Structural reliability analysis with implicit limit state functions

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1 J. Miranda Structural reliability analysis with implicit limit state functions 1 Structural reliability analysis with implicit limit state functions Jorge Miranda Instituto Superior Técnico, University of Lisbon, Portugal ABSTRACT: This study aims at developing reliability applications and methods in MATLAB language for safety assessment of structural elements with implicit and explicit limit state functions. Reliability analyses using FORM, crude Monte Carlo and importance sampling Monte Carlo are carried out. The assessment is based in a limit state function, composed by the element strength and the applied load. Explicit limit state functions, defined through semi-empirical design equations are addressed. Implicit limit state functions are also used, requiring a connection to the finite element method aiming the numerical element strength calculation. The results obtained by the semi-empirical equations and by the finite element method are compared and, on the basis of this comparison, a new semi-empirical equation is proposed, derived from one of the studied models and then the associated error are assessed. The response surface method is also adopted in order to convert implicit limit state functions in explicit ones. Several response surfaces with polynomial representation are obtained, which predict the strength of the element. Then the quality assessment of these predictions are assessed with the measure of the associated error. A MATLAB code is developed, which allows to obtain response surfaces, based in random samples. The response surfaces can be then implemented as explicit limit state functions. Reliability analysis are carried out, using explicit or implicit limit state functions or even response surfaces. Well detailed case studies are developed wherein a reliability commercial program is used and it is also tested a non-commercial program (developed within an investigation project), applying the studied models. Finally, a reliability program in MATLAB language is also developed and tested with the case studies developed. 1 INTRODUCTION The probabilistic safety assessment of a structural element is carried out taing into account two components: the applied load and the strength of the element. This element can be considered appropriated (do not occur a fail) only if the structural strength is greater than the applied load [1]. The structural strength of the element can be obtained by many ways, being the most simple and direct to use an explicit design equation, which characterize the structural behavior, relating the many variables, defined by probabilistic distributions. In case of an explicit function capable of defining the behavior of the structure is not available, it is possible to use the finite element method to compute the behavior of the structure. This process is of high computational complexity, and time consuming. In the first case the limit state function used to assess the structural safety is defined as an explicit form instead the second case where limit state function is implicit. The safety assessment procedure with implicit functions can be simplified, by obtaining an alternative explicit limit state function which allows a significant decrease of the response calculation time. This function can be obtained through the Response Surface Method by fitting a surface to various realizations, for a set of variables, based on a Monte Carlo simulation (using an implicit limit state function provided by the finite element method). In this wor the response is assessed by three different approaches: a semi-empirical design equation (explicit limit state function), a finite element model (implicit limit state function) and a response surface model, obtained through a Monte Carlo simulation, using several realizations of variables and responses generated by finite element method (explicit limit state function). Alternative tools will be used: a commercial program (Comrel) and also the reliability non-commercial program, started by Teixeira [] and further developed in an investigation project, named ProRel. Another program, with MATLAB language will be also developed. The results will be assessed as well as the efficiency of the response surface instead of implicit limit state function. FIRST ORDER RELIABILITY METHODS (FORM) Freudenthal et al. [1] has defined the basic structural reliability problem involving the load effect (L) and the structural strength (R).

