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1 Doument downloaded from: Thi paper mut be ited a: Bonet Senah, JL.; Romero, ML.; Miguel Soa, P. (2011). Effetive flexural tiffne of lender reinfored onrete olumn under axial fore and biaxial bending. Engineering Struture. 33: doi: /j.engtrut The final publiation i available at Copyright Elevier

2 Effetive flexural tiffne of lender reinfored onrete olumn under axial fore and biaxial bending J.L. Bonet a, M.L. Romero a, *, and P.F. Miguel a a Intituto de Cienia y Tenología del Hormigón (ICITECH). Univeridad Politénia de Valenia, Spain ABSTRACT Mot of the deign ode (ACI and Euro Code ) propoe the moment magnifier method in order to take into aount the eond order effet to deign lender reinfored onrete olumn. The auray of thi method depend on the effetive flexural tiffne of the olumn. Thi paper propoe a new equation to obtain the effetive tiffne EI of lender reinfored onrete olumn. The expreion i valid for any hape of the ro etion, ubjeted to ombined axial load and biaxial bending, both for hort-time and utained load, normal and high trength onrete, but it i only uitable for olumn with equal effetive bukling length in the two prinipal bending plane. The new equation extend the propoed EI equation in the Biaxial bending moment magnifier method by Bonet et al [6], whih i valid only for retangular etion. The method wa ompared with 613 experimental tet from the literature and a good degree of auray wa obtained. It wa alo ompared with the deign ode ACI-318 (08) and EC-2 (2004) improving the preiion. The method i apable to verify and deign with uffiient auray lender reinfored onrete olumn in pratial engineering deign appliation. Key word: reinfored onrete, olumn, biaxial bending, trength, deign. Correponding author: Tel: (ext:76742) Fax: , addre: mromero@me.upv.e 1

3 1 INTRODUCTION The ode ACI-318 [1] and Euro Code 2[2] propoe the moment magnifier method in order to take into aount the eond order effet to deign lender reinfored onrete olumn. The auray of thi method depend on the effetive flexural tiffne EI of the olumn. Suh parameter depend on raking, reep and non-linear material behaviour. Over the lat three deade, many author and national ode have propoed different method to determine the olumn tiffne for hort-time and utained load. Thu, the ACI-318 (08) ode [1] propoe an equation that i independent from the load applied to the olumn. However, the EC-2 ode [2] and mot author, uh a Mavihak and Furlong [3], Mirza[4], Weterberg [5], Bonet et al[6], Tikka and Mirza[7],[8] and o on, laim that flexural tiffne EI depend on the load applied by mean of the relative eentriity or ele through the axial load. Table 1 ompare the different EI equation from the literature and the deign ode. A it i hown, there i no homogeneity between the different propoal regarding the variable analyzed and the funtion ued. Mot of the propoed EI equation by thee author are only appliable for retangular ro etion. Only, Ehani et al [9] and Sigmon et al [10] propoe an equation of EI for irular etion and intantaneou load. Suh author agree that the EI formula propoed by the ACI-318 (08) [1] i very onervative for thi type of olumn. Furthermore, mot of the EI equation were obtained for normal trength onrete. Sine, the mehanial behavior of high trength onrete annot be extrapolated from the normal trength one, it i neeary to update the appliability of uh expreion to any range of trength. 2

4 In thi paper a new equation to alulate the tiffne EI in reinfored onrete olumn with any ro etion hape ubjeted to axial load and uniaxial bending i propoed. It attempt to fill the gap in the equation preented in the bibliography beaue they are only valid for retangular and normal trength onrete, and in pratie there are etion with different hape: retangular, irular, ovoid, ro hape, hexagonal or thin-walled box. Moreover, many reinfored onrete etion are ubjeted to biaxial bending and axial load a a reult of their poition in the truture, the hape of the ro-etion or the oure of the external load. For thoe ae, the ACI-318 (08) ode [1] amplifie the firt order bending moment in eah flexure plane independently. The deign of the ro-etion of the olumn i baed on thee magnified fore. Although the EC-2 ode [2] alo magnifie the bending moment eparately in eah diretion, the deign i performed uing the load ontour method by Breler [11]: M M tx ux γ M + M ty uy γ 1 (1) where M ux, M uy are the nominal bending moment trength around the x and y axe, repetively. M tx, M ty are the nominal bending moment that are applied in the ritial ro-etion of the olumn onidering the eond order effet. γ axial load ontour exponent. It depend on the hape of the roetion. For biaxial bending thoe method an produe unafe ituation of deign for axial load level loe to the the ultimate axial load of the olumn if the mot important bending fore orrepond to the diretion of the lower lenderne (bending with 3

