This article was downloaded by: [46.143.219.234] On: 29 March 2013, At: 11:03 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Civil Engineering and Management Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcem20 Influence of partial safety factors on design reliability of steel structures ‐ probability and fuzzy probability assessments a Zdeněk Kala a Faculty of Civil Engineering, Institute of Structural Mechanics, Brno University of Technology, Veverí Str. 95, Brno, 602 00, Czech Republic E-mail: Version of record first published: 14 Oct 2010. To cite this article: Zdeněk Kala (2007): Influence of partial safety factors on design reliability of steel structures ‐ probability and fuzzy probability assessments, Journal of Civil Engineering and Management, 13:4, 291-296 To link to this article: http://dx.doi.org/10.1080/13923730.2007.9636449 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 1. Keywords: steel. Valid normative criteria emanat. According to of a steel profile IPE 140 of grade S235 under tension is our understanding. which are of ran. Partial safety factors are basic indicators. fuzzy. The probabilistic evaluation of the design procedure tive to and which it is not sensitive to [4. The reliability of steel structures design is also dependent on the partial safety factor values. non–specificity and conflict. probabilistic and sensitivity studies performed as an illustration. Vol XIII. fuzzy set theory. 602 00 Brno. . theory. The probabilistic assessment of reliability is performed as a parametric study in the first part of the numerical analysis. however. within which the appara- assessing of the reliability of steel structures. ing from the static respectively dynamic analysis of a are more frequently utilised. proaches in the analysis of uncertainty in design is dis- For this purpose material and geometric characteris. only en. simulation.0 The aforementioned approaches.k dom character and for which information on the statistical is the characteristic value of material yield strength and characteristics from measurements on ample samples is γM – the material safety factor. 291–296 INFLUENCE OF PARTIAL SAFETY FACTORS ON DESIGN RELIABILITY OF STEEL STRUCTURES – PROBABILITY AND FUZZY PROBABILITY ASSESSMENTS Zdeněk Kala Brno University of Technology. A number of factors influencing reliability however are not of random character. classical stochastic methods other alternative approaches sign of real load bearing steel structures are based on the to the representation of uncertainty of model prediction. 95. The probability of failure is analysed in dependence on values of partial safety factors of material. E-mail: kala.jcem. ISSN 1392–3730 print / ISSN 1822–3605 online JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT http:/www. tion characteristics of structural design reliability. 5]. An is the nominal sectional area. Faculty of Civil Engineering. where Rd is design resistance calculated according to able the quantification of uncertainties.234] at 11:03 29 March 2013 geometric characteristics evaluated by samples obtained from frequently manufactured products. which are specified by the design standards. No 4. The random variability of structural dimensions and of material characteristics does not present the only source of uncertainty present during design according to standards.lt 2007. imperfection. permanent loading and long time variable loading.vutbr. Veveří Str. random. The implementation of alternative ap- of the real properties of load bearing structural members. reliability. possibilities and the Dempster–Shafer theory) [6–14]. Partial safety factors are considered as fuzzy numbers with triangular membership functions in the second part of the numerical analysis. Partial The basic theory with elaborately worked out rules is the safety factor values cannot be verified without knowledge fuzzy set theory. γM 1. represent an important tool for the development and veri. Czech Republic. Membership functions of failure probability are evaluated using the general extension principle. design. which include vagueness. f yk ⋅ An 235 ⋅ 10 6 ⋅ 1. member. Probabilistic study of tensile member of reliability analysis is the sensitivity analysis determin- ing which variables the monitored output is most sensi. limit state methodology. The most important results are then summarised and additional applications of the verifications of limit states design criteria are described.z@fce. accepted 03 Oct 2007 Abstract. fy.vgtu. (1) provisions utilised in steel design practice. Institute of Structural Mechanics. An integral part 2.64 ⋅ 10 −3 fication of more sophisticated but yet simple and lucid Rd = = = 385.219. probability theory. structure. which deter- mine structural dimensions in relation to loading.4 kN. They are investigated within model of the real structural system are verified when the framework of five theories. Material and geometric characteristics are random variables represented by histograms during the probabilistic