2 J. Miranda Structural reliability analysis with implicit limit state functions The failure occurs when the strength (R) is less than the applied load (L) [3]. The fail probability p f is: p f = P(R L) = P(R L 0) = P ( R L 1) p f = P[g(R, L) 0] where g is named limit state function. Cornell [4] has developed the initial FORM approaches for the case where the limit state function is linear and the random variables follow a normal distribution: he has defined a reliability index (β C ) given by the quotient between the mean and standard deviation values of g (μ g and σ g, respectively). However, there is a problem in this approach when the limit state function is replaced by an equivalent one (lac of invariance) [5]. As such, Hasofer and Lind [6] have proposed a new approach, when the limit state function is non-linear, using the transformation U i = (X i μ Xi )/σ Xi, which transforms basic independent variables (with normal distribution probability X i ) into a set (U) of random standard normal variables (U i ) with mean zero and unitary variance. Hasofer and Lind have shown that a linearization about project point u* should be performed. The Hasofer-Lind reliability index is defined as the minimum distance between the origin and the limit state surface. (1) the normal tail transformation proposed by Ditlevsen [7] is required, which can be defined as: p = F X (x) = Φ(u) u = Φ 1 [F X (x)] where p is a probability content associated with X = x and hence U = u ; F X is the marginal cumulative distribution of X and Φ( ) is the cumulative distribution function for the standardized normal random variable Y. 3 MONTE CARLO SIMULATION Basically, Monte Carlo simulation method consists of generating a set of sample values x i of each basic variable X i of the problem, according with their probability distribution function F Xi (x i ) and compute the limit state function to this set, assessing if g(x ) 0. If this process is repeated N times, the failure probability is given by: N p f 1 N I[g(x j) 0] j=1 where I(g(x )) is a binary function which taes the value 1 if g(x ) 0 and 0 for g(x ) > 0 as shown in Figure (a). (3) (4) (a) Figure 1. Standard normal space of basic random variables and Hasofer-Lind reliability index for reliability problem. The failure probability is then given by the expression: P f Φ( β HL ), com β HL = u* () Where u* is obtained by the following optimization procedure: minimize u { subjected to g(u) = 0 (b) Figure. Monte Carlo simulation method: (a) crude and (b) importance sampling approaches When the limit state function is non-linear and the random variables follow a non-normal distribution,

3 J. Miranda Structural reliability analysis with implicit limit state functions 3 The process above is the crude form of this method which is not efficient in a structural reliability analysis since the failure probability of structures are typically low, implying an high number of simulations. In this context arises another approach of this method named importance sampling which is based in the integral N p f 1 N {I[g(v j) 0] f X(v j) h v (v j) } j=1 where v j is a vector of sample values obtained from the importance sampling function h v ( ), whose optimum case the function can be written: with h v (v) = I[g(v) 0] f X(x) J J = E {I[g(v) 0] f X(v) h v (v) } = E (If h ) The importance sampling Monte Carlo simulation is represented in Figure (b). 4 RESPONSE SURFACE METHOD In some structural reliability problems, the limit state function g(x) is unable to define the limit state of the problem as an explicit function of basic variables X due to the high complexity of the problems. In order to save computational costs, the response surface method (RSM) arises, transforming the implicit limit state function into an explicit one. Monte Carlo simulation or reliability methods (first or second order) can be then used to predict the failure probability, associated to the new limit state function [], [8], [9]. The original limit state function is approximated to a polynomial one η(x), derived by regression analysis from a set of results of deterministic analyses with selected values from the random analysis. The basic formulation of RSM is presented in [10], [11], in which is mentioned that the general concept of regression, is to minimize the error in sense of an expected value. Assume that the output quantity z is related to the n input variables x 1,, x n that are assembled into a vector x by a functional relation f( ) with the formal representation: (5) (6) z = f(p, x) (7) where function f depends on the parameter vector p = [p 1, p,, p v ] T whose values have yet to be determined. That parameters are determined by a least square estimate. The parameters p can be obtained through the equation system: where m Q p = q (8) Q ij = g i (x () ) g j (x () ) ; m =1 q j = z () g j (x () ); i, j = 1,, v =1 The quality of the approximation of the response surface can be assessed by the coefficient of determination: R = ρ E[Y Z] YZ = ( ) σ Y σ Z v ; y = p i g i (x) (9) (10) When the number of data sample is (almost) equal to the number of regression parameters, R values tend to be over-optimistic. Hence, it is generally accepted to adjust R values such as to tae into account the limited sample size [1]: R adj = R v 1 m v (1 R ) (11) where m is the sample size and v is the number of unnown parameters in the regression model. The adjusting correction term vanishes as the sample m increases. In stochastic analysis problems (for example for the reliability assessment) the first or second polynomials RSM models are the most used [8], [10], [13] [15]. Polynomial equations with linear terms, linear and interaction terms and also with linear, interaction and quadratic terms are frequently used to define a limit state function:. n η = θ 0 + p i x i + p ij x i n 1 n j=i+1 x j n + p ii x i linear interaction quadratic (1) 5 ULTIMATE STRENGTH OF STRUCTURAL ELEMENTS 5.1 Semi-empirical design equations It is mentioned in [16] that several semi-empirical design equations are available to predict the ultimate strength of the plate elements [17]. It is common to normalize the expressions build upon the elastic bucling strength σ c by the yield stress, σ y.

