Estimating Joint Failure Probability of Series Structural Systems

Transcription

1 Estimating Joint Failure Probability of Series Structural Systems Yan-Gang Zhao, M.ASCE 1 ; Wei-Qiu Zhong ; and Alfredo H.-S. Ang, Hon.M.ASCE 3 Abstract: The failure probability of a series structural system theoretically involves multidimensional integration and is usually difficult to calculate. The search for efficient computational procedures for estimating system reliability has resulted in several approaches, including bounding techniques and efficient Monte Carlo simulations. For the narrow bound method, the joint failure probability of every pair of failure modes needs to be calculated. In the present paper, in order to improve the accuracy of the narrow bound estimation method, a computationally effective point estimation method for calculating the joint probability of every pair of failure modes is proposed and examined for series systems. Based on the computational results of several illustrative examples, it can be seen that the results by the present method are in good agreement with those obtained through integration. DOI: /ASCE :5588 CE Database subject headings: Structural reliability; System reliability; Joints; Failures; Integrated systems. Introduction The evaluation of system reliability for structures has been an active area of research for over three decades. According to the logical relationship of the failure modes of structures, structural systems can be divided into three types: series systems, parallel systems, and hybrid structural systems. Of interest here is the reliability assessment of series systems, which is encountered most frequently in practical design and analysis. The failure probability of a series structural system theoretically involves multidimensional integration, which is usually difficult to evaluate, especially for structures of practical significance. The search for efficient computational procedures for estimating system reliability has resulted in several approaches, including bounding techniques and efficient Monte Carlo simulations MCS. The bounding methods include the wide bound estimation method and the narrow bound estimation method. For the narrow bound method, the joint failure probability of every pair of failure modes needs to be calculated. In this study, in order to improve the accuracy of the narrow bound estimation method, a point estimation of the joint failure probability of series structural systems is proposed and examined. 1 Associate Professor, Dept. of Architecture, Nagoya Institute of Technology, Gokiso-cho, Shyowa-ku, Nagoya , Japan. Assistant Professor, Venture Business Laboratory, Nagoya Institute of Technology, Gokiso-cho, Shyowa-ku, Nagoya , Japan. 3 Research Professor, Dept. of Civil and Environmental Engineering, Univ. of California, Irvine, CA Note. Associate Editor: Arvid Naess. Discussion open until October 1, 007. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 1, 006; approved on October 9, 006. This paper is part of the Journal of Engineering Mechanics, Vol. 133, No. 5, May 1, 007. ASCE, ISSN /007/ /$5.00. Review of System Reliability of Series Systems Consider a series structural system with k possible failure modes, and let the performance function for failure mode i be given by g i X = g i x 1,x,,x n ; i =1,,,k 1 where x 1,x,...,x n are the basic random variables and g i is the performance function. Define the failure event for failure mode i as E i = g i X 0 Since the occurrence of any failure event E i will cause the failure of the structure, the failure event E of the structure is the union of all the possible failure modes, which can be expressed as E = E 1 E E k In structural reliability theory, the failure probability P f of a series structural system corresponding to the occurrence of the event E of Eq. 3 theoretically involves the following integration: P f = E 1 E E k f x1,x,,x n x 1,x,,x n dx 1 dx dx n where f is the pertinent joint probability density function. The evaluation of the above multidimensional integration is often difficult, especially for structures of practical dimensions. For this reason, approximate methods have been proposed and developed. These include the following wide bound technique for the failure probability of series structural systems e.g., Cornell 1967: max 1ik P fi P f 1 1 P fi where P fi is the failure probability of the ith failure mode. Since only the failure probability of a single failure mode is considered and the correlation of the failure modes is neglected, k i= / JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007

