,, ) lk f c = (1 ) rn υδlk rr l k υ ηlrk ηkrl π υ r (6a) (6b) where r is the distance between the source point and the field point. Linear elastic 2 domains can be analyzed by evaluating Eq. (5) on the elements located on the boundary of the structure. However, for solids with cracks, use of this equation for the discretization of all boundaries produces a singularity, due the fact that crack faces are located at the same geometric position. Many formulations are proposed in the literature to deal with crack problems in boundary element formulation. Among these, the dual boundary element formulation is distinguished, which is adjusted to model arbitrary crack growth process. In the dual boundary formulation, four algebraic relations are required at each node along the crack line. To avoid adopting redundant relationships, the four relations are obtained from two integral equations of displacements and from two integral equations of tractions. The hiper-singular integral representation of the boundary, in terms of tractions, can be obtained from Eq. (5). This must be derived to obtain the integral representation in terms of strains. After that, Hooke s law is applied to write the integral representation in terms of stresses and finally, using the equilibrium conditions, Eq. (7) is obtained: 1 P( f) + η S ( f,) cu() cdγ = η ( f,) cp() cdγ (7) 2 j k kj k k kj k Γ Γ where is the principal value integral of Hadamard, terms S kj and kj contain the derivatives of P ij * and u ij *, respectively (Portela et al., 1992). In this paper, only linear boundary elements are used. The singular integrals are evaluated in numerical form, using the sub-element procedure, while the hiper-singulars integrals are calculated by analytical expressions. This procedure was found to be sufficiently accurate and efficient in arbitrary crack growth problems. The two crack faces are described by the same geometry and no singularity appears. 4. STRUCTURAL RELIABILITY FORMULATION

3 The set of random or uncertain problem parameters is given by vector X. In crack propagation problems, vector X includes material crack growth parameters C and n, loading or stress cycles, initial crack size, which are significant sources of uncertainty. The performance or safety of the structure can be described by limit state functions g(x)=0, which divide the random variable domain in safety and failure domains. The boundary between safety and failure domains is known as limit state surface. The failure probability is given by the multi-dimensional integral: Pf = fx( x) dx (8) g( x ) 0 where f X (x) is the joint probability density function of the problems random variables. This problem can be solved directly, via Monte Carlo simulation, or using approximate solutions. By far, the most popular approximate solution is the First Order Reliability Method (FORM). In this solution, the vector of random variables X and the limit state functions are mapped to the standard normal space, Y, in a transformation that involves rotation of coordinates and evaluation of equivalent normal distributions. In the space of Y, the joint probability density function f Y (y) presents radial symmetry, and the most probable failure point, or design point y*, can be found by solving the following optimization problem: minimize: d 2 t = y y subject to: g( y )=0 (9) The (minimal) distance between design point y* and the origin of Y space is known as Reliability Index (β). A linearization of the limit state function at the design point yields the first order failure probability approximation, P =Φ β, where Φ is the standard normal cumulative distribution function. The optimization problem (Eq. 9) can be f [ ] solved by any optimization algorithm. The Hassofer-Lind-Rackwitz-Fiessler algorithm (HLRF) is frequently used. This algorithm, or any other gradient-based algorithm, requires the gradient of the limit state function in Y space: g( y ). 5. RESPONSE SURFACE METHO The Response Surface Method (RSM) is a widely established method for the solution of complex problems, which finds applications in many fields of computational mechanics. The RSM allows replacement of a complex (numerical) model by an approximate, surrogate polynomial model. In application to crack propagation reliability problems, the RSM is used to construct polynomial approximations to the limit state function g(x)=0. The polynomial variables, in this case, are the random variables of the reliability problem, or vector X. Each point of the experimental design represents one realization of the problems random variables (vector X). Any shape of polynomial surface can, in principle, be considered. In this paper, a complete quadratic polynomial, defined in an n-dimensional space, is used to approximate the structural response: 0 n n n 2 ii i ij i j i= 1 i= 1 j= 1 j i g( x) RS( x) = a + a x + a x x (10) where a are the coefficients to be determined. The set of points where the numerical model is evaluated is called the experiment design. Based on the response at these points, a polynomial surface is adjusted by using, for example, least square regression. It is important to note that evaluation of the response at each point of the experimental design requires a complete fatigue crack growth analysis, and this represents a significant computational cost. Hence, experiments should be optimally designed in order to reduce the numerical effort. An experience plan is a pre-defined systematic way of selecting the points where the numerical solution is to be computed. ifferent experience plans are available in the literature. In this paper, 6 plans are considered: star, hipercube, minimum plan, composite plan, 13 points and 8 points. The last two experience plans are not cited in literature, and have been constructed by the authors. The approximate polynomial surface represents a local approximation of the true structural response, valid in a vicinity of the experience plan center. This represents an additional degree of difficulty in the application to reliability problems, since the response surface approximation has to be constructed iteratively, following the search for the design point. Since the design point is the point of maximum likelihood among all points in the failure domain, it turns out that any approximation to the failure probability (of first or second order) is only good when performed around the design point. Three distinct iterative schemes are used in this paper to construct local approximations of the limit state function. The first scheme simply follows the design point approximation found in the previous iteration, and centers the new