4 J. Miranda Structural reliability analysis with implicit limit state functions 4 φ = σ c σ y = 3.6 λ (13) with λ as the plate slenderness, given by λ = b t σ y E (14) where b is the width, t is the thicness and E is the Young modulus of the material. Faulner [18] had proposed an expression which is applicable to simple supported plates, with initial distortions but free of residual stresses: φ F = σ u = σ y λ 1 λ for λ > 1.0 (15) characteristics of the applied stress are then represented by a random variable C a, following a Weibull distribution with mean value 0.6 and coefficient of variation (cov) of 0.1 that affects directly the 5% of the characteristic value of the yield stress. Table 1 presents a summary of the stochastic models of the random variables. Combining (14) with expression (15) it is obtained the first semi-empirical design project to be analyzed. The method proposed by Faulner was extended by Guedes Soares [19], obtaining an equation that tae into account explicitly initial distortions and residual stresses. Aiming to apply that equation to simplified limit state functions, it was neglected the residual stresses effects, obtaining the simplified version of that equation: φ GS = ( a 1 λ a λ ) (1 (a 3 a 4 λ) w t ) σ y (16) with parameters: a 1 =.16, a =-1.08, a 3 =0.66 and a 4 = The equation (16), with constant values above, will be used as semi-empirical design equation. It will be also modified in its parameters to fitting the plate model studied with better accuracy. 5. Plate Model A plate model developed by Teixeira & Guedes Soares [0] was used in this study. This model considers simply supported square plates (a/b=1) and slenderness b/t=50 with longitudinal edges against transverse displacement. The ultimate strength of the plate corresponds to the maximum value of the longitudinal stress-displacement curve of the plate under in plane longitudinal compression obtained by nonlinear finite element analysis. The applied load is uniformly distributed (δ), applied along the transverse edge of the plates with width b=1000mm, as illustrated in Figure 3. The random variables considered in this model was the thicness (t), the Young s modulus (E), the yield stress (σ y ) and the amplitude of the initial distortions of the plate (w max ). The applied load (σ a ) is taen, in average, the value of 60% of the characteristic value of the yield stress (σ y c ) used in the design of the structural element. The probabilistic Figure 3. Boundary conditions and loading of the plate model Table 1. Stochastic models of random variables (X) Variable Unity Characteristic Dist. St. Mean cov type Dev. value (X c ) F x (X c ) t mm Normal E MPa Lognorm σ y MPa Lognorm w max mm Lognorm C a - Weibull The limit state function of this problem is given by: g(x) = R(X) σ a = R(t, E, σ y, w max ) C a σ y c (17) Within this study, it was necessary to define a stochastic model concerning to the amplitudes of the initial distortions of the plate (w max ). This model was based in a sample of measurements of fabrication distortions of ship plating [1]. The initial geometrical imperfections shape (w 0 ) has represented by only one component of the Fourier series, w 0 = w max sin ( πx ) sin (πy a b ) (18) where a and b are the plate dimensions and w max is the amplitude of the initial imperfection shape. A sample of the amplitude of initial distortions normalized by the thicness of the plate (w max /t) has been used to build a stochastic model used in the reliability study, assuming that this parameter can be defined as following a lognormal distribution. Table shows the parameters of the obtained distribution for a set of square plates with relation b/t=50. Table. Stochastic model of the amplitude of initial distortions of the plates (w max /t) with t=0mm and a/b=1, b/t=50 Distribution Mean St. dev. cov Sew Lognormal

