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Practical Method To Determine Load And Resistance Factors Used In Limit State Design PRACTICAL METHOD TO DETERMINE LOAD AND RESISTANCE FACTORS USED IN LIMIT STATE DESIGN Yasuhiro MoriÉ, Tsuyoshi TakadaÉÉ, and Hideki IdotaÉÉÉ É Graduate School of Environmental Studies, Nagoya University, Nagoya Japan ÉÉ Graduate School of Engineering, the University of Tokyo, Tokyo Japan É ÉÉ Dept. of Civil Engineering and Architecture, Nagoya Institute of Technology, Nagoya Japan Abstract In reliability-based limit state design, the performance level of a structure can be con- trolled by diãerentiating load and resistance factors. In order to take fully the advantage, Draft Recommendation of Limit State Design by AIJ proposes a practical method for evaluating the factors. In the practical method, the combination of time variant loads are considered using Turkstra's rule and non-lognormal random variables are approximated to an equivalent lognormal random variables for which load and resistance factors can be expressed in a closed form. This paper investigates the accuracy of the practical method using numerical examples. It also discuss the potential improvement of the practical method. Keywords: Limit State design, Load Factors, Resistance Factors, Sensitivity Factors, Reliability Index, Practical Method INTRODUCTION The safety problem in structural engineering can be treated more rationally with probabilistic methods. These methods provide basic tools for evaluating structural safety quantitatively. Uncertainties in loads, material properties, and construction practice, which have been tradi- tionally dealt with by empirical safety factors, can be taken into account explicitly and con- sistently in probabilistic safety assessment. Based on such methods, a å rst-generation of limit ed state design codes were developed; the safety checks are associated with a speciå limit state probability or reliability. The design format in these codes is given with load and resistance factors or partial safety factors with which structural design can be carried out semi-deterministically without complicated probability analysis. In most cases, only single set of the factors is provided in the code. Conceptually, the structural performance level can be controlled by diã erentiating the factors [1], which makes reliability-based design an ideal tool for performance based design. However, intensive knowledge in the theory of probability and statistics is required to determine the appropriate factors. 1 In order to implement fully the advantage of reliability-based design, sub-committee on Limit State Design, Architectural Institute of Japan (AIJ) published Draft Recommendation for Limit State Design for Building Structures in 2001 and presented three ç ows to determine load and ed resistance factors in accordance with target reliability level [2]. In the simpliå method (Flow I), tables of load and resistance factors are provided for typical load combinations assuming statistical characteristics of various kinds of load events at a typical site in Japan. The target reliability level is set to equal 1.0, 2.0, or 3.0 for the reference period of 50 years for ultimate limit states and for the period of one year for serviceability limit states. ed The load and resistance factors presented in the simpliå method are intended to be used to design a building at any site in Japan. However, there are many cases that an engineer feels the necessity of determining the factors by himself/herself, e.g., when statistical characteristics of load events at a construction site are diãerent from that used in the draft to determine the factors, when he/she has up-to-date information on the statistical characteristics of load and resistance, and when he/she wants to control the target reliability level in detail. Although it can be carried out using probability analysis such as AFOSM (Flow III), it is complicated for ordinary engineers. The Draft Recommendation provides simple formulae as a practical method (Flow II) for approximately evaluating