4 response surface in this point. In other words, a response surface is constructed around an initial point, following one of the experiment plans. The next, candidate design point is found using the HLRF algorithm, and another response surface is constructed around this point. This is done iteratively until convergence. The second scheme is similar to the first, but the distance between the points of the experimental plan is reduced as the algorithm converges to the design point. When the distance between the design points of two consecutive iterations is smaller than a specified tolerance, the distance between experience plan points is reduced. Hence, the first experimental design provides broader information about the problem, and successive approximations provide greater accuracy of the response in the vicinity of the design point. The third scheme is similar to the second, but the distance between the points of the experimental plan is reduced progressively, for each step in search of the design point. The algorithm converges when the distance between successive design point approximations is smaller than a specified tolerance, and the distance between experience plan points is smaller than another specified tolerance. This third (progressive) iterative scheme resulted in good convergence, as will be seen in the sequence. 6. NUMERICAL GRAIENTS TECHNIQUE One alternative that completely avoids the iterative construction of response surfaces is called herein the numerical gradients technique (NG). In this scheme, the gradients of the limit state function are evaluated numerically, via finite differences, and directly from the numerical boundary element response. The gradients are evaluated in the original design space (X) and transformed to the standard Gaussian space (Y). The gradients are used to compute the next candidate design point, following the HLRF algorithm. In this point, the new gradient is computed numerically, and so on, until convergence. The numerical gradients technique leads to great reduction on the number of limit state function calls, as will be shown in the examples to follow. 7. NUMERICAL EXAMPLES 7.1 Plane structure with three holes This example involves reliability analysis of a plane structure containing three holes and a notch at its inferior face, as shown in Figure 1. eterministic problem parameters are presented in Table 1 and random variable parameters are presented in Table 2. ue to the complex geometry, different crack paths are obtained for different realizations of the problems random variables. Four different experience plans where considered in the iterative response surface solution: Minimum, Composite, 13 Points and 8 Points. Two distinct iterative schemes where considered: the adaptive and the progressive schemes discussed in Section 5. F f ao i Figure 1: Geometry of the plane structure of example 7.1, dimensions in cm. Table 1: eterministic variables of example 7.1. Variable Symbol Value Unit Modulus of elasticity E 2.68 GPa Poisson s coefficient υ Critical stress intensity factor K IC 2.56 GPa mm 1/2 Hole diameter mm Crack propagation exponent n Number of load cycles N load cycles Table 2: Random variables of example 7.1. Variable Symbol istribution μ X σ X Unit

