ࡱ> Zbjbj *rr';ab!!$$$$$$8"%&$Qx())))I I I PPPPPPP_UXP$I  ^I I I P!!))MQgggI ! )$)PgI PggB  $E)PÈC*D$PQ0QNDXXHEEX$F I I gI I I I I PPI I I QI I I I XI I I I I I I I I  :PROBABILISTIC ANALYSIS OF BOLTED RHS END-PLATE JOINTS IN AXIAL TENSION Josip Piakovia, Darko Dujmovia,c, Boris Androib,c, a Department of Structural Engineering, Faculty of Civil Engineering, University of Zagreb, Croatia b IA Projektiranje Structural Engineering Ltd., c Croatian Academy of Engineering, Zagreb, Croatia  HYPERLINK "mailto:jpiskovic@grad.hr" jpiskovic@grad.hr,  HYPERLINK "mailto:dujmovic@grad.hr" dujmovic@grad.hr,  HYPERLINK "mailto:androic@iaprojektiranje.com" androic@iaprojektiranje.com Introduction Bolted end-plate joints have always been a suitable solution for Rectangular Hollow Section (RHS) tension members jointed on building site. In such joints bolting can be provided only along two sides or along all four sides of a hollow section. This paper deals with the joints bolted along two opposite sides (Fig.1). EN 1993-1-8 [1] does not give recommendations for calculation of resistance of such joints. The suitability of EC3 model used for I and H (IHS) section joints with extended end plate to design of RHS joints is questionable. The main difference between bolted IHS and RHS end plate joints is yielding of RHS adjacent to the hogging plastic hinge and participating in the general failure mechanism. Design models from relevant AISC [2] and CIDECT [3] Design Guides can be used with modification of partial factors according to Eurocode. In this paper joints with intermediate plate behaviour are investigated. Joint resistances are calculated using models from [1], [2] and [3] and compared with test results from literature [4], [5]. Indicator of model accuracy is mean of test to predicted ratio and coefficient of variation. A better comparison can be achieved by calculating reliability index ( of joint by means of probabilistic analysis and comparing with target reliability index (target. In reliability analyses it is important to consider the uncertainties in the calculation model as well as the uncertainties in the resistance parameters and applied loads.  Fig. 1. End plate joint with bolts along two sides of RHS Resistance OF BOLTED RHS END-PLATE JOINTS IN AXIAL TENSION Experimental tension resistance Kato and Mukai [4] performed two (labeled S1 and S2), and Packer et al. [5] performed series of 16 tests (labeled LB1-LB16) on bolted RHS end-plate joints in axial tension with bolts along only two opposite sides of the RHS. Geometrical and mechanical characteristics of specimens and failure loads from these two experimental studies are used in this paper. Some of the specimens developed end plate mechanism, (thin plate behavior), but all eventually failed by bolt fracture. Namely, end plate mechanism is a theoretical failure mode and does not represent the ultimate joint resistance. The end plate is able to resist additional forces because of strain hardening and membrane effects. The specimens that failed by first failure mode (thin end plate configuration) according to Eurocode (LB2, LB13 LB14) are not considered in this study as well as specimens failed by tube tearing (LB11, LB12, LB16) [5]. For all specimens considered in this study, measured ultimate joint resistances are reported in Table 1. EN 1993-1-8 model for I/H section joints According to Eurocode EN 1993-1-8 [1] three failure modes have to be checked and the lowest is governing. The resistances of all joints considered are governed by second mode, i.e. bolt failure with yielding of end plate given in Eq. (1)  EMBED Equation.3  (1) Where  EMBED Equation.3  is design plastic bending resistance of end plate,  EMBED Equation.3  is design tension resistance of bolt, e1, e2, lw defined in Figure 1 and Table 3. The characteristic resistances calculated using Eurocode model compared to test results are given in Table 1. Resistance according to AISC In AISC Design Manual [2] a modified T-stub [6] design procedure is used to evaluate the connection limit states. In Struik and de Back s T stub prying model [7] the term  has been used to represent the ratio of the (sagging) bending moment per unit plate width at the bolt line, to the bending moment per unit plate width at the inner (hogging) plastic hinge. Thus, for the limiting case of a rigid plate,  = 0, and for the limiting case of a flexible plate in double curvature with plastic hinges occurring both at the bolt line and the edge of the T stub web,  = 1.0. Hence, the term  in Struik and de Back s model was restricted to the range 0 (  ( 1.0. For bolted RHS end-plate joints, this range of validity for  was changed to simply  ( 0. This implies that the sagging moment per unit width at the bolt line is allowed to exceed the hogging moment per unit width, which was proposed because the RHS member tends to yield adjacent to the hogging plastic hinge and participate in the general failure mechanism. Also, the distance e1 was adjusted to e1, where: e1=e1-(d/2)+t1 and e2=e2+(d/2). The limit states for the end-plate connection bolted on two sides, are: yielding of the end plate and tensile resistance of the bolts, including prying action Eq.(2). Partial factors for actions and resistance according to Eurocode are adopted. After substitution and rearrangement the expression resistance verification can be written as.  EMBED Equation.3  (2) where FT,Ed is design tension force acting on joint Ft,Ed is design tension force acting on bolt (including prying) all other variables are defined in Table 3 The characteristic resistances calculated using AISC model compared to test results are given in Table 1. Since the action force per bolt is calculated, and not the resistance, Resistance according to CIDECT In CIDECT [3] the modified T-stub design procedure is used. Advantage of this model compared to AISC is that design resistance of bolt is used instead of action force. For  = 0 (no prying action), with the bolts loaded to their tensile strength, the required end-plate thickness is "tc". The appropriate value of "" for a connection was expressed by comparing the end-plate thickness "tp" of a connection with "tc". After substitution and rearrangement the expression can be written as:  EMBED Equation.3  (3) The characteristic resistances calculated using CIDECT model compared to test results, are given in Table 1. Table 1. Comparison of actual to predicted joint resistances TESTEN 1993-1-8AISCCIDECTIDJoint FTBolt FtJoint resistance FT,2,RkFT/ FT2,RkBolt tension Ft,EkFt /Ft,EkJoint resistance FT,RkFT/ FT,RkkNkNkNkNkNLB-14431364131,071590,863791,12LB-36222216350,982201,005831,07LB-47932027781,022120,957601,04LB-58602098970,962011,048551,01LB-695521010600,901931,099770,98LB-797120110900,891911,0510120,96LB-897420111200,871891,0610220,95LB-97952218330,952191,017411,07LB-107951388360,951520,917251,10LB-156801356930,981470,926181,10S15181645250,991800,914721,10S26501756650,981990,885751,13Mean0,960,971,06 Stdev.0,060,080,07Probabilistic approach The tension resistance of bolted joint is considered according to three presented analytical models. Random variables of resistance and action are identified with the appropriate distribution function, mean value and standard deviation. Reliability analysis of joint subjected to permanent and snow load is conducted. The aim of a component reliability analysis is to estimate the probability of failure  EMBED Equation.3  (4) where X is a random vector which contains all uncertain basic variables; fx(x) represents the joint probability density function of the basic random variables; and g(x) is the limit state function corresponding to the failure mode considered and defined such that the failure event corresponds to g(x) ( 0. Reliability methods are described in a number of papers. The reliability analysis has been performed using STRUREL (8(. The reliability indices obtained from the analysis for different failure modes enabled to estimate the reliability degree and to compare studied design models mutually. Limit state functions Limit state functions are defined for three design models Eq(5), Eq(6) and Eq(7). They can be expressed in general form as follows: gi(x)=ri(x)"e(x), where ri(x) and ei(x) are the vectors of basic variables of resistance and action effect respectively.  