5 repet to the trong axi). Suh effet wa onfirmed experimentally by Pallare et al [12]. Suh method do not take into aount the interation that both urvature have in the trutural behavior of the member. Hene, Bonet et al [6] propoed the Biaxial bending moment magnifier method, where an equation of the effetive flexural tiffne wa introdued for biaxial bending and retangular etion. It inluded the interation between both axe of bending. Thi paper extend the propoed EI equation by Bonet et al [6] whih wa valid only for retangular etion to any hape of the ro etion. The novelty again i foued in the addition of the interation between both axe of urvature, in ditintion from what the method from the ACI-318 ode [1] and the EC-2 ode [2] do. A new equation of EI for biaxial bending i propoed, beaue it doe not exit in the literature for a general ro-etion hape. The method will be limited to the ae where the effetive bukling length of the olumn i equal in the two prinipal bending plane. It will be appliable if there are one or two axe of ymmetry (retangular, thinwalled box, ovoid, or C-hape ro etion), but alo if there i not any ymmetry ( L ro etion p.e.). The biaxial bending moment magnifier method wa baed on the magnifiation of the firt order bending moment applied in the ritial etion of the olumn: M t = δ M (2) n d where M t i the total vetor modulu for deign M = M + M (3) t 2 tx 2 ty M d vetor modulu of the firt order bending moment M = M + M (4) d 2 dx 2 dy δ n magnifiation fator 4

6 δ 1 n = > 1 1 N N (5) d r N d deign axial load N r ritial bukling load, whih i a funtion of the flexural tiffne of the olumn EI and of the effetive length (l p ) N r 2 π EI = (6) l 2 p The effetive flexural tiffne EI of the olumn repreent the equivalent tiffne of a fititiou olumn with ontant tiffne, whoe effetive bukling length (l p ) and ritial axial load (N r ) agree with thoe of the real olumn. Suh olumn flexural tiffne EI repreent the global behaviour of the total element and not of jut one partiular etion. The flexural tiffne EI equation wa inferred from the reult obtained with the numerial imulation deribed in the next etion. The adjutment of the propoed equation wa ompared with 613 experimental tet from the literature. 2. NUMERICAL SIMULATION The flexural tiffne EI of the olumn wa obtained from the utiliation of a general method of trutural analyi for reinfored onrete uing finite element. Thi numerial method inlude the following main iue: 1-D finite element with non-ontant urvature: the finite element ha 13 degree of freedom (d.o.f ), Marí [13]. Thi element ha three node, with 6 d.o.f. in the initial and final node (three rotation and three diplaement), while the mid-pan node ha only one degree of freedom in the axial diretion to apture the variable urvature of the element, Figure 1.a. 5