5 Load F log-normal kn Crack propagation rate C log-normal mm/cycle istance between holes and crack f normal mm Convergence histories for random variables C f and for reliability index β are presented in Figure 2, where the progressive iterative scheme is identified by suffix Prog. Convergence results show agreement between the different solution schemes employed in this problem. It can be seen that convergence of the progressive scheme is smoother, in comparison to the adaptive scheme, where distance reduction for experiment plan points is only made after convergence. Moreover, convergence for the progressive scheme seems independent of the experimental plan used in the solution. The same is not true for the adaptive scheme. Convergence of the numerical gradients solution, in this problem, is similar to the adaptive response surface solutions. Solutions using the composite experience plan failed to converge in this problem. One possible explanation for this behavior is the poor representation of the true limit state function, obtained for this experience plan. The computational cost (or efficiency) of different solution schemes is compared in Figure 3. It can be observed that, although the rate of converge is apparently the same (as seen in Fig 2), the numerical gradients scheme is by far more efficient than the iterative response surface schemes. The numerical gradients scheme, in this problem, takes less than a quarter of the number of limit state function calls (24 calls) required by the most efficient iterative response surface scheme (minimum experience plan, progressive or adaptive distance reduction, with 110 function calls). istance between crack and holes (in) 2,016 2,014 2,012 2,010 2,008 2,006 2,004 2,002 2,000 1, Points 8 Points 13 Points Prog 8 Points Prog Numerical Gradients Minimum Minimum Prog Iterations Figure 2: Convergence histories for f (left) and for β (right). Reliability Index 1,05 1,00 0,95 0,90 0,85 0,80 13 Points 8 Points 13 Points Prog 8 Points Prog Numerical Gradients Minimum Minimum Prog Iterations Number of Mechanical Model Calls Model Adopted 13 Points 8 Points 13 Points Prog 8 Points Prog Numerical Gradients Minimum Minimum Prog Figure 3: Number of limit state (mechanical model) calls required to obtain each solution. 7.2 Plane Structure with Hole This example involves reliability analysis of a plane structure containing a single hole and a notch at its left face, as shown in Figure 4. Stress is applied at the upper face. eterministic problem parameters are presented in Table 3 and random variable parameters are presented in Table 4. In this problem, only the numerical gradients scheme was used, due to the high cost of computing solutions. The numerical gradients scheme was able to find a solution to this highly non-linear problem, involving 5 random variables. The numerical gradients solution required 13 iterations and 78 limit state (mechanical model) function calls.

6 3,00 f 1,20 P y a0 0,7 0,3 x Figure 4: Geometry of the plane structure studied in example 7.2, dimensions in m. Table 3: eterministic variables of example 7.2 Variable Symbol Value Unit Modulus of elasticity E 30 GPa Poisson s coefficient υ Critical stress intensity factor K IC 36.4 u=mpa.(mm) -3/4 load Number of load cycles N cycles Table 4: Random variables of example 7.2 Variable Symbol istribution μ X σ X Unit Load P log-normal kn/m Crack propagation rate C log-normal mm/cycle Hole diameter normal mm istance between hole center and base f normal m Initial crack size A 0 log-normal mm 8. CONCLUSIONS This paper addressed reliability analysis of structures subject to fatigue crack propagation. Problem mechanics was modeled using the boundary element method, which avoids any re-meshing due to crack propagation. The paper discussed coupling of reliability and mechanical models, through use of response surfaces and via a numerical gradients scheme. Response surface solutions considered in the paper included a number of different experience plans and different iterative schemes. The study of three example problems has shown that the numerical gradients scheme is stable, and was able to converge to the same solution provided by response surface techniques. The scheme was also shown to be very efficient, when comparing the number of limit state function calls required for convergence. The numerical gradients scheme required a much smaller number of limit state function calls than any of the response surface schemes, regardless of adopted experience plan or iterative scheme. 9. ACKNOWLEGEMENTS Sponsorship of this research project by the São Paulo State Foundation for Research - FAPESP and by the National Council for Research and evelopment - CNPq is greatly acknowledged. 10. REFERENCES Portela, A; Aliabadi, M.H; Rooke,.P. (1992). ual boundary element method: Efficient implementation for cracked problems. Int. J. Num. Meth. Engn., V.33, Paris, P.C, Erdogan, F. (1963). A critical analysis of crack propagation laws. Journal of Basic Engineering, 85, Wirsching, P,H; Torng, T,Y; Martin, W,S. (1991). Advanced fatigue reliability analysis. Int. Journal of Fatigue. 13: Zhao,Y,X. (2000). A methodology for strain-based fatigue reliability analysis. Rel. Eng. and System Safety, V.70, Sobczyk, K; Spencer, B, F. (1992). Random Fatigue: From ata to Theory. Academic Press. Cisilino, A.P; Aliabadi, M.H. (1999). Three-dimensional boundary element analysis of fatigue crack growth in linear and non-linear fracture problems, Engineering Fracture Mechanics, 63, Yang, B; Mall, S; Ravi-Chandar, K. (2001). A cohesive zone model for fatigue crack growth in quasibrittle materials, International Journal of Solids and Structures, 38, Khatir, Z; Lefebvre, S. (2004). Boundary element analysis of thermal fatigue effects on high power IGBT modules. 44, Jivkov, A.P. (2004). Fatigue corrosion crack extension across the interface of an elastic bi-material, 71, RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.

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