LR factors taking additional information such as above into account. This paper describes the practical method in detail and investigates the accuracy of the method based on the reliability level achieved using the load and resistance factor evaluated by the practical method. It also discusses the potential improvement of the practical method. PRACTICAL METHOD FOR LOAD AND RESISTANCE FACTORS Load and Resistance Factors In general a structure is subjected to more than one time varying load processes. Design format of limit state design with load and resistance factor can be expressed as, X 1 R ûÅ n = i S ç Å ni (1) i=n in which R is the resistance of a structural component, Si (i = 1; Å Å n) is the load eã having Å; ect the same dimension with R, û and ç are load and resistance factors, respectively, and Xn is a i nominal value of X. Several method have been developed to handle the combination of time varying load process [3, 4, 5, 6]. Among them Turkstra's rule [7], is one of the most simple method, with which the combination can be approximated as the sum of time invariant random variables. Then the load and resistance factors can be evaluated by AFOSM. When all the basic random variables are lognormally distributed and statistically independent of one another, load and resistance factors can be expressed in closed forms as, p 1 ñR û = å õ exp(ÄãR Å T Å ln R ) (2) 2 1 + VR Rn p 1 ñSi çi = å õ exp(ãSi Å T Å ln Si ) (3) 2 1 + VSi Sni in which å is the target reliability index, ñX , õ and VX are the mean, standard deviation, T X and coeécient of variation (c.o.v.) of X, and ãX is the sensitivity factor of X expressed as, 2 v u õ R Å ñR ûÅ u X ln ãR = (4) t(õ R Å ñR )2 + n ln ûÅ (õ Si Å i Å Si )2 ln ç ñ i=1 v u õ Si Å i Å Si ç ñ u X ln ãSi = (5) t(õ R Å ñR )2 + n ln ûÅ (õ Si Å i Å Si )2 ln ç ñ i=1 in which û, ç , and ñR are unknown quantities. When only one load act on a structure, the i sensitivity factors can be evaluated analytically as õ R ln õ S ln ãR = ; ãS = (6) õ M ln õ M ln p in which õ M = ln (õ R )2 + (õ S )2 . ln ln when a structure is subjected to more than one load process, Eqs.(4) and (5) have to be evaluated iteratively. The Draft Recommendation provides simple formulae to approximate the sensitivity factors as described in the next section. Sensitivity Factors When R and S1 ; Å Å Sn are all normal random variates, sensitivity factors can be expressed as Å; [1], õR õ S ãR = ; ãSi = i (7) q õM õ M in which õ = õ2 + õ2 1 + Å Å õ2 n . M R S Å+ S When a structure is subjected to more than one load process, both resistance, R, and the sum of the all load eãect, Q, are approximated to be lognormally distributed, and the their sensitivity factors are evaluated by Eq.(6). Then the sensitivity factors for Si 's, ãSi , are evaluated by Eq.(7), replacing õ with õ . The procedure is described in the following. M Q v 1) Determine the statistics of Q from those of Si 's. u n X uX ñSi ; õ = t n õQ ñQ = Q õ i 2; S VQ = (8) i=1 i=1 ñQ 2) Determine standard deviation of ln R, ln Q and ln M . q Ä Å õ R= ln ln 1 + VR 2 ' VR (9) q Ä Å õ Q= ln ln 1 + VQ 2 ' VQ (10) q q õ M = ln õ2 R + õ2 Q ' ln ln VR 2 + VQ 2 (11) 3) Determine ãR and ãQ from Eq.(6) 3 Pn p 4) Determine ãSi 's from Eq.(7) and normalize them so that ãSi 2 = 1 Ä ãR 2 . q i=1 õi S 2 ãSi = Å 1 Ä ãR (12) õQ Substituing Eqs.(8)-(11) into Eqs.(6) and (12), ãR and ãSi can be evaluated by ãR = v u ~R õ u 2 X 2 Åu (13) tõ + n ~R õi S i=1 ãSi = v u õi u 2 X 2 S Åu (14) tõ + n ~R õi S i=1 P in which õ = VR Å n ñSi . u is the safety factor to consider an error of the approx- ~R i=1 imation considered in the practical method, and u = 1.05 is recommended in the Draft Recommendation. In the Draft Recommendation Eqs.(13) and (14) are expressed in terms of c.o.v. rather than standard deviation for use in practice. Non-lognormal Load Eãect ect In most cases R is modeled as a lognormal random variate; however, load eã such as earth- ect ect quake load eã and wind load eã is often described by an extreme value distribution. As load and resistance factors depend on the probability distribution of load eã ect, it is necessary to approximate a load eã ~ ect, Si , as a lognormal random variate, Si , to evaluate the factors by Eqs.