5 J. Miranda Structural reliability analysis with implicit limit state functions 5 The plate model is then defined by the random variables and the limit state function (17). 5.3 Semi-empirical equations for strength plate prediction The semi-empirical equations above can be applied to the studied plate model and the results for ultimate strength of the element obtained through those equations can be compared with the nonlinear finite element analysis. A Monte Carlo simulation with 500 realizations of the stochastic models of Table 1 for an implicit limit state function (FEM solved by Ansys) was performed in order to assess the quality of semi-empirical models. Figure 4 and Figure 5 present that assessment of quality. The Guedes Soares equation has the best fitting to the FEA model. However, it is possible to manipulate the equation (16) in its parameters a in order to improve this fit. Comparing the ultimate strength values obtained by nonlinear FEA and by the semi-empirical equation proposed by Guedes Soares and solving equation (16) for the Ansys values (at the same 500 Monte Carlo realizations used before) and see to minimize the error, new parameter values were obtained for the equation (16): a 1 =.46, a = , a 3 =0.64 and a 4 =-0.1. The new equation obtained was named as Guedes Soares corrected equation and represented by φ GS. Figure 6 shows the quality of that equation. It is possible to observe the values of ultimate strength with greater accuracy of the numerical values. The relative error inherent to each equation was also studied through another sample, again with 500 realizations, in order to obtain different values of the parameters determination, avoiding a false reduction of the error of Guedes Soares corrected expression and allowing the error assessment with the same sample to the three equations. The relative mean error of each equation was obtained by the error of each of the 500 realizations. Then, the mean of all obtained values was calculated, as shown in equation (19). 500 E Mean = ( φ i ANSYS i ) 500 (19) ANSYS i σ u (Faulner) Figure 4. Ultimate strength of plates obtained by nonlinear FEA and semi-empirical Faulner expression (Eq. (15)) σ u (GS) Figure 5. Ultimate strength of plates obtained by nonlinear FEA and semi-empirical Guedes Soares expression (Eq. (16) with: a 1 =.16, a =-1.08, a 3 =0.66 and a 4 =-0.11) σ u (GS') Figure 6. Ultimate strength of plates obtained by nonlinear FEA and semi-empirical Guedes Soares corrected expression (Eq. (16) with: a 1 =.46, a = , a 3 =0.64 and a 4 =-0.1) Table 3 shows the mean errors of the three equations showing that the obtained values are according to the expected, i.e. lower mean error of φ GS. Table 3. Relative errors of the ultimate strength for each semiempirical equations φ F φ GS φ GS Mean Error 6.81% 3.4% 1.34%

6 J. Miranda Structural reliability analysis with implicit limit state functions 6 Elasticity 6 STRUCTURAL RELIABILITY ANALYSIS 6.1 Reliability analysis with explicit limit state functions A FORM reliability analysis with explicit limit state function is carried out with the two semi-empirical design equations above-mentioned: Guedes Soares equation and Guedes Soares corrected equation. The limit state function becomes, g(x) = φ L (0) with φ corresponding to one of both equations: GS or GS and L is the load applied. Both equations were tested with the stochastic variables presented in Table 1 using programs Comrel and ProRel aiming to establish a comparison between both programs. Table 4 and Table 5 presents the results of the reliability analysis carried out. The obtained results of both programs are very close, with residual errors only. Concerning to each equation, it is possible to see some differences not only on the design point U but also in α-values. Figure 7 shows graphically the differences of sensitivity factors (α) of each random variable in both limit state functions: g(x) for φ GS strength and g (X) for φ GS strength. Table 4. Reliability analysis with Guedes Soares equation Comrel ProRel β P f 1.78E E-04 Random Variables U α U α t E σ y w C a Table 5. Reliability analysis with Guedes Soares corrected equation Comrel ProRel β P f 7.69E E-05 Random Variables U α U α t E σ y w C a w σ (a) E t (b) Figure 7. α-values of random variables of both functions: (a) g(x) and (b) g (X) Table 6 shows the elasticities of mean and standard deviation for both equations and to all random variables. There are some differences between the elasticities in both equations which corresponds to importances of each variable (less impact on β when occurs an increase of 1% in the variable). That matter is also shown graphically in Figure 8. Table 6. Mean and St. dev. elasticities of random variables for both equations Mean St. dev. Elasticities g(x) g (X) g(x) g (X) t E σ y w C a Ca Figure 8. Elasticities of the mean value of the random variables Figure 9 shows a study on the variation of the reliability index, β, with different percentages of load. The β-value is the same around 90% of load, from which the separation becomes steeper with the increase of load, with limit state function g eeping the high value. w Ca t E σ w Ca σ t GS GS' E