7 The numerial integration of the ro etion i performed uing the Green theorem, Bonet et al. [14], Figure 1.b. Non-linear onrete behaviour (Model Code-90[15], CEB-FIP [16]) Non-linear teel behaviour: bilinear diagram. (ModelCode-90[15]) Geometri non-linearity: The geometri tiffne matrix and the update of the diplaement are inluded in the definition of the model. Time-dependent effet: reep and hrinkage (CEB[17],[18]) The numerial model wa verified with 613 tet from the bibliography ([3], [19-41]). The experiment orrepond to reinfored onrete olumn pinned-pinned ubjeted to axial load and both to uniaxial and biaxial bending. In thoe tet, the magnitude and the diretion of the eentriity are fixed, evaluating the maximum axial load of the olumn. The hape of the ro-etion are retangular, quare, box with one or two ell, ovoid, C -etion or L -etion. The length of the olumn and the ize of the ro-etion are the ame than the experiment. A even number of finite element were ued beaue the applied load wa ymmetri. Moreover, it wa verified that with a length of the finite element equal to the height of the etion the reult obtained had reaonable auray. Table 2 how the variation of parameter tudied in the experiment. The auray of the numerial model i evaluated through the ratio between the axial load of the tet N tet and the axial load from the numerial imulation N NS. Table 3 preent the auray of the numerial model for both the type of load (hortterm and intantaneou) and the type of the ro-etion (retangular or nonretangular). Table 4 how the auray for the type of urvature (uniaxial or biaxial bending) and for the type of load. It an be een that that an average ratio of 1.06 (afe ide) and a variation oeffiient of 0.13 wa obtained when all the ae are analyzed. 6

8 The atter of the reult i the typial for thi type of laboratory experiment. It wa verified that the degree of auray i imilar for all the parameter onidered in thi tudy. The previou alibrated numerial model wa ued here to perform the analyi of the main variable that exert an influene on the tiffne EI. Table 5 how the analyed parameter and their variation oeffiient, whih when ombined produed 7360 numerial tet. 3. PROPOSAL OF A FLEXURAL STIFFNESS EI a) Flexural Stiffne of a olumn for axial load and uniaxial bending under hortterm load. The etimation of the tiffne EI of the olumn ubjeted to hort-term load i obtained through the well-known equation: EI = α E I + E I (7) where E i the hort-term eant elati modulu of the onrete and equal to ( f /10) (in MPa), where f m m i the mean ompreive trength of onrete (in MPa); E i Young modulu of reinforement and equal to MPa; I, I are the moment of inertia of the gro etion of onrete and of the longitudinal reinforement with repet to the entre of gravity of the gro etion and, in thi reearh, α i termed effetive tiffne fator. Thi oeffiient need to be adjuted againt numerial reult. The onrete and teel elati modulu were obtained from the Euro Code-2 [2]. A mean trength f m equal to the trength of the onrete from the numerial tet f wa hoen to perform the fitting of the oeffiientα. If a numerial imulation (N.S.) i performed for a lender olumn (λ m 0) ubjeted to axial load and uniaxial bending, the ultimate firt order bending moment (M 1 ) NS an be 7

9 obtained for a partiular axial load N i. Likewie, it i alo poible to ompute the ultimate bending moment (M u ) NS of the ro-etion of the olumn (λ m = 0) for the ame axial fore, Figure 2. From both value the effetive tiffne fator α an be alulated by performing the following tep in equene: a) Firt, the magnifiation fator i obtained: ( δ ) = ( M ) ( M ) (8) n NS t NS d NS b) Thi value allow the ritial bukling load of the olumn to be omputed by reordering equation 5: (N r ) NS N i = 1 1 ( δ ) (9) n NS ) The flexural tiffne of the olumn an be omputed from equation 6: 2 ( Nr ) NS l p ( EI ) NS = (10) 2 π d) Finally, the effetive tiffne fator α an be obtained from equation 7: ( EI ) E I NS (α ) SN = (11) EI Figure 3 preent (a an example) the α oeffiient graphially in term of the firt order relative eentriity η and of the mehanial lenderne λ m, for the partiular ae of a irular etion with 12 reinforing bar, mehanial reinforement ratio (ω) equal to 0.5, and for a onrete trength of f =30 MPa. The firt order relative eentriity an be omputed a: η ( M ) /N 4 i 1 NS i 0 = = (12) e 4 i 8