(2) and (3). ~ In the Draft Recommendation, normalized mean value of ln Si (mean value of ln(Si =ñSi )), ~ É ~ ~i , VS , are determined to satisfy the following ñln Si , standard deviation, ln õ Si , and c.o.v. of S ~ ~ln i conditions. FSi (s50 ) = GSi (s50 ) (15) FSi (s99 ) = GSi (s99 ) (16) ~ in which FSi (s) and GSi (s) are probability distribution function (cdf) of Si and Si , respectively, s50 and s99 are the value of s satisfying FSi (s) = 0:5 and FSi (s) = 0:99（0.999 if Si is described by Type II extreme value distribution and å ï 2:5), respectively. As it is not practical to T solve the nonlinear simultaneous equations of Eqs.(15) and (16), the Draft Recommendation presents the following regression formulae. ñÉ Si = e0 + e1 Å Si + e2 Å Si 2 + e3 Å Si 3 ~ln V V V (17) q Ä2 Å 2 3 ~ln õ Si = s0 + s1 Å Si + s2 Å Si + s3 Å Si V V V (18) ~ VSi = exp õ Si Ä 1 ~ln (19) ect in which VSi is the c.o.v. of the annual maximum value of load eã before approximated to be lognormal random variate, and ej and sj are provided in Tables 1 and 2 based on the probability distribution of the annual maximum value. The reference period of serviceability limit state design and ultimate limit state design are assumed to be one year and 50 years, respectively. The annual maximum value of the non-principal loads and principal load of serviceability limit 4 Table 1: ej in Eq.(17) and sj in Eq.(18) to approximate annual maximum value cdf of ej sj annual maximum value e0 e1 e2 e3 s0 s1 s2 s3 Normal 0.00 0.00 0.00 0.00 0.01 0.85 -0.49 0.14 Type I 0.00 -0.16 -0.01 0.00 0.02 1.13 -0.67 0.20 Type II å î 2:5 T 0.00 -0.28 -0.05 0.07 0.00 1.44 -0.98 0.26 å > 2:5 T 0.00 -0.28 -0.05 0.07 0.00 1.68 -1.14 0.30 Table 2: ej in Eq.(17) and sj in Eq.(18) to approximate 50-year maximum value cdf of ej sj annual maximum value e0 e1 e2 e3 s0 s1 s2 s3 Normal 0.02 1.89 -1.05 0.30 0.02 0.34 -0.32 0.11 Lognormal -0.01 2.34 -1.07 0.16 0.00 0.61 -0.13 0.00 Type I 0.04 2.32 -1.43 0.43 0.05 1.59 -0.60 0.22 Type II å î 2:5 T 0.01 2.82 -2.15 0.62 0.00 1.44 -0.98 0.26 å > 2:5 T 0.01 2.82 -2.15 0.62 0.00 1.68 -1.14 0.30 Table 3: Probabilistic model（Annual maximum and single load） mean c.o.v. S 5.0 0.1～1.0 R å =1.0, 2.0, 3.0 T 0.2 state design are approximated as lognormal random variate using parameters in Table 1, while 50 year maximum value of the principal load of ultimate limit state design is approximated using parameters in Table 2. ACCURACY OF PRACTICAL METHOD In the practical method described above, 1) load eãects are approximately described by equivalent lognormal cdf (lognormal approx- imation), 2) the sum of lognormal random variates are assumed to be a lognormal random variate for evaluating sensitivity factors, and ect 3) the combination of time variant load eã is considered by Turkstra's rule. The accuracy of these approximations depends strongly on the statistical characteristics of basic random variables. The reliability level achieved using load and resistance factors determined by the approximation method could be far oã from the target reliability level. Many studies have been conducted about Turkstra's rule and it is pointed out that the rule is accurate enough in practice when the principal load is well dominant [6, 8]. Thus, the above approximations 1) ect and 2) are investigated here. It is assumed in the following that non-principal load eã and resistance are lognormally distributed and that u=1.0 in Eqs.