Acta Polytechnica Vol. 43 No. 112003 Reliability of Concrete Elements Designed for Alternative Load Combinations Provided in Eurocodes M. Holickf, J. Markov6 The basic European standnrd for design of buildings and other engineering worhs, EN 1990 "Basis of structural dai,gn", proaid.es alternatiae design procedures, for uhich national choice is alloued. One of the most irnportant questi,ons concer"ns three fundnmental combinations of actions for persistent and transienl design situations in the Ultimnte limit states. Simple examples of reinforced concrete elements show, that the alternatiae load combinations ma) lead to considerably dffirent relktbility leaek. Probabilistic methods of structural reliability theory are used to ilcntifi characteristic features of each combination and to formuLate recommendations. Houeoer, further calibration studies are urgently needed in order to prepare NationaL annexes to EN I 990 on time. Keyutords: Eurocodes, cornbination of actions, partiaL factors, reinforced concrete beam, coLwnn, reLiabiLity index. I Introduction -fhe operational European standards for the design of structures, so called Eurocodes, are currently in an advanced stage of development. During the next few years, most Europ- ean countries should be prepared to use newly developed l-rarmonised standards for structural design. The original intention of the European Committee for Standardisation CEN that the design may differ in individual countries only in the numerical values of some parameters, such as partial factors and characteristic values of climatic actions, has not been fulfilled. Recently available documents show that the foreseen high level of harmonisation has not been reached, and that the national institutions will be free to choose not only the numerical values of the various reliability elements, but in some cases also one of the alternative design procedures. Since April 2002, two basic documents EN 1990 [l] and EN 1991-l-1 [2] have been available. These tlvo standards resulted from transformation of the relevant prestandards [3] and [4]. The final draft of the standard EN 1992-l-l [5] for the design of concrete structures, which relates to the prestandard [6], is now also available. In the next hvo years, these transformed documents will be implemented in the Member States of CEN, together with the National annexes' as operational national standards. Existing valid standards, that are in contradiction with the CEN documents, will grad- ually be modified or withdrarvn. -l-his process will require important national decisions that should be based on well- -prepared calibration. The National annexes should include the recommend- ation of one of the alternatives indicated in EN 1990 for a fundamental combination of actions in the Ultimate limit states and partial factors y,, and 1n for permanent and variable actions. It will be shown that the choice of these nationally determined parameters may significantly affect the resulting reliability level. The reliability of two basic reinforced concrete elements, a beam and a column, are analysed taking into consideration alternative load combinations and possible variation of the partial factors y. and yq. In this way the presented study extends a previous similar study [7] that concerns structures made from different materials. However, contribution [7] is related to an earlier version of Eurocode ENV 1991-l [3]. Additional updated reliability studies are therefore required [..g., 8, 9] to develop comprehensive background materials enabling national decisions concerning load combinations and partial factors to be applied in accordance with the present version ofEN 1990 [l]. 2 Fundamental load combination In the following analysis, the combination of three actions is considered: permanent action G, imposed load Q fleading) andwind I4l(accompanying). EN 1990 [l] for the fundament- al combination of these loads in permanent and transient design situations introduces three alternative procedures denoted here A, B and C. Assuming linear behaviour of structural members, actions G, QandW and their characteris- dc values G1, Q1 and tr42. denote generally appropriate load eflects (not the original actions). A. Considering the formula (6.10)in EN 1990 [l], the design value of action effect E6 is given as Ea =ycGu + yq Qr. + Tw,!twWu.. (1) B. An alternative procedure is provided in EN 1990 [l] by twin expressions (6.10a) and (6. 