assessment of reliability. which represents basic calibra. The reliability assessment of a steel member is based on extensive experimental research on real material and Downloaded by [46. tic are monitored [1–3] and probabilistic assessments of steel members reliability are performed. Introduction available. The paper is aimed at the analysis of the influence of partial safety factors on the reliability of designing steel members. tus for their quantification is developed (the crisp set The current Eurocode approach is based on the par.cz Received 02 July 2007. Lately apart from Common design procedures utilised during the de. cussed in the conclusion of the presented paper. theory of tial safety factor method. EUROCODE 3 [15].143. This is Pfd = 7. zero skewness and kurtosis. (3) G k + Qk For permanent action. 0. which is given as: Qk δ= .3 are depicted strength fy obtained from experimental research were in Fig 1. sufficiently satisfactory design reliability.5 are values listed in loading γG and γQ were chosen as further calibration standards [15. expressed by the sum of partial effects G. The yield strength has the following are very low and thus insignificant and attest reliable statistical characteristics: mean value 297. is interesting. Gk.21 Qk was considered.234] at 11:03 29 March 2013 which.3 published in [1]. (2) 1) γ M ∈ 1. The design mentally obtained histograms utilising the Monte Carlo could otherwise be very unsafe. which increases rapidly with decreasing partial safety factor γQ value. the histogram cannot be ap.1 was assumed according to [16].2E-5.0. γG = 1.215 and kurtosis 3. depicted in Figs 1–3. γG = 1. is also the mean value. The material partial safety factor γM was chosen as the first calibration quantity. which takes into for which δ < 0.35. The secured provided that γM > 1.076. γ Q ∈ 1.54. Pf < 7. ie Fd = Rd. γQ = 1. should not be lower than approximately 1.8 MPa. . tosis of the yield strength and sectional area the realisa. (5) Histograms of the random sectional area A and yield Only failure probabilities Pf for δ>0. Reliable design Pf < 7. 292 Z. γQ=1. Parametric probabilistic analysis fied problem of tensile member loaded by permanent load combined with single variable load is considered [16]: Three variants of partial safety factor values γ were considered in the parametric study. The partial safety factors of Values γM=1. skewness 0.5. Reliability of design is determined dard deviation 16. stated that the partial safety factor γM = 1. Fd = γ G ⋅ Gk + γ Q ⋅ Qk . For variable action. The analysis of the member reliability is based on the condition that the random load-carrying capacity R is greater than the effects of external forces F.0. The values were obtained from the evaluation of value given by the standard [16] for reliable design as 562 samples obtained from a third of the flange. No 4.5.0 guarantees The random sectional area has the following pa. 1) values is proximated by the Gaussian probability distribution. the Gumbelian distribution with mean value mQ = 0. the Gaussian probability dis- tribution will be assumed.1.2E-5 is guaranteed by δ ∈ 0. simulation method.55) .35. 1. skewness –0. The non-linear behaviour of failure probability. The 2) γM = 1. standard deviation It is evident from Fig 2 that reliable design 53. Failure probabilities Pf pro parameters δ ≤ 0.219. 5 million simulation runs of this It is apparent from Fig 3 that for δ < 0.15. Due to non– sus partial safety factor γM in Fig 1 is very interesting. Q: R >G+Q .35 . γ G ∈ 1. Failure probability Pf versus partial safety factor γM R = A⋅ fy .8 (which method were used. the characteristic value of Downloaded by [46. rameters: mean value 1681 mm2. it can be account the influences of skewness and kurtosis. characteristic values Gk and Qk can be determined ac- cording to (3) in dependence on the chosen ratio δ. 291–296 For the elaboration of the parametric study a simpli. The probability that (4) is not fulfilled holds for majority of light steel structures) a sufficiently was evaluated. reliable design is guaranteed by the partial safety factor γQ = 1. γQ = 1. Vol XIII. γG=1. 1. simplified loading case according to (2) will be consid- 3) γM = 1.32 and kurtosis by comparing obtained values with the so–called target 2. the standard procedure prescribed by the standards for The non-linear behaviour of failure probability ver- quality control of metallurgical products. Partial safety factor γG for commonly occurring structures tions of these variables were simulated from experi. Kala / JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT – 2007. ered. Results of the probabilistic study are quantities.6 Qk and standard deviation SQ = 0.15 .3 MPa. 3.1. If the partial safety factors γG and γQ are known. provides a better approximation of the histogram. design of the member.143.8 (majority of real structures).2 mm2.2E-5 for all δ ∈ 0. It is unsafe In view of the non–zero values of skewness and kur. It was assumed that the structure is designed for maximum exploitation (economic design). (4) where R is the load-carrying capacity of tensile member: Fig 1. 