TECHNICALSCIENCESAbbrev.: Techn. Sc., No 11, Y 2008DOI10.2478/v10022-008-0008-xTHE PLASTIC EQUALIZATION METHODFOR BENDING MOMENTS AND THE ROTATIONCAPACITY OF PLASTIC HINGES IN CONTINUOUSREINFORCED CONCRETE BEAMS IN LIGHTOF EUROCODE REQUOREMENTSMarek Jdrzejczak1, Micha Knauff21Department of Civil Engineering and Building StructuresUniversity of Warmia and Mazury in Olsztyn2The Institute of Building StructuresWarsaw University of TechnologyK e y w o r d s: civil engineering, reinforced concrete constructions, redistribution of bendingmoments, plastic hinges.A b s t r a c tThe method worked out by JDRZEJCZAK and KNAUFF in 2007 has been used to calculate rotation inplastic hinges of the continuous reinforced concrete beams. It has been stated that when the degree ofredistribution of bending moments does not exceed the allowable limits described in the Eurocode,the rotationdoes not exceedthe limit values indicatedinthe normeither. Using the plasticequalization method for bending moments according to the Polish norm without limiting the degreeof redistribution leads to excessive rotation.METODA PLASTYCZNEGO WYRWNANIA MOMENTW A ZDOLNO DO OBROTUW PRZEGUBACH PLASTYCZNYCH ELBETOWYCHBELEK CIGYCH W WIETLEWYMAGA EUROKODUMarek Jdrzejczak1, Micha Knauff21Katedra Budownictwa i Konstrukcji BudowlanychUniwersytet Warmisko-Mazurski w Olsztynie2Instytut Konstrukcji BudowlanychPolitechnika WarszawskaS o w ak l u c z o w e: budownictwo, konstrukcje elbetowe, redystrybucja momentw zginajcych,przeguby plastyczne.A b s t r a k tDo obliczania ktw obrotu w przegubach plastycznych elbetowych belek cigych zastosowanosposb opracowany przez JDRZEJCZAKA i KNAUFFA (2007). Stwierdzono, e jeeli stopie redystrybucjinie przekracza dopuszczalnychgranic okrelonychwEurokodzie, to kty obrotu rwnie nieprzekraczaj wartoci granicznych wskazanych wtej normie. Zastosowanie plastycznego wyrwnaniamomentw wedug polskiej normy bez ograniczania stopnia redystrybucji wywouje nadmiernekty obrotu.IntroductionAccording to the Polish norm PN-B-03264:2002 it is possible to use severalmethodsof nonlinearanalysisof reinforcedconcreteconstruction. Generalprinciples of the plastic method of analysis and the elastic analysis with limitedredistribution of moments in the Polish norm have been taken from Eurocode 2(1992). Additionally, thePolishnormalsocontains themethodof plasticequalizationformomentsinonewayspanningslabsuniformlyloadedandsecondary continuous beams (point 9.1.2.3 in the norm). This method has beenbased on the 1970s editions of the norm.The principles of the general, nonlinear method based on realistic models ofreinforced concrete are not described in the Polish norm, but can be found inEurocode2(from1999and2008)andamanual(Podstawyprojektowania...2006).The literature concerned with the application of the theory of plasticity tocalculateconcreteslabs, beamsandframesisveryextensive. ThebookbyTICHY and RAKOSNIK (1971) contains 108 items listed in its bibliography. Butthe book of the greatest practical value in Poland was, for a long time, the onewritten by KOBIAK and STACHURSKI (part one issued in 1973, and part 2 issuedin 1979). Both parts of the book, and the numerous articles included in it, didnot, however, paytoo much attention to the problem ofrotation capacity inplastic hinges.Due to the development in reinforced steel production and concrete tech-nology, we now use materials that are much stronger but less ductile than theonesusedat thetimewhentheapplicationof thetheoryof plasticityinreinforced concrete structures was rather rudimentary. This is why it is notwise to use methods assuming the redistribution of bending moments (e.g. incontinuousreinforcedconcreteslabs andbeams) incontemporarydesignswithout checking rotation capacity in plastic hinges.The Polish norm PN-B-03264:2002 states that if redistribution of momentscalculated according to the linear elastic method is applied, it is necessary toprovide the critical cross-sections with the rotation capacity, so that they canThe Plastic Equalization Method... 