10b) Ea =ycGl +f etteQa.+twVwWu (2) Ea =6ycGt + yqQr +^twrywWu.. (3) The less favourable action efl'ect from (2) and (3) should be considered. C. In addition EN 1990 [] allows further modification of alternative B, simplifing equation (2) by considering perma- nent loads only, thus the load effect is then (4)Ea =lcGr 29 Acta Polytechnica Vol. 43 No. ll200g The less favourable action effect resulting fiom (3) and (4) is then considered. In addition to the combinations A, B, C provided in EN 1990 [] (for recommended values f c= 1.35, Ts= 1.5) an additional combination D is also considered in the analysis as indicated in Figs. I and 3. The load combination D is the modified combination A (equation 6.10 in []) when reduced partial factors (y"= t.2,TS= 1.4) recommended in the Czech implementation of ENV l99l-l [3] are considered. This combination should illustrate the sensitivity of the result- ing reliability level to partial factors, and the possible effect of reducing them. If the leading action is windW!then in equations (l) and (2) instead of reducing wind action W by factor Va the imposed load Q should be reduced by the appropriate factor Vq. Factors y", yq and y, denote the partial factors ofactions G, Q and W (the partial factors for both variable actions are equal, yr=y*). To investigate resulting load effects under various in- tensities of variable actions, the characteristic values of Gu, Q1 and l4lu are related using quantities 1 given as the ratio of variable actions Qn+Wuto total load Go+Qu+Wr, andratio k of accompanying action l{zu to the main action Q. x =(Qr. +w1,)l(Gy + Qr + Wk), k =WxlQ'.. (5) Note that a realistic range of 1 is from 0.1 to 0.6. However in some cases the load ratio ;g may be very low if not zero (e. g. underground garages). For a given design value of the load effect Eo the characteristic values of individual actions Go, Qu, Wu can be expressed using variables 26 and ft as follows Gl= (€)vc + ((*e)te + h(vilvw)x (t+a)(l-1) o, = xck '1k =G;;X-' (6) W,^=kQv. The factors (, 1" and yq indicated in the first relationship of (6) in brackets are applied in the sameway (eitheryes or no) as in equations ( I ) to (a) for alternative combination rules Ao B and C. For alternative A, equation (l) is valid in the whole range 0 Eo). In the reliability analysis a structure is usually considered to be safe if resistance rR is greater than load eflect E, both considered as random variables. Thus, when the limit state function (reliability margin) C(X) = n - E is greater than 0, X being the vector ofbasic variables. In the case ofa beam the limit state function can be written as ( A" f.,\ g(x) = K nArfvl h -dr-0s-::t l- xu(c + Q+w) , (r0)'\. oJ, ) where Kp and K6 are coeflicients of model uncertainties for resistance -R and action effects E. Note that all variables in equation (10) are considered as random variables having a certain type of probability distribution. 3.2 Reinforced concrete column The design value of resistance .Ro for the short reinforced concrete column assuming very small (negligible) eccentricity and cross-sectional dimension h x b is given as Rd =A't'- a gg1t6 df'u (l l) Ym The column parameters are determined considering again the "economic design" when Itr=Eo. In the case of the column the limit state function can be written as a condition given as s(x) = K *(A,f, + os hbf,) - x n(G + Q + w), (12) where K4 and K6 are again coe{ficients of model uncertainties for variables R and E. |, 4*) * =*l h-drobr' il | ,n,t 'u"+:) F. . y6(t - l)(t + *) xri^'c =, "11 - ;y1 a PS 1 y ra + yry M' (8) Acta Polytechnica Vol. 43 No. 112003 4 Principles of reliability analysis The probability of failure P,. is the basic reliabilitv measure used in this study. It can be expressed on the basis of a limit state (performance) function g(tr) defined in such a way that a structure is considered to survive if g(D > 0 and to fail if C (I0 < 0. An example of the function g(X) is given by equa- tions (10) and (12). In a general case failure probability Prcan be determined using the