(13) and (14) unless otherwise speciå ed. 5 β β 1.5 1.5 β T = 1.0 β T = 1.0 1.0 1.0 0.5 VS 0.5 VS 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2.0 β T = 2.0 2.0 2.0 1.5 VS 1.5 VS 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3.0 β T = 3.0 3.0 3.0 □ Normal □ Normal ◇ Type I × Lognormal ○ Type II ◇ Type I 2.5 VS 2.5 VS 0 0.5 1.0 0 0.5 1.0 t L = 1 year t L = 50 years Figure 1: Reliability level when single load act on a structure (S ò non-lognormal) Equivalent Lognormal Random Variate ect When a structure is subjected to a single load and the load eã is lognormally described, the reliability level achieved using the load and resistance factors determined by the practical method, å , agrees with the target reliability level, å . The diã a T erence between å and å a T ect when load eã is not lognormally distributed is due to the lognormal approximation. Fig.1 ect illustrates å and å as a function of the c.o.v. of the annual maximum load eã using the a T probability models shown in Table 3. Reference period, tL , is assumed to be 1 year or 50 years. As 50-year maximum value, S50 is also described by Type II distribution with the same c.o.v. if annual maximum value, Sa is described by Type II, only annual maximum is illustrated for Type II distribution. å is evaluated accurately and eéciently using a probability analysis using a Fast Fourier transform [9]. Fig.1 show that å by the approximation method depends strongly on the type of cdf of load a eã and target reliability level. When Sa is described by a Type I distribution function, å ect a is unconservative for å î 2:0 and VS ï 0:3 if tL =1 year, and the error increases with increase T 6 Table 4: Probability model (two load act on a structure) mean c.o.v. Case1 S2 5.0 0.1～1.0 Case2 S2 10 0.1～1.0 Case3 S2 25 0.1～1.0 S1 5.0 0.1, 0.4 All cases R å = 1.0, 2.0, 3.0 T 0.2 Table 5: Probability Model (Combination of 4 Loads) r.v. mean c.o.v. S2 1.5 0.4 CaseA S3 1.5 0.4 S4 5.0 0.1～1.0 S2 5.0 0.4 CaseB S3 5.0 0.4 S4 5.0 0.1～1.0 S2 5.0 0.4 CaseC S3 5.0 0.4 S4 25 0.1～1.0 S2 5.0 0.4 CaseD S3 5.0 1.0 S4 25 0.1～1.0 S1 5.0 0.1 All cases R å = 1.0, 2.0, 3.0 T 0.1, 0.2, 0.4 of VS . On the other hand, if tL =50 years, the target reliability is roughly achieved. When Sa is described by a Type I distribution function, S50 is also described by a Type I distribution function; however, VS50 in only 0.24 even if VSa =1.0 When Sa is described by a Type II distribution function, å is a little conservative estimate a for å =1.0 regardless of a reference period, and decreases with increase of å . When Sa T T is a lognormal random variate and tL =50 years, å decreases with increase of å , becomes a T unconservative side å =3.0 if VS ï 0:4. When Sa is a normal random variate, å cannot be T T achieved for å î 2:0 if tL =1 year. å comes to be very unconservative estimate if VSa is large. T a On the other hand, å is a little conservative estimate, if tL =50 years. a On the course of the analyses, it was found that the accuracy increases a little with increase of VR . As VR increases, R=S comes to be closer to a lognormal random variate. Load Combination When a structure is subjected to more than one load, Eqs.(13)-(14) comes to be approximation even if all the basic random variables are lognormally distributed. In order to investigate the accuracy and applicability of this approximation, the combinations of two loads and four loads are considered in this section. Lognormal random variates Fig.2 illustrates å when all of R, S1 , and S2 (principal load) are lognormally distributed using a the probability model presented on Table 4. For Case 1, in which ñS2 = ñS1 , where ñX is the 7 β β 1.5 1.5 β T = 1.0 × Case1 β T = 1.0 × Case1 □ Case2 □ Case2 ○ Case3 ○ Case3 1.0 1.0 0.5 VS 0.5 VS 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2.0 β T = 2.0 2.0 2.0 1.5 VS 1.5 VS 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3.0 β T = 3.0 3.0 3.0 2.5 VS 2.5 VS 0 0.5 1.0 0 0.5 1.0 VS1 = 0.1 VS1 = 0.4 Figure 2: å under combination of twoload（S ò lognormal) a mean of X, and the principal load is weakly dominant, å is unconservative. å is very close a a to å if VS1 =0.1. On the other hand, å come to be unconservative when VS1 =0.4 and S2 is T a weakly dominant. On the course of analyses, it was found that such trends are fairly lightly dependent on VR . The dependency of å on the \dominance of the principal load" under the combination of four a load S = S1 + S2 + S3 + S4 is illustrated in Fig.3 using probability model presented in Table 5. In general, the principal load is dominant if the mean value and/or c.o.v. of the principal ect ed load eã is larger than the other