109adapt to the predicted redistribution. The norm does not contain any informa-tion about the allowable plastic rotation (such information can be found in theEurocodesfrom1992,1999 and2008) orcalculation methodsforsuchrota-tions. TheonlycriterionmentionedinthePolishnormPN-B-03264:2002(quoted after the 1999 edition of the Eurocode) in point 4.4.2 are the conditionsreferringto the allowable degree of the moments redistribution. Detailsconcerning calculating the rotation and the allowable degree of redistributionchanged with each subsequent edition of the Eurocode (1992, 1999, 2008), and,as we have already mentioned, the Polish norm permits the use of the methoddevelopedfortyyearsago. Thus, theset of thecurrentrulesisnot reallycoherent.The method of plastic equalization of bending moments is of great practicalimportance in Poland. This method may be considered a very specific case ofthe linear-elastic method with redistribution, in which the moments in spansand supports have been equalized. In our article (JDRZEJCZAK, KNAUFF 2002)showed that using this method, the conditions concerning the allowable degreeof the redistribution of moments are often (but not always) fulfilled and thenit is not necessary to control the plastic rotation capacity. There is, however,a question if the rotation appearing as a result of the redistribution predictedwith the use of this method do not exceed the values permitted by Eurocode 2(2008). The answer to this question is the subject of this article.The calculation method of rotationThe rotation in a plastic hinge in a continuous beam may be calculatedwith a good approximation with the use of the method presented by (JDRZEJ-CZAK, KNAUFF 2007), by using the following formula =1 1 lW (1)12W dwhere:Wcoefficient depending on static structure, i.e. the number of spans andthe distribution of live load, the degree of redistribution, i.e. the ratio of the redistributed momentto the elastic bending moment,l/d span slenderness ratio (l the length of span, d the effective depth),W coefficientdependingonthereinforcementratioandthecharacter-istics of concrete and steel.Marek Jdrzejczak, Micha Knauff 110Coefficient w is calculated using the formula:W = gg+ qq(2)p pwhere:g is dead load, q live, p total, g and q are coefficients to calculate extrememoments (e.g. according to Winklers table).Table 1 shows the values of W over supports B (g = 0.105, q = 0.119)and C (g = 0.079, q = 0.111) in a five-span beam and over support B ina three-span beam (g = 0.100, q = 0.117) for different ratios of loads q/g.Table 1Values of wq/g 0.5 1.0 2.0 4.0 8.02 3 1A B CA five-span beamw (B) 0.110 0.112 0.114 0.116 0.117w (C) 0.090 0.095 0.100 0.105 0.1072 1A B CA three-span beamw (B) 0.106 0.109 0.111 0.114 0.115Coefficient W is calculated using the following formulaeW = eff(1 0.5eff) fcd, jII =JII, eff = fyd(3)EcdjIIbd3fcdwhere:JII is the moment of inertia of the cross-section in a clear phase II. The valuesof W are presented in diagrams a), b) and c) of Figure 1.The Plastic Equalization Method... 1110.05 0.15 0.25 0.35xeff1234W103C50/60A-III steel10a b53512340.05 0.15 0.25 0.35xeffC12/15A-IIC50/60C12/15W103c d12340.05 0.15 0.25 0.35xeff0.05 0.15 0.25 0.35xeffC12/15W103C50/60A-Iclass C steelclass B steelm C50/6015202530q pl,d103Fig.1.Diagramsforchecking theallowableplasticrotation a, b, c)thevaluesofexpressionW;d)allowable plastic rotation pl,dbasedonfig.5.6NinEurocode2 (2008); (steel classeswhich aredescribed in PN-B-03264:2002 except A-IIIN fulfill the requirements appropriate for B ductility classaccording to the Eurocode 2 from 2008)Figure 1 d) shows the allowable plastic rotation pl,d based on figure. 5.6Nin the Eurocode (2008) calculated for shear slenderness = 3,0. If the shearslenderness is different, then the value of allowable plastic rotation pl,d takenfrom figure1d) hasto bemultiplied bykobtainedfrom: k= /3,where = aq/d, and aq is the distance between point zero of the moments diagram andthe support. For the sake of simplification, it can be assumed that = MSd/(Vd).Rotation in plastic hinges after equalizing the momentsInthemethodof plasticequalization, bendingmomentsarecalculatedusing the following formulae:Marek Jdrzejczak, Micha Knauff 112in the first and last spansand on the second and last but one supports:M1 = (g + q)l2eff = 0.0909 pl2eff(4)11in intermediate spans and in intermediate supports:M2 = (g + q)l2eff = 0.0625 pl2eff(5)16If live is equal to dead load (with different load ratios q/g, the values aredifferent as well, but as show in Table 1 the influence of the q/g ratio has nosignificant meaning), then the coefficients w and the degree of redistribution as well as the coefficients of shear slenderness for the five-span beams are:over the second and last but one supports (B):w = 0.112, =MR= 0.0909pl2eff= 0.0909= 0.8116Me(p) wpl2eff0.112 = MR= 0.0909pl2= 0.1818lVd 0.5pld dover the intermediate support (C):w = 0.095, = 0.0625pl2eff= 0.0625= 0.6579wpl2eff0.095 = 0.0625pl2= 0.125l0.5pld dThe corresponding values for the three-span beams are:w = 0.109, = 0.0909= 0.8339, = 0.1818l0.109 dThe Plastic Equalization Method... 