integral cad and Matlab software products were used to develop the simple programs applied in the following analysis. Note that there are commercially available software products (e.g., VaP [0], COMREL tl ll), which can be used to determine the failure probability P,. in more complicated cases than those considered here (when expression (14) can- not be used). These software products were used in this study to check results obtained by numerical integration based on expression (14). An alternative measure of reliability level is the reliability index B (see Annex C of EN 1990 [1]), which is related to the probability of failure P,. as P/ = @(- B) (r5) rvhere O is the cumulative distribution function of the standardised normal distribution. The reliability index B is fiequently used, as its numerical values are more comfortable to handle than values of failure probability Pp' The relation- ship beween Pr and B is illustrated by the numerical values indicated in Table 1. 'fable 1: Relation between B and Pi Pt i ro-' l0-2 l0-i I 0-4 l0-5 I 0-6 l0-7 p 1.28 2.32 3.09 3.72 / cra 4.75 5.20 EN 1990 [] recommends for the Ultimate limit states of buildings designed for a fiftv year period a target value of reliabilitv index p,= 3.8 (for a one year period 9,= 4'7), which corresponds to the probability of failure P- 7 .24'l}-t' . 5 Probabilistic models of basic variables As rnentioned above in the reliability analysis all basic variables are considered as random variables having a certain tvpe of probabilitv distribution. The probabilistic models of basic varliables X used in this study are summarized in Thble 2. Pr = Prob(g(x) <0) = f q*(xlax, c(x)<0 ( 13) rvhere guQQ denotes the joint probability density distribution of the vector of basic variables X, which may, horvever, not be available. Assume that both the resistance R()Q and the load efTect [()Q represent a single variable Z used to analvse structural performance (e.g., axial fbrce, bending moment or stress tlrat is reprcsented by R(Z) and E(Z)). -I'hen the integration indicated in expression (13) may be sirnplified and the prob- ability P, can then be expressed as: |4\ where q1(Z) denotes the probability density function of E(Z)' On(Z) the distribution of R(Z). -I'o use equation (14) both the probability density function gp(Z) and the distribution function On(Z) rnust be known (at least in an approximate form). A procedure based or.r expression (14) is used in this study. Simplified, a tirne independent load model is accepted, using'lurkstra's rule (the nlain action G is described by an extreme value distribution for the considered design life of' a structure, the accompanying wind load W is approximated by the distribution of the annual extremes) is accepted. N'lath- Table 2: Probabilistic models of basic variables f Pr = I'r,,b(g(Z ) < 0)= I q n(z\ @ R(7.)dZ, J -@ Type of variable Symbol X Basic variable Distr. Units Charact.value tr o Action c Permanent action N MN/m Gk Gk 0.1 Gk o Imposed (50 years) GUM MN/m2 O,. o'6 Qt 0.2lQk W Wind (l year) CUM MN/m' 1,1/'. 0.3 wk 0.15Wk W Wind (50 years) GUM MN/m: wr 0.7 wk 0.245Wk Marerial Properties A Reinforc. area DE-f 2m nolTl nom 0 T Concrete strength LN MPa 20 28 4 t, Reinforc. strcngth LN MPa +.)5 5r)u 30 Geometric data h Beam height N m 0.6 0.6 0.008 h,b Column dimensions N m 0.3 0.3 0.0r )t,l Reinforc. distance GAM m 0.03 0.03 0.006 Model uncertainties KF, Load uncertainty LN 1.0 1.00 0.05 KR Resistance uncert. LN 1.0 r.10 0. l5 cl Acta Polytechnica Vol. 43 No. 112003 The models of basic variables indicated in Thble 2 are chosen taking into account data provided byJCSS [2]. Note that the beam width b:0.3 rn and coefficient of long-term concrete strength ct, = I are considered as deterministic values. 