10 where i i the radiu of gyration of the onrete etion with repet to the axi of bending and e 0 i the firt order eentriity. The effetive tiffne fator α preent a non-linear behaviour in term of the relative eentriity η and the lenderne λ m. In fat, the effetive tiffne fator α i independent of the lenderne λ m if the relative eentriity η i equal to 0.2, a an be dedued from Figure 3. It an alo be inferred from thi figure that the performane of α i different if the relative eentriity η i lower or higher than 0.2. Thu, if η i higher than 0.2 α dereae and i appreiably independent of the lenderne and an be approximated by only one traight line. Otherwie, α depend trongly on the lenderne and ha to be approximated by traight line whoe lope i non-ontant in term of the mehanial lenderne λ m. For high value of the relative eentriity (η > 0.2), the failure i produed by the ultimate trength of the etion. Conequently, α i not influened by the lenderne. In thi ae, when η i inreaed, the ro-etion of the olumn reahe higher deformation and it produe a dereae in the tiffne of the olumn. However, for mall value of η and high lenderne, the failure i produed by the intability of the olumn. Therefore, α depend on the lenderne. For thi ae, when the lenderne i inreaed (maintaining ontant the eentriity) the poibility to reah an untable poition i higher and, in onequene, the ro-etion i le deformed and the tiffne inreae. Finally, for mall value of the relative eentriity and lenderne, the fator α dereae in term of the relative eentriity. In thi ae the olumn i very ompreed and the failure i due to the ultimate trength of the etion. Thu, when the relative eentriity η dereae, the olumn ha higher ompreion and the differene 9

11 between the real elati modulu of the material and the tangent elati modulu adopted in equation 7 i higher. The parameter α orret thi differene. The leat quare adjutment of the line α- η from the numerial imulation enable the following equation to be propoed for the effetive tiffne fator (α): f α = ( λm ) ( η 0.2) </ 225 f f α = (0.2 η) </ for for η < 0.2 η 0.2 (13) Figure 4 how a omparion between the method propoed in thi paper and the ode ACI-318[1], Euro Code 2 [2], and alo with the method propoed by Weterberg[5] and Tikka and Mirza [8] for the ame ro-etion (ued in Figure 3). In order to apply the equation from Tikka and Mirza [8] to a irular etion an equivalent height of the etion (h eq ) wa ued, equal to 12 time the radiu of gyration of the onrete irular ro etion (i ). It an be notied that both deign ode propoe an effetive tiffne fator α independent of the relative eentriity η and the mehanial lenderne λ m. However, the other author inlude the dependene of α in term of the eentriity or the lenderne. In general, thee author propoe an equation of α that dereae withη. Only the formula from Weterberg [5] how that for mall value of η, the parameter α inreae with thi parameter. The propoal from thee author onfirm the non-linear behavior of the parameterα. b) Flexural Stiffne of a olumn for axial load and uniaxial bending ubjeted to utained load. 10

12 The tiffne EI equation for utained load i ahieved in a imilar manner to the equation from the previou etion: E E = (14) EI α I + I 1+ ϕ 1+ ξϕ ( λm, ϕ) where ϕ i the reep oeffiient and ξϕ i a redution funtion of the tangent teel elati modulu for utained load. In the equation 14, the elati modulu of onrete (E ) i redued through the expreion E /(1+ϕ). Moreover, the deign value of the modulu of elatiity of the reinforement (E ) i redued with the fator ξϕ. It i a redution funtion that aording to the numerial imulation depend on the mehanial lenderne (λ m ) and on the reep oeffiient (ϕ), in uh a way that if the reep oeffiient i inreaed and the lenderne i dereaed, the teel deformation i redued. Thi funtion i obtained by leat quare from the reult of the numerial imulation: ξ ϕ =.9 ϕ exp( λ 25) (15) 1 m In the end, taking into aount the effet of the reep for mall eentriitie (η<0.2) gave rie to appreiable modifiation in equation 13: f α = ( λm 0.25 ϕ) ( η 0.2) </ 225 f f α = (0.2 η) </ if if η < 0.2 η 0.2 (16) For the ae where the permanent load applied to the olumn i different to the total load, the reep oeffiient (ϕ) from equation 14, 15 and 16 will be replaed by the effetive reep ratio (ϕ eff ). Aording to the laue from the Euro Code 2[2], thi oeffiient i the reep oeffiient time the ratio between the firt order bending 11