113Table 2 shows rotation over supports B and C in the five-span beam andover support Binthethree-spanbeams, as well as theallowableplasticrotationaccordingtotheEurocode(2008) for four values of effandfourslenderness ratios of the spanl/d. Calculations have beendone for B25concrete and A-III steel. Table 3 shows a sample calculation for support B ofa five-span beam, when eff = 0.05 and = 2.0 (formulae according to Podstawyprojektowania... 2006).Table 2Rotation over supportsFive-span beamBC[mrad] [mrad]Three-spaneff beamB [mrad]k pl,d[mrad](1) (2) (3) (4) (5) (6)0.051.0 1.80 2.03 7.82 6.702.0 3.59 4.06 15.65 9.473.0 5.39 6.10 23.47 11.604.0 7.19 8.13 31.30 13.390.101.0 1.92 2.17 8.36 8.022.0 3.84 4.34 16.73 11.343.0 5.76 6.52 25.09 13.904.0 7.68 8.69 33.46 16.050.151.0 2.00 2.27 8.72 7.512.0 4.00 4.53 17.43 10.613.0 6.01 6.81 26.15 13.004.0 8.01 9.07 34.87 15.010.201.0 2.07 2.34 9.01 6.642.0 4.14 4.68 18.02 9.393.0 6.21 7.02 27.03 11.504.0 8.27 9.36 36.04 13.280.251.0 2.12 2.39 9.21 5.662.0 4.23 4.79 18.43 8.003.0 6.35 7.18 27.64 9.804.0 8.46 9.57 36.85 11.320.301.0 2.15 2.43 9.37 4.562.0 4.30 4.87 18.73 6.453.0 6.45 7.30 28.10 7.904.0 8.60 9.37 37.47 9.120.351.0 2.18 2.46 9.48 3.462.0 4.35 4.92 18.95 4.903.0 6.53 7.39 28.43 6.004.0 8.70 9.85 37.90 6.93Marek Jdrzejczak, Micha Knauff 114Table 3Sample calculationEcd = EcmEcd =30= 25GPac1.2e =Ese =200= 8Ecd25= efffcd= 0.05 13.3= 0.0019fyd3501 = e 1 = 8 0.0019 = 0.0152II = 21 + 21 1II = 0.01522+ 2 0.0152 0.0152 = 0.1598jII = 3II+ 1 (1 II)2jII = 0.15983+ 0.0152 (1 0.1598)2= 0.01213 3W = eff (1 0.5eff) fcdW = 0.05(1 0.5 0.05) 13.3= 2.145 10-3jIIEcd0.0121 25 103= MR= 0.0909pRl2= 0.1818ll= 5.5= 5.5 2 = 11Vd 0.5pRld d dk=k=2= 0.8163 3=1 1 lW kpl,d=1 1 0.8116 11 2.145 10-3= 4.06 10-312w d 12 0.112 0.8116 0.816 11.6 10-3= 9.47 10-3ConclusionAsshownintheanalysis(resultspresentedinTable2),whileusingtheplastic equalization of bending moments, rotation in plastic hinges, which maythen appear, are unacceptable according to Eurocode 2 (2008) (i.e. the rotationin columns 3, 4 or 5 are bigger than in column 6 of Table 2). Thus, this method even though the norm PN-B-03264:2002 presents no objections to it cannotbe used without checking the rotation in plastic hinges.Comparing the results with the conclusion of the article by JDRZEJCZAK,KNAUFF (2002), we have to state that the simple conditions offered in the normPN-B-03264:2002,whichexemptusfromcalculatingtherotationinplastichinges (with regard to the linear-elastic method with redistribution), are alsoappropriate for the methodof plastic equalizationfor bending moments,because the values of rotation presented in Table 2 are exceeded only when thenormPN-B-03264:2002doesnot exemptusfromtheobligationtocontrolThe Plastic Equalization Method... 115these rotations. Exceeding the allowable plastic rotations appears, for example,in a three-span beam with significant reinforcement (eff = 0.35, when 3)and in a five-span beam over the central support C, when q/g > 0.5.ReferencesEurocode 2: Design of Concrete Structures. Part 1-1: General Rules and Rules for Buildings. 1992.Eurocode 2:Design ofConcrete Structures. Part 1-1: General Rules andRules for Buildings, July1999.Eurocode 2: Designof Concrete Structures. Part 1-1: General Rules andRules for Buildings,December 2008.JDRZEJCZAKM., KNAUFF M. 2002. Redystrybucjamomentwzginajcychwelbetowychbelkachcigych zasady polskiej normy na tle Eurokodu. Inynieria i Budownictwo, 8.JDRZEJCZAK M., KNAUFF M. 2007. Kty obrotu w przegubach plastycznych elbetowych belek cigychw zalenoci od stopnia redystrybucji momentw. Inynieria i Budownictwo, 12.KOBIAK J., STACHURSKI W. 1973. Konstrukcje elbetowe cz 1. Arkady, Warszawa.KOBIAK J., STACHURSKI W. 1987. Konstrukcje elbetowe cz 2. Arkady, Warszawa.PN-B-03264:2002. Konstrukcje betonowe, elbetowe i sprone. Obliczenia statyczne i projektowanie.Podstawy projektowaniakonstrukcji elbetowychi spronychwedugEurokodu2. 2006. Pracazbiorowa koordynator M. Knauff. Dolnolskie Wydawnictwo Edukacyjne, Wrocaw.TICHY M., RAKOSNIK J. 1971. Obliczanie ramowych konstrukcji elbetowych z uwzgldnieniemodksztace plastycznych. Arkady, Warszawa.Accepted for print 2.08.2008 r.Marek Jdrzejczak, Micha Knauff 116