6 Results of reliability analysis The results of the reliability analysis, indicated by the reliability index p (equation (14)) are shown in trigs. I and 2 for the reinforced concrete beam, and in Figs. 3 and 4 for the column. In both cases, the load ratio of variable actions I = 0, which proves to be more critical than I > 0, as shown in [9]. The resulting reliability levels shown in trigs. I to 4 should be considered only as indicative and relative values that are obviously dependent on assumed characteristics of basic vari- ables shown in Table 2. Particularly the model uncertainties K^ and Ku have significant effects. Characteristics of these variables are chosen taking into account available data provided byJCSS [12], where the mean 1.20 (greater than Fig I . l0 given in Table 2 for K*) and the coeflicient of variation 0.15 (the same as given in Table 2) are indicated. Note that if "0 Fig. l: Reliability index B for the reinforced concrete beam versus the load ratios l for reinforcement ratio P = 0,01 and I = 0; A, B, C - alternative combinations according to EN 1990 (y"= 1.35, yq= 1.5)' D - combination Awith reduced part- ial factors (\c= 1.2, Ya= 1.4). p Fig. 2: Reliability index B for the reinforced concrete beam versus reinforcement ratio p for ratios I=0.3 and &=0; A, B - alternative combination according to EN 1990 [l] Q6= r'35, Ye= l'5)' 32 the mean 1.20 is used instead of 1.10, would increase considerably (reliability crease by about 0.5). 5 k qF ./l ^lUI I't" I X,rim, c I X,lim. B x 3: Reliabilitv index B for the reinforced concrete column versus the load ratios 1 for reinforcement ratio P = 0.04 and ft = 0; A,B,C - alternative combinations according to EN 1990 [] (yc= 1.35, yg= 1.5), D - combination A with reduced partial factors (^yc= | .2, f q= 1.4). v Fig. 4: Reliability index B for the reinforced concrete column versus reinforcement ratio p for X, = 0.3 and t = 0; A' B' - alternative combination according to EN 1990 [l] (Yc = 1.35, Ys= 1.5) Another important aspect of the obtained results is the dependence of the reliability of reinforced concrete elements on their reinforcement ratio p. As indicated in Figs. 2 and 4, the reliability of the beam GiS. 2) increases with increasing reinforcement ratio p (reliability index p may easily increase by about l), the reliability of the column (Fig.4) decreases with increasing reinforcement ratio p (reliability index p may decrease by about 0.5). For this reason the reinforcement ratio p = 0.01 is considered for the beam in trig. I, and p = 0.04 is considered for the column in Frg. 3. tigures I and 3 show that the reliability level of both the beam and the column determined for load combination A is greater than the reliability resulting from load combinat- ions B, C and D. The lowest reliability level is provided by load combination C, allowed in EN 1990 [l], and by load combination D when reduced Partial factors for actions are applied (Yc=1.2, Ts=l.S instead of Tc= 1.35, yn= t.51. A comparison of ligs. I and 2 further indicates that the reliability level of the beam (trig. l) is slightly lower than the the reliability level index B would in- 0.80.60.40.2-0 ^< 0.40.2 I I -A -,'R L/4 'c t-g.:.'Y- Pt- _:\ ; | -..,.. ... 3.8 I Xlim,c I I 1?(ti.,t B 3.8F' Acta Polytechnica Vol. 43 No. 112003 reliability level of the column (Iig. 2). Reliability index B for the beam may be lower (by about 0.3) than p for the column. This frnding is however strongly dependent on the model uncertainties K* and K", and should not be generalised. For both the beam and the column the load combinations C and D lead to reliability approaching the level recom- mended in EN 1990 (F,= 3.8), in particular for very low load ratios 1 (when the permanent load is dominant) and for load ratios 1 greater than 0.6 (when the imposed load is more significant than the permanent load). Alternative B provides the most uniform reliability level within the expected load ratios X (a realistic range of 1 is from 0.1 to 0.6) and from this point of view seems to provide the best load combination. 7 Concluding remarks The newly available EN 1990 provides alternative design procedures and parameters that should be unambiguously specified in the National annexes of Member States of CEN. These alternative design procedures lead in some cases to significantly different reliability levels. Preparation of Nation- al annexes is therefore a complicated task for each Member State. Furthermore, the Eurocode standards recognise the responsibility of the regulatory authorities in each Member State and safeguard their right to determine values related to regulatory safety matters at national level. Simple examples of reinforced concrete elements con- firm the results of