2 the above wide bound estimation method is simple to evaluate; however, the bounds can be very wide, especially for a complex system. A narrow bound estimation method for the failure probability of series systems is also available Ditlevsen 1979 k P f1 + i= i 1 maxp fi j=1 k P fij,0 P f i=1 k P fi i= maxp fij ji 6 where P fij is the joint probability of the simultaneous occurrences of the ith and jth failure modes. The left- and right-hand sides of Eq. 6 are, respectively, the lower bound and upper bound of the failure probability of a series structural system with k potential failure modes. Observe that because the joint probability of simultaneous failures of every pair of failure modes must be evaluated, the resulting bounds of Eq. 6 are narrower than those of Eq. 5. As is well known, P fij can be expressed by Ang and Tang 1984 i P fij = i, j, ij = j x i,x j, ijdx i dx j where 1 x i,x j, ij = exp 1 1 ij xi + x j ij x i x j 1 ij 8 The reliability indices i and j correspond to the ith and jth failure modes, respectively; ij is the correlation coefficient between the ith and jth failure modes; and and are the probability density function and cumulative distribution function, respectively, of D standard normal distribution. 7 Eq. 7 is an accurate expression for P fij. To obtain the results, however, numerical integrations would be needed. To avoid such numerical integrations, further approximations are often adopted involving further bounds for P fij. Specific formulas for evaluating the lower and upper bounds of the joint failure probability P fij were proposed by Ditlevsen 1979 as follows: where maxpa,pb P fij PA +PB ij P fij minpa,pb ij 0 PA = i j ij i 1 ij PB = j i ij j 1 ij 10 Since Eq. 9 is a bound rather than a specific value, it is not convenient to use in Eq. 6. Feng 1989 gave a point estimate for the joint failure probability P fij as P fij = PA +PB1 arccos ij / 11 where the definitions of PA and PB are the same as those in Eq. 9 and can also be calculated by Eq. 10. Since Eq. 11 is a specific value rather than a bound, it is convenient and considered to have high accuracy to be used in Eq. 6 for obtaining the narrow bounds of the system reliability Wu and Burnside 1990; Song 199; Penmesta and Grandhi 00; Adduri et al As described by Feng 1989, when the correlation coefficient ij =0 or 1, Eq. 11 gives accurate solutions, whereas when 0 ij 1, the calculational accuracy is reasonably high, especially when ij 0.6. However, as will be shown later, the lower bound Fig. 1. Probability calculation of P f1 in standard normal space JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007 / 589

3 obtained with Eq. 11 can sometimes be lower than the lower bound given by Eq. 9. The present paper proposes an alternate method for estimating the joint failure probability, P fij. Proposed Point Estimate of Joint Failure Probability To express the formulas more conveniently, 1 and are used to represent i and j, respectively, and is used to represent ij. Without loss of generality, assume 0 1. Let Z 1 and Z be the limit state functions in standard normal space corresponding to 1 and ; then, the geometrical relationship between Z 1 =0 and Z =0 can be depicted in Fig. 1a when 1 / and in Fig. 1b when 1 /. Let the angle between OB 1 and OC be v; then = arccos 1 In Fig. 1, the crossing point of Z 1 =0 and Z =0 is point A. Define the length of the line segment OA as crossing index 0, and denote the angle between OA and OB as v 1 and the angle between OA and OC as v ; then v 1 and v can be expressed as 1 = arccos 1 / 0 13a = arccos / 0 13b With the aid of the geometrical relationships of 0, 1, and, 0 can be given as see Appendix I 0 = Let V 1 denote the area of the failure zone between ray OA and Z 1 =0, and V denote the area of the failure zone between OA and Z =0, shown as the respective shaded zones in Fig. 1. The angle OAB is equal to / v 1 and the angle OAC is equal to / v. Since the joint failure probability P f1 is the area of the failure zone between Z 1 =0 and Z =0, P f1 can be given as the following equation according to the geometrical relations in Fig. 1: P f1 = V 1 + V 1 / 15 P f + V 1 V 1 / In particular, when 1 / =, 0 =. Then one can see that V = P f / from both Fig. 1a and Fig. 1b, which means that both the formulas in Eq. 15 give the same results for 1 / =. When v m /4 where m=1,, V m of Eq. 15 can be obtained by constructing two perpendicular lines, DD and EE, through point A; the crossing point of Z 1 =0 and Z =0. Let the angle DAO= EAO=/4, as shown in Fig.. Obviously, if EE and DD are considered to be limit state lines, both of their corresponding reliability indices would be 0 /. Since EE and DD are perpendicular, the probability associated with the area enclosed by EAD gray area in Fig. can be obtained as 0 /. Since the angle between OA and Z m =0 the shaded zone in Fig. that corresponds to V m is /-v m, we have Hence V m can be given as / 0 / / m V m 16 V m 0 / m 1 ; m /4 where m =1, 17 When v m /4 where m=1,, with the horizontal line FF through point A, the reliability index corresponding to limit state line FF is equal to 0 m and the probability corresponding to the angle between Z m =0 and FF, shown as the gray area in Fig. 3a, can be given as m 0 m, because the lines Z m =0 and FF are perpendicular. Denote the probability corresponding to the angle OA and FF as V the shaded zone in Fig. 3a, then V m can be given by the following equation according to the geometrical relations in Fig. 3a V m m 0 m V In order to obtain V, draw two perpendicular lines DD and EE through point A. Let the angle DAO= EAO=/4, as shown in Fig. 3b. Obviously, both of the reliability indices corresponding to the limit state lines EE and DD would be 0 /, and the probability corresponding to the area defined by the angle EAD gray area in Fig. 3b can be obtained as 0 / because EE and DD are perpendicular. Since the angle between OA and FF the shaded zone in Fig. 3b that corresponds to V is equal to v m, one obtains and then Therefore Fig.. Geometrical relations for v m /4 / m 0 / V V 0 / m 590 / JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007