EMBED Equation.3  (5)  EMBED Equation.3  (6)  EMBED Equation.3  (7) Basic variables of actions In the process of evaluation of the basic variables of action effects, the proportion of the permanent and snow design action effect is 40% and 60 % respectively (FEd,g = 0.4 FEd, FEd,s = 0.6 FEd) . The joint is fully utilized ie. the action effect FEd is equal to the design resistance of joint FRd. The basic variable of permanent action has been adopted with mean value equal to characteristic value and variance of 0,01. The snow load has been adopted according to [9]. The characteristic value of snow load is 98% fractal value, and the mean value is calculated. Table 2. Basic variables of actions EC3AISCCIDECTFEd (kN)355295321gsgsgsFEk (kN)10514287,6118,295,1128,4( (kN)10576,687,663,795,169,2( (kN)1,0525,2780,87621,020,95122,836DistributionPermanent (Feg)Normal; Snow(Fes) -GumbelBasic variables of resistance Mean values, standard deviations and the corresponding distribution functions of these basic variables are provided in Table 3. Table 3. Basic variables of resistance IDDistributionMeanStdevPlate widthbpNormal (Gauss)1500,18Plate thicknesstpNormal (Gauss)20,20,1RHS wall to bolt centere1Normal (Gauss)44,50,28Plate edge to bolt centere2Normal (Gauss)41,50,26Bolt holed0Normal (Gauss)17,50,2Plate yield strengthfypLognormal30021Bolt ultimate strengthfubLognormal75038Bolt diameterdNormal (Gauss)15,90,06Weld leg sizelwNormal (Gauss)101Number of boltsnConstant4-Model uncertainty Since in developing resistance and action models certain influences are neglected, deviations between analytical and test results are to be expected. This fact is considered by introducing a model uncertainty in resistance fmr and actions fme. The parameters of fmr are mean and standard deviation of the ratio of test to predicted resistance for each analytical model. As for the uncertainty in actions, fme, mean values are adopted as 1 and variance of 0,05. In AISC analytical model, model uncertainty in action effects is introduced with the mean and standard deviation of the ratio of test to predicted bolt load with ( =0,97 and ( = 0,08. Table 4. Basic variables of model uncertainty EC3AISCCIDECTResistance fmr (Normal)0,96-1,060,06-0,07Action fme (Lognormal)1,000,971,001,000,050,080,050,05Results and discussion of probabilistic analysis It can be seen that correlation between actual joint resistance and predicted joint resistance is very good for all models, with ratio having mean (=0,96 and standard deviation ( =0,06 for EC3, (=0,97 and ( = 0,08 for AISC and ( =1,06 and ( = 0,07 for CIDECT. For EC3 model the error is slightly on the un-conservative side. For AISC and CIDECT model the error is on conservative side. Probabilistic analysis has been conducted by STRUREL [8] according to FORM (First Order Reliability Analysis) for three design models. The design working life of the structure was assumed to be 50 years. Thus, according to [10], the target value of reliability index for ULS is adopted, target = 3.8. The reliability index values obtained according to FORM for ULS and different analytical models are given in Table 5. Reliability indices for all failure models are higher than target value of the reliability index (target = 3.8. Table 5. Reliability indices ( Analytical modelEC3AISCCIDECTReliability index (3.9514.7434,120 FORM also provides a sensitivity factors. Generally written as a percentage, these factors allow the ranking of the basic random variables according to their importance in the reliability analysis. The sensitivity factor of the basic variable FE,s has the greatest influence on the reliability index. An extremely high influence of FE,s on the reliability index is anticipated considering the stochastic character of this basic variable. Among basic resistance variables, the basic variable fyp and fub have the highest influence on the reliability index.  EMBED MSGraph.Chart.8 \s  Fig. 2. Sensitivity factors, (i , of basic variables Conclusion Using probabilistic analysis taking into account the variability of basic variables of resistance and actions, the reliability of bolted RHS end-plate joints in axial tension designed according 3 analytical models has been investigated. The systematic development of mechanical joint models by a probabilistic approach to the evaluation of the experimental results may improve the existing models, and thus reliability level of component method may come closer to the required one. 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