the earlier studies that the reliability of structures, designed according to the alternative combination rules provided in EN 1990 by expressions (6. l0), (6. l0a) and (6.10b) may vary considerably. Expression (6.10) leads to the most reliable but in some cases to uneconomical structures. Twin expressions (6. l0a) and (6.10b) provide a lo*'er but comparatively most uniform reliability level for all load ratios. Moreover, they seem to fully comply with EN recommend- ations (reliability index 3,8 for a 50-year time period). The lowest reliability is obtained from the third alternative, given by modified expression (6,10a) and expression (6.10b). This alternative seems to lead to a rather low reliability level, particularly for structures exposed rnainly to a permanent load. An important aspect of reliabiliry of reinforced concrete elements is the reinforcement ratio p. It appears that the reliability of the beam increases considerably with increasing reinforcement ratio p (reliability index B may easily increase by about l), the reliability of the column decreases with increasing reinforcement ratio p (reliability index B may de- crease by about 0.5). In order to make an unambiguous recommendation for National annexes to EN 1990, further investigations are urgently needed. Obviously more complicated structural ele- ments, made of various materials, should be analysed and compared. Such a calibration activity should preferably be organised on an international level. The short-term objective of these activities should be to develop the necessary back- ground materials for preparation of the National annexes' The long-term objective should be to further harmonise the alternative design procedures to be considered during the next revision ofthe present generation ofEurocodes. Acknowledgement This study has been partly prepared at the Klokner Institute of the CTU in Prague, Czech Republic as a part of the research project CEZ: J04l98 : 2 I 0000029 "Risk Engine- ering and Reliability of Technical Systems" supported by MSMT References t I I EN I 990: 2002 Eurocodz - Basb of StnrcturaL Design. Euro- pean Committee for Standardisation, April 2002, p. 87. i2l EN l99l-l-l: 2002 Eurocofu 1: Actioru un Structures - Part l-l: Gmeral Actions - Dmsitizs, Self-weight, and. Irn- posed Loa.ds for Buildings. European Committee for Standardisation, April 2002, p. a6. t3l ENV l99l-l: Basis of Design and Actions on Sttuctures. Part 1: Ba"sis of Design European Committee for Stan- dardization, 1994. t4l ENV I 991-2- I : Actions on Structures - Parl 2- I : Gmeral Ac- tions - Densitizs, Self-ueight and Imposed Ina.ds. European Committee for Standardization, 1995. t5l prEN 1992-l-ll. Design of Cancrete Structures. Gmeral Rules and Rules for Buil.dings. CEN/IC 250/SC2, Draft for Stage 49, July 2002, p. 46. t6l ENV 1992-l-l : Design of Cancrete Structures. General Rules and Rules for Buillings. European Committee for Stan- dardization, 1993, p. 46. t7l SAKO;Joint Committee of NKB and INSTA-B . Basis of Duign of Structures. Proposal for Modifuation of Partial Safetl Factors in Eurocodes. 1999, p. 55. t8l Holicki, M., Markov6, J.: Verifuation of Load. Factors for Concrete Components b1 Reliabili$ and Optirnimtion Armlysb: Bachground Documsnts for Implemmting Eurocodcs. Prog- ress in Structural Engineering and Materials, Vol. 2, 2000, No. a,p.502-507. tgl Gulvanessian, H., Holickf, M., Markov6, J.: Cahbration of Eurocodz Relinbility Elemmts Cunsidering Steel Mernbers. In: Proceedings ofThird European Conference on Steel Structures, Volume II; Antonio Lamas and Luis Simoes da Silva, CMM - Associacao Portuguesa de Construcao Metalica e Mista, Guimaraes (Portugal), ISBN 972-98376-3-5, p. l5ll-1520. [0] VaP, Variable Processor, version I.6, ETH Ztirich, 1997. [1] COMREL, version 7.10, Reliability Consulting Pro- grams, RCP MUNICH, 1999. tl2l JCSS, Joint Committee for Structural Safety, http ://www jcss.ethz.ctr/, 2002. Prof. Ing. Holickf Milan, DrSc. Phor.re: +420 224 353 842 Fax: *420 224355232 e-mail: holicky@klok.cvut.cz Ing. Markovii Jana, Ph.D. Department of Reliability Czech Technical University in Prague, Klokner Institute A r.Sollnova / 166 08 Prague 6, Czech Republic 33 Scan 29 Scan 30 Scan 31 Scan 32 Scan 33