7 J. Miranda Structural reliability analysis with implicit limit state functions 7 β % load Figure 9. Variation of β according to the applied load g g' method (RSM) can be developed aiming to build the expressions which predict the strength component. It is necessary to have a sample of variables and responses as a basis. Thus, the first step of the procedure is to obtain two random samples of variables, through a Monte Carlo simulation and obtain the response by FEM (implicit limit state function). The load associated variable is not used. As such, only the first four random variables of Table 1 are considered. A sample of 100 realizations and another with 500 realizations were generated aiming to assess the effect of the sample size on the prediction of the different response surfaces. 6. FORM analysis with implicit limit state function (with Ansys connection) The case study presented before is modified so that its limit state function is now implicit, i.e. ProRel connects to Ansys to perform a FEA in each limit state function evaluation. The implicit limit state function can be written as: g i (X) = Response t 35 C a (1) where the term Response is the strength evaluated by FEA. Since the response from Ansys is a force it is necessary to divide this term by the transversal area of the plate (b t, with b=1000mm) and the load is defined as in the limit state function of equation (17). Considering the stochastic models of the Table 1 applying to the limit state function (1) in a reliability analysis, the results for g i (X) in Table 7 are obtained. The table shows also a comparison of the reliability with the explicit limit state function g (X) with special attention to the significant difference between reliability indexes and to the α-values (the α-values of each random variable in implicit limit state function has some mared differences). Table 7. Comparison between reliability analysis with implicit and explicit limit state functions g i (X) g (X) β P f 3.84E E-05 Random Variables U α X U α X t E σ y w C a Response Surface expressions The four first variables of Table 1 are, in this case named as X 1,, X 4, together with the respective response of each sample. Three types of response surfaces are considered: with linear terms only (Eq. ()), with linear and interaction terms (Eq. (3)) and composed by linear, interaction and quadratic terms (Eq. (4)). η 1 = θ 0 + p i X i η = θ 0 + p i X i + p ij X i X j 1 j=i+1 η 3 = θ 0 + p i X i + p ij X i X j 1 j=i+1 + p ii X i () (3) (4) The quality of the fitting can be measured through the calculation of statistical parameters concerned to multiple linear regression: the coefficient of determination and the adjusted coefficient of determination. For a good regression model, this coefficients should be high. A MATLAB program, which implements the least square method, was developed to build this equations. The number of realizations used is designated by Nsample. Hence, six surface responses expressions are build: η 1, η and η 3 for Nsample=100 and Nsample=500. The quality of the developed expressions are illustrated in Figure 10 and assessed with the coefficients of determination (Table 8). 6.3 Reliability analysis using response surface Reliability analysis can also be performed with response surface. For this purpose, response surface