4 Fig. 3. Geometrical relations for v m /4 Write V m m 0 m 0 / m ; m /4, where m =1, 18 P m = m 0 m P 0 = 0 / Then Eqs. 17 and 18 can be written as V m =P 01 P m P 0 m m m 4 4 m where m =1, 19 0 Eqs. 15 and 0 are the proposed formulas for estimating the joint failure probability P f1. When v m =/4, according to Eq. 13, 0 = m, then P m = P 0 = m. In this case, the two formulas in Eq. 0 give the same results. The formulas also can be written as follows: For 1 / And for 1 = / P f1 where + P 01 1 P f1 =P 4 4 P 1 + P P 0 1 P f P , P f P + P , 4 P f + P 1 P P 0 P 1 = P = 0 3a 3b in which P 1 and P can also be expressed as follows see Appendix II: P 1 = a P = 1 1 4b Eqs. 4 are almost the same as PA and PB in Eq. 10 if one uses 1,, and to represent i, j, and ij, respectively, in Ditlevsen s formula. In particular, when =0, it can be seen that v=arccos0 =/ and 0 = 1 + ; then P 1 = P = 1 Since v 1 +v =v=/, and v 1 v, one can see that v 1 /4. Then P f1 is given by P f1 = 1 ; =0 5 Another special case is when =1. Obviously, P f1 cannot be directly given by the equations given above since 0 is not defined when =1. In this case, when 1, we have see Appendix III lim P f1 = P f 1 Examination through Specific Examples 6 In order to evaluate the advantage or superiority of the proposed method, a number of series structural systems are examined. JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007 / 591