loads. The dominance can be quantiå by the ratio of the ect standard deviation of the principal load eã to that of the combination of the others, rõ. rõ = p õ4 S (20) õ 12 + õ 22 + õ 32 S S S It is assumed that all load eã are lognormally distributed, and VR = 0:2. Safety factor u in ect 8 β β 1.5 1.5 β T = 1.0 CaseA β T = 1 .0 CaseA CaseB CaseB CaseC CaseC × CaseD × CaseD 1.0 1.0 0.5 rσ 0.5 rσ 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2 .0 β T = 2 .0 2.0 2.0 1.5 rσ 1.5 rσ 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3 .0 β T = 3.0 3.0 3.0 2.5 rσ 2.5 rσ 0 0.5 1.0 0 0.5 1.0 VR=0.2, u=1.0 VR=0.2, u=1.05 Figure 3: å under Combination of Four Loads (VR = 0:2) a Eqs.(13) and (14) is set to be 1.0 or 1.05. In Cases A and C, the mean value of the principal load eã ect, ñS4 , is relatively large, and accordingly the principal load is strongly dominant. In Case B, ñS4 is equal to the mean values of the other loads. In Case D, VS3 is increased from that of Case C to 1.0, and accordingly the dominance of the principal load is weakened. Except for Case B, å for all the cases takes similar values when principal load is dominant a es (rõ ï 3). Although å is a little unconservative when u = 1:0, it roughly satiså the target a level using u = 1:05. For Case B, å takes similar values to the other cases when å ï 2:0. a T However, the accuracy decreases to the unconservative side with increase of rõ when å = 1.0. T Fig.4 illustrates å assuming VR =0.1 or 0.4 and using u = 1:05. Similar to the case when a VR = 0:2, the target reliability level can be roughly achieved when rõ ï 4 using u = 1:05. Non-lognormal load eãects Fig.5 illustrate å when a structure is subjected to the combination of two loads and when S2 a 9 β β 1.5 1.5 β T = 1.0 CaseA β T = 1 .0 CaseA CaseB CaseB CaseC CaseC × CaseD × CaseD 1.0 1.0 0.5 rσ 0.5 rσ 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2 .0 β T = 2 .0 2.0 2.0 1.5 rσ 1.5 rσ 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3 .0 β T = 3.0 3.0 3.0 2.5 rσ 2.5 rσ 0 0.5 1.0 0 0.5 1.0 VR=0.1, u=1.05 VR=0.4, u=1.05 Figure 4: å under Combination of Four Loads (VR =0.1，0.4) a is described by Type I or Type II extreme value distribution using the same probability models used for Fig.2 except that VS1 = 0:1. å depends strongly on the type of probability distribution a and target reliability level. The error is similar to that due to lognormal approximation shown in Fig.1, implying the necessity of more accurate lognormal approximation. REFINEMENT OF LOGNORMAL APPROXIMATION The possibility of reåning the practical method presented in the Draft Recommendation of Limit State Design is described in this section. Lognormal Approximation Considering åT For lognormal approximation, two conditions are necessary to determine two parameters of lognormal distribution function. In the Draft Recommendation Eqs.(15) and (16) are used for simple approximation. However, numerical examples in the previous section implied that å T should be considered in the approximation. One of the possibility is to use Eq.(21) in stead of 10 β β 1.5 1.5 β T = 1.0 × Case1 β T = 1.0 □ Case2 ○ Case3 1.0 1.0 × Case1 □ Case2 ○ Case3 0.5 VS 0.5 VS 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2.0 β T = 2.0 2.0 2.0 1.5 VS 1.5 VS 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3.0 β T = 3.0 3.0 3.0 2.5 VS 2.5 VS 0 0.5 1.0 0 0.5 1.0 S2～Type I S2～Type II Figure 5: å under Combination of Two Loads (S2 - non-lognormal) a Eq.(16) so that å agrees with å when a structure is subjected to a single load. a T å =å a T (21) ~ The c.o.v. of the approximated random variable, VS , å = 1:0; 2:0; 3:0 is illustrated in Figs.6 T and 7 for tL = 1 year and 50 years, respectively. It is assumed in the illustration that VR =0.1, 0.2, 0.3，or 0.4. The solid lines in the å ~ gures are the regression formula for VS as a cubic ~ function of 1=å . VS depends strongly on not only the type of probability distribution of S but T also å . However, it depends fairly lightly on VR except for a couple of cases. T Fig.8 illustrates å using lognormal approximation described in this section using the probability a models presented in Table 4. The accuracy of å is improved substantially from Fig.5 and a compatible with that when all the basic random variables are lognormally distributed (see Fig.2). 