1. in the event that δ approaches zero and γG approaches 1. 16]. stan.1.35.65 . For light steel structures Hermite four parametric distribution.0. The variation coeffi- cient 0.5. The fuzzy analysis was performed separately for Fig 4. Fuzzy number of partial safety factor γM each input fuzzy number γM. Z.1.65) . Each partial safety factor is assigned a linear degree of membership representing input information for the quantification of the dependence of Pf versus γ. The extension principle may be illustrated on fuzzy number γ ⊆ R. γ G ∈ 1. containing elements with degree of membership μ2 obtained as the supremum of all γ for which Pf = f(γ) holds [17]. No 4. Vol XIII.35) . γ Q ∈ 1.0. Fig 5. Fuzzy probability analysis For purposes of the quantification of the influence of partial safety factors on the failure probability partial safety factors were considered as fuzzy numbers with linear membership functions (Figs 4–6). 291–296 293 Downloaded by [46. 1. γG. Fuzzy number supports are assumed similarly as in preceding parametric analysis γ M ∈ 1. Let γ be a convex fuzzy number with membership function μ1 and let it be defined by the func- tion Pf = f(γ). 1. Fuzzy number of partial safety factor γG .0.15) .219. Failure probability Pf versus partial safety factor γG Fig 3.234] at 11:03 29 March 2013 Fig 2. 1. Failure probability Pf versus partial safety factor γQ 4. The degree of membership μ2 of fuzzy number Pf can then be evaluated according to (6): μ 2 (Pf ) = V μ1 (γ ).143. (6) Pf The result is fuzzy number Pf. γQ utilising the general extension principle [17]. Kala / JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT – 2007. 291–296 Fuzzy numbers of failure probabilities correspond- ing to the fuzzy number of partial safety factor γG. Fig 6. The verification of partial safety factors using reli- ability analysis methods was presented in the paper. It is evident that with decreas- ing δ the support of the fuzzy number decreases. 5. Fuzzy number of partial safety factor γQ Downloaded by [46. as does the kernel value of failure probability. Fuzzy numbers of failure probability due to γG calibration of partial safety factor values. It is apparent from Figs 7–10 that the output mem- bership functions are non-linear despite linear input mem- bership functions (Figs 4–6). Kala / JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT – 2007. No 4. 294 Z. Fuzzy numbers of failure probability due to γQ Fuzzy numbers of failure probabilities correspond- Fig 7. Very effective Fuzzy numbers of failure probabilities correspond. Before long member countries of CEN will design struc- Fig 8. Limitation to purely probabilistic methods is misleading. So far reliability is significantly unbalanced. Vol XIII. Differences in obtained failure probabilities point out the need for further calibra- tion of partial safety factors. Further refinement of Eurocodes and desired unification of currently alternative ap- proaches is expected. It is evident that with decreasing 14]. The output fuzzy number for δ = 1 (purely variable load) is a singleton. implementation of all available methods of the general ing to the fuzzy number of partial safety factor γM. 9. Fuzzy numbers of failure probability due to γM ing to the fuzzy number of partial safety factor γQ (see Fig 6) are depicted in Fig 10. The Eurocodes will even- tually become valid normative standards and the current standards of individual member countries will be can- celled. . (see Fig 5) are depicted in Figs 8. because the partial safety factor of permanent load action γG has no influence on the value of variable load. This is apparent from the nonlinear behaviours and sup- port sizes.143. The intended high degree of unification of alternative approaches has however not been achieved yet. Probabilistic assessment of reliability pointed out the need for further Fig 9.234] at 11:03 29 March 2013 Fig 10. (Fig 4) uncertainty theory is feasible in reliability analysis [6– are depicted in Fig 7. as does the kernel value of failure probability. Increase in reliability and structural durability and also an increase in material consumption can be expected in the Czech Republic. In the case of δ = 0 (permanent load) the failure probability is signifi- cantly influenced by the fuzzy uncertainty of factor γG. δ the support of the fuzzy number decreases. Fuzzy numbers of failure probability due to γG tures according to a unified methodology. Conclusions Eurocode creation has achieved significant success and acknowledgement even outside the European context.219. and UTKIN. Design characteristics of structural Practically output membership functions were nonlinear steels based on statistical analysis of metallurgical pro- in all cases. and ROZLÍVKA.. which are fuzzy data. W. Structural Safety and Reliability ICOSSAR'05. Journal of Constructional Steel Research. W. NOVÁK. HOFFMANN. L. 2004. K. 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