5 Example 1 Consider first a series structural system with only two failure modes. Several cases are examined to compare the results by different methods as follows. When the reliability indices of the two failure modes are the same, the results for P f1 are presented in Figs. 4a c with 1 = =; 1 = =3; and 1 = =4, respectively, and for various correlation coefficient. Figs. 4a c show the solutions obtained by integration, the Ditlevsen s bounds, Feng s point estimation, and the point estimation proposed here. From the figures, it can be seen that 1 Ditlevsen s bounds correctly bound the integration results; the results by Feng s method are quite close to the integration results; and 3 the results given by the present method have a better agreement with the integration method than those by Feng s method. When the reliability indices for the two failure modes are different, the variations of the joint failure probability P f1 with respect to the correlation coefficient are depicted in Figs. 4d f, respectively, for 1 =.5, =3.5; 1 =, =4; and 1 =1, =5. From these figures, it can be seen that the integration results are always located between the narrow bound solutions. Both the results by Feng s method and the present method have good agreement with the integration method when =0 and =1. However, whereas the results by the present method have a good agreement with those by integration for all three cases, the results given by Feng s method tend to be lower than the lower bound solution, especially for large, and this discrepancy becomes very large for significant differences in the two reliability indices Fig. 4f. The variations of the joint failure probability P f1 with respect to the difference between the reliability indices for two failure modes are depicted in Figs. 5a and b, respectively, for 1 =, =0.5, and for 1 =, =0.8. From these figures, it can be seen that the larger the difference between the two reliability indices, the narrower the reliability bound width. Feng s method gives good results for the small difference between the two reliability indices, and the present method gives good approximation of the integration results for the whole investigation range. Example Consider next a series structural system with four failure modes, in which the first-order reliability indices for the four individual failure modes have been obtained as 1 =.5, =.5, 3 =3.0, 4 =3.5, and the correlation coefficient between every pair of failure modes is assumed to be =0.86. The joint failure probability results, P fij, calculated by different methods are listed in Table 1, and the corresponding results for the system failure probability, P f, are listed in Table. From Tables 1 and, it can be seen that the results by the present method are between the lower and upper bounds and are in good agreement with those obtained by numerical integration. Also, the bounds obtained with the present method are the narrowest among all the methods. Example 3 Consider a one-story one-bay elastoplastic frame shown in Fig. 6 after Ono et al The loads M i and member strengths S i are independent log-normal random variables with mean values of M1 = M =500 ft kip, M3 =667 ft kip, S1 =50 kip, S =100 kip and standard deviations of M1 = M =75 ft kip, M3 =100 ft kip, S1 =15 kip, S =10 kip. The performance functions that correspond to the six most likely failure modes obtained from stochastic limit analysis are listed below, with the FORM reliability index for each mode given in parentheses to show the relative dominance of the different modes: g 1 =M 1 +M 15S 1 F = 3.47 g = M 1 +3M +M 3 15S 1 10S F = g 3 =M 1 + M + M 3 15S 1 F = 3.56 g 4 = M 1 +M + M 3 15S 1 F = 3.56 g 5 = M 1 + M +M 3 15S 1 F = a 7b 7c 7d 7e g 6 = M 1 + M +4M 3 15S 1 10S F = f Using the performance functions listed in Eq. 7, the correlation matrix = is as follows: C and the joint failure probability for each pair of failure modes are given in the following matrix: P fij = from which the lower and upper bounds of the system failure probability are obtained, respectively, as and The corresponding MCS solution using a 10-million sample size is with a COV of 1.75%. One can see that the MCS result is outside the indicated bounds. This is because the FORM reliability indices used in calculating the above bounds are not accurate for each performance function. Using the 4M approach Zhao and Ang 003, the reliability indices are more accurately obtained as 3.93, 3.63, 3.69, 3.69, 3.871, 3.957, corresponding to the six respective performance functions of Eq. 7. With these latter reliability indices, the joint failure probability for each pair of failure modes is then obtained as follows: P fij = from which the bounds of the system failure probability become and Then we can observe that the MCS solution for the system failure probability of is clearly bounded by the narrow bounds. 59 / JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007

6 Fig. 4. Variations of joint failure probability P f1 with respect to correlation coefficient JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007 / 593

7 Fig. 5. Variations of joint failure probability P f1 with difference between two reliability indices Example 4 Finally, consider the simple elastoplastic beam-cable system shown in Fig. 7 after Ang and Tang The performance functions of the potential failure modes are listed below with the respective FORM reliability indices indicated in parentheses g 1 =6M L / F = 3.3 g = F 1 L +F L wl F = g 3 = M + F L wl / F = a 8b 8c g 4 =M + F 1 L wl F = d where M, F 1, F, and w are normally distributed with mean values of w = kip/ft, F1 =60 kip, F =30 kip, and M =100 ft kip and COVs of V w =0. and V F =V M =0.1. Using the performance functions listed in Eq. 8, the correlation matrix is obtained as = C and the joint failure probability of each pair of failure modes are given in the following matrix P fij = from which the bounds of the system failure probability are and The result obtained by numerical integration is One can see that the bound width is quite narrow, and both the lower and upper bounds are close to the solution obtained through numerical integration. Conclusion In order to improve the narrow bounds of the failure probability of a series structural system, a point estimation method is proposed for calculating the joint probability of every pair of failure modes of the system. Based on the computational results of the illustrative examples, the following conclusions can be observed: 1. With the proposed point estimation of the failure probability for each pair of failure modes in a series structural system, the results of the narrow bound method can be improved.. When the correlation coefficient, =0, or =1, the proposed method gives accurate solutions, whereas when 01, the present method yields results that are quite close to those obtained by numerical integration and are consistently located between the lower and upper bound solutions. 3. Sometimes accurate estimates of the reliability indices of the Table 1. Calculation of Joint Failure Probability P fij 10 4 Method Present method Feng s method Lower bound Ditlevsen s method Upper bound Numerical integration P f P f31, P f P f41, P f P f / JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007