11 ~ ~ VS VS β T = 1.0 Regress. 1.0 1.0 ○ VR=0.1 ◇ VR=0.2 □ VR=0.3 × VR=0.4 0.5 Regress. 0.5 ○ VR=0.1 ◇ VR=0.2 □ VR=0.3 × VR=0.4 VS β T = 1 .0 V 0 0 S 0 0.5 1.0 0 0.5 1.0 β T = 2 .0 1.0 1.0 0.5 0.5 VS β T = 2 .0 V 0 0 S 0 0.5 1.0 0 0.5 1.0 ~ ~ VS VS β T = 3 .0 1.0 1.0 0.5 0.5 VS β T = 3 .0 V 0 0 S 0 0.5 1.0 0 0.5 1.0 S～Normal S～Type II ~ Figure 6: VS Considering å (tL = 1 year) T CONCLUSIONS This paper investigate the accuracy and applicability of the practical method for evaluating load and resistance factors introduced in the Draft Recommendation of Limit State Design for Building Structures by the subcommittee of Limit State Design of AIJ. For ultimate limit state design the reliability level achieved using the load and resistance factors evaluated by the es practical method generally satiså the target level. However, it would be very unconservative for serviceability limit state design with tL = 1 year, and when å ô 1:0, principal load is T described by normal distribution or Type I distribution, and its c.o.v. is large. It would also be very unconservative when the principal load is weakly dominant. As the accuracy of Turkstra's rule is also in question when the principal load is weakly dominant, special attention should be made to apply the practical method to such cases. This paper also presented that the accuracy of the practical method could be improved by reåning the conditions of lognormal approximation. As it is impractical to solve nonlinear simultaneous equations during design process, simpler approximation formulae needs to be 12 ~ ~ VS VS Regress. β T = 1.0 ○ VR=0.1 ◇ VR=0.2 0.3 0.4 □ × VR=0.3 VR=0.4 0.2 Regress. 0.2 ○ VR=0.1 0.1 ◇ VR=0.2 □ VR=0.3 β T = 1.0 V × VR=0.4 VS 0 S 0 0 0.5 1.0 0 0.5 1.0 ~ ~ VS VS β T = 2 .0 0.3 0.4 0.2 0.2 0.1 β T = 2 .0 V VS 0 S 0 0 0.5 1.0 0 0.5 1.0 ~ ~ VS VS β T = 3 .0 0.3 0.4 0.2 0.2 0.1 β T = 3 .0 V VS 0 S 0 0 0.5 1.0 0 0.5 1.0 S～Lognormal S～Type I ~ Figure 7: VS Considering å (tL = 50 years) T developed for use in practice. ACKNOWLEDGMENT c Support of this research through Grant-in-Aid for Scientiå Research (C) from the Ministry of Education, Science, Sports, and Culture and Japan Society for the Promotion of Science is gratefully acknowledged. REFERENCES [1] Melchers RE. Structural reliability; analysis and prediction - Second edition. West Sussex, UK: JohnWiley and Sons, 1999 [2] Draft Recommendation of Limit State Design for Building Structures, 2nd Edition. Sub- committee on Limit State Design, AIJ, Japan. 2001.12. (in Japanese) [3] Wen YK. Statistical combination of extreme loads. J. Str. Div., ASCE Vol.103, No.5, pp.1079-1093, 1977.5 13 β β 1.5 1.5 β T = 1.0 × Case1 β T = 1.0 × Case1 □ Case2 □ Case2 ○ Case3 ○ Case3 1.0 1.0 0.5 VS 0.5 VS 0 0.5 1.0 0 0.5 1.0 β β 2.5 2.5 β T = 2.0 β T = 2.0 2.0 2.0 1.5 VS 1.5 VS 0 0.5 1.0 0 0.5 1.0 β β 3.5 3.5 β T = 3.0 β T = 3.0 3.0 3.0 2.5 VS 2.5 VS 0 0.5 10 0 0.5 1.0 S2～Type I S2～Type II Figure 8: å of Reå a ned Method（S2 ò non-lognormal） [4] Larrabee RD, Cornell CA. Combination of various load processes. J. Str. Div., ASCE, Vol.107, ST1, pp.223-239, 1981 [5] Kohno M, Sakamoto J. Exceedence probability of stochastic load combinations. Proc. ICASP 6. Mexico: Mexico City, pp.622-629, 1991. [6] Mori, Y. and K. Murai. Load and resistance factors taking combined load processes into account. Applications of Statistics and Probability, Proc. ICASP8, 1999.12, pp.973-980. [7] Trukstra CJ, Theory of structural design decision, Study No.2. Solid Mechanics Division. Univ of Waterloo, Waterloo, Ontario, Canada. 1970 [8] Pearce HT, Wen YK. Stochastic combination of load eã ects. J. Str. Div., ASCE, Vol.110, No.7, pp.1613-1629, 1984.7 [9] Sakamoto J, Mori Y, Sekioka T. Probability analysis method using Fast Fourier transform and its application. J. Structural Safety, Vol.19, pp.21-36, 1997. 14

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