8 J. Miranda Structural reliability analysis with implicit limit state functions σ u (RSM η=1) σ u (RSM η=1) σ u (RSM η=) σ u (RSM η=) σ u (RSM η=3) σ u (RSM η=3) (a) Figure 10. Ultimate strength obtained by nonlinear FEM and by the RSM expressions for: (a) Nsample=100 and (b) Nsample=500 (b) Table 8. Coefficients of determination of RSM expressions η Nsample=100 Nsample=100 i R R adj The coefficients of determination values are coherent with the assessment of the predicted values shown on Figure 10: grater proximity of points to the line = σ u (RSM) corresponds to coefficients of determination closer to 1. Another assessment can be based on the relative mean error, obtained with the same method used when calculated the semi-empirical equations (Eq. (19)). Table 9. Mean relative errors associated to the RSM equations Mean Error Nsample η 1 η η % 1.3% 0.78% % 1.00% 0.76%

9 J. Miranda Structural reliability analysis with implicit limit state functions 9 The best RSM expression to define the strength of the element is composed by linear, interaction and quadratic terms and built based in a sample of 500 realizations (Nsample=500) FORM analysis using response surfaces The response surfaces obtained above were used to define an explicit limit state function in a FORM analysis, allowing to write: g(x) = η i 1000 t 35 C a (5) where η i represents the RSM expressions developed, for i = 1,,3 and considering Nsample=100,500, with the stochastic models contained in Table 1. The FORM analysis performed is presented in Table 10 which shows large fluctuations namely in the reliability index with the increase of the complexity of the equation. Moreover, in spite of Nsample have some influence in obtained values, it do not present significant oscillations for the same degree equation. index, with residual errors only, whereby it is realized that the program is a good tool for reliability analysis. Table 11. β-values comparison between the three limit state functions and three types of simulation GS GS' η 3 Nsample=500 β ProRel MATLAB Relative Error FORM % MC % MC % FORM % MC % MC % FORM % MC % MC % Concerning to the Monte Carlo simulations it is verified that importance sampling needs much less realizations to achieve acceptable results when compared with crude Monte Carlo simulation. Table 10. FORM analysis with response surface RSM, η i Nsample = 100 Nsample = 500 i β P f β P f E E E E E E-03 As expected, also the σ-values suffer many variations between all the variables and for the six equations. 7 RELIABILITY CODE DEVELOPED IN MATLAB A reliability code implemented in MATLAB language is developed aiming to carry out the reliability analyses of the above case studies. This new tool has capacity to perform FORM and Monte Carlo simulations (crude and importance sampling) analyses. The limit state functions are explicit and can include also response surfaces, obtained before. After implemented the three algorithms are used to performed FORM analysis, crude Monte Carlo with realizations (MC1) and importance sampling Monte Carlo with realizations (MC) and applied to the explicit limit state functions g, g and η 3 with Nsample=500. A comparison with Pro- Rel for β results are presented in Table 11. It is perceptible a really close estimation of the reliability 8 CONCLUSION This wor has assessed the structural safety of elements using different structural reliability analysis methods. Explicit, implicit and response surface limit state functions were considered. Semi-empirical design equations were used and one of them was modified to fit with more accuracy the FEM results and then used to assess the differences between them in the reliability analysis. It was implemented a response surface method to obtain new explicit limit state functions, to replace the implicit ones. It was concluded that the best approximation is achieved with the expression with more terms and built with a larger sample. The reliability analysis performed with this equation revealed a relative error close to the semi-empirical ones. However, the computational cost of this method is much less when compared with the use of implicit limit state function. Finally, a reliability code was developed in MATLAB, for three reliability analysis methods and to three explicit limit state functions and the achieved results were similar to the results already obtained with ProRel. REFERENCES [1] Freudenthal, A. M., Garrelts, J. M., & Shinozua, M., The Analysis of Structural Safety, J. Struct. Div, Am. Soc. Civ. Eng., vol. Vol. 9, pp , 1966.

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