8 Table. Calculation Results of System Failure Probability P f 10 3 Method Present method Feng s method individual failure modes is necessary for determining the correct narrow bounds of the system failure probability. This is illustrated in Example The method of Feng1989 gives good results when the reliability indices for the pair of failure modes are the same; however, when the reliability indices of the two failure modes are different, Feng s method generally gives results that are below the lower bound for a relatively large correlation coefficient. Moreover, this error increases for a larger difference in the two reliability indices. Appendix I. Crossing Index 0 Ditlevsen s method According to Fig. 1, OA= 0, OB= 1, OC=. In the triangle OBC, according to the cosine law BC = OB cos BOCOBOC + OC = 1 cos 1 + = Numerical integration Lower bound Upper bound P f =9.891 Bound width OCA = OBA = / Points O, A, B, C lie on the same circle with the center of the circle being at the midpoint of line segment OA. In the same circle, OAB= OCB. In the triangle OAB, according to the sine law 0 sin / = 1 sin OAB In the triangle OCB, according to the sine law Appendix II 1 sin OCB = BC sin OAB = OCB 0 sin / = BC sin BC = 0 sin = 0 1 cos = = = In the present paper, according to Eq = = 1 1 Then Hence Similarly 0 = 1 1 P = 1 1 P 1 = Fig. 6. One-story, one-bay frame of Example 3 Appendix III. Limit of P f1 for \1 When =1, according to Eq. 1, one has lim 1 v =lim 1 arccos=0 1. If 1 / =1, one can see that v 1 +v =v=0, and 0 = 1 =, then according to Eq. 19, one has P 1 = 1 0 = 1 1 Therefore P = 0 = 1 = P 1 Fig. 7. Beam-cable system of Example 4 P f1 = P 1 + P P 0 = = P f JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007 / 595

9 . If 1 / 1, according to Eq. 14, one can see that when 1, 0, and hence 0 1, 0, and 0 /. According to Eq. 19 lim 1 P 1 = lim =0; lim 1 P = lim = 0; and lim 1 P 0 = lim 1 0 / =0. Therefore, according to Eq., one obtains lim 1 P f1 = P f. References Adduri, R. P., Penmestsa, R., and Grandhi, R Estimation of structural system reliability using fast Fourier transforms. 10th AIAA/ ISSMO Multidisciplinary Analysis and Optimization Conf., Albany, New York. Ang, A. H-S., and Tang, W Probability concepts in engineering planning and design. II: Decision, risk, and reliability, Wiley, New York. Cornell, C. A Bounds on the reliability of structural systems. J. Struct. Div., 931, Ditlevsen, O Narrow reliability bounds for structural systems. J. Struct. Mech., 74, Feng, Y. S A method for computing structural system reliability with high accuracy. Comput. Struct., 331, 1 5. Ono, T., Idota, H., and Dozuka, A Reliability evaluation of structural systems using higher-order moment standardization technique. J. Struct. Constr. Eng., 418, in Japanese. Penmestsa, R., and Grandhi, R. 00. Structural system reliability quantification using multipoint function. AIAA J., 401, Song, B. F A numerical integration method in affine space and a method with high accuracy for computing structural system reliability. Comput. Struct., 4, Wu, Y. T., and Burnside, O. H Computational methods for probability of instability calculations. AIAA/ASME/ASCE/AS/ASC 31st Structures, Structural Dynamics and Materials Conf., A Zhao, Y. G., and Ang, A. H-S System reliability assessment by method of moments. J. Struct. Eng., 1910, / JOURNAL OF ENGINEERING MECHANICS ASCE / MAY 007