13 moment in quai-permanent load ombination, SLS (M 0Eqp ) and the firt order bending moment in deign load ombination, ULS (M 0Ed ). ) Flexural Stiffne of a olumn ubjeted to axial load and biaxial bending. It i important to notie that if the olumn i ubjeted to axial load and biaxial bending, the magnifiation of the bending moment i performed in aordane with the bending plane (Figure 5, equation 2). The equation of the olumn tiffne EI for axial load and uniaxial bending wa expanded for the biaxial ae: E E = (17) EI α Ie + Ie 1+ ϕeff 1+ ξ ϕ where α i the effetive tiffne fator: f α = ( λm 0.25 ϕ eff ) ( η 0.2) </ f f α = (0.2 η) </ if if η < 0.2 η 0.2 (18) η i the firt order relative eentriity: e 4 i 0 d η = = (19) M N 4 i d where e 0 i the firt order eentriity e 0 = M d N d (20) M d i the vetor modulu of the firt order bending moment (Figure 5.b) M = M + M (21) d 2 dx 2 dy M dx, M dy firt order bending moment with repet to the axe of oordinate x and y of the etion, repetively N d deign axial load 12

14 i ritial radiu of gyration of the ro-etion (Figure 5.a). The minimum radiu of gyration of the gro etion with repet of the prinipal axe of inertia i eleted (i u, i v ) λ m mehanial lenderne of the olumn i = min ( i, i ) (22) u v λ = l i (23) m p I e equivalent moment of inertia of the gro etion I e equivalent moment of inertia of the reinforing bar ξϕ redution fator of tangent teel modulu E for utained load ξ ϕ =.9 ϕ eff exp( λ 25) (24) 1 m The equivalent moment of inertia of the gro etion (I e ) and of the reinforement (I e ) are obtained by interpolating the moment of inertia of the etion: ( ) I = I δ + I 1 δ (25) e u v where I u, I v are the moment of inertia with repet to the prinipal trong and weak axi repetively (Figure 5.a) and δ i an interpolating funtion. In order to ompute the diretion whih orrepond to the prinipal trong axi of inertia (Figure 5.a) with repet to the x axi, the following equation mut to be olved: ( I1 I2 ) in( 2 θ p ) 2 o( 2 θ p ) Ixy = 0 θ p (26) where: I 1, I 2 are the maximum and minimum moment of inertia of the etion with repet to the x and y axe of the onrete etion, repetively. 13

15 I 1 = max( I x, I y ); I 2 = min( I x, I y ) (27) I x, I y moment of inertia of the etion with repet to the x and y axe of the onrete etion I xy produt of inertia of the etion with repet to the x and y axe of the onrete etion If the moment of inertia of the etion with repet to the x and y axe are equal, then from equation 26 it i obtained thatθ = π / 4. p Otherwie: 1 2 Ixy θ = p atan (28) 2 I1 I2 The angle (θ p ) i poitive for ounter-lokwie (Figure 5.a) and the mehanial propertie are alulated with repet to the entre of gravity of the onrete etion. In the equation 26, it wa onidered that the prinipal axi with higher inertia agree with the x axi (I 1 = I x ). Otherwie, the angle θ p will be inreaed in π/2. The prinipal moment of inertia of the etion with repet to the trong axi (I u ) and weak axi (I v ) an be obtained with the following equation: I I u v = 0.5 = 0.5 ( I + I ) x x y 2 ( I + I ) + I x y y + I I The radii of gyration of the onrete etion with repet to the prinipal axe of inertia are alulated with: x I 2 I 2 y I 2 xy xy (29) where A i the area of the onrete etion 14 i = I A ; i = I A (30) v u u v

16 The moment of inertia of the reinforement (I u, I v ) with repet to the prinipal axe of inertia u and v are obtained with: I I u v = 0.5 ( I = 0.5 ( I x x + I + I y y ) ( I ) ( I x x I I y y ) o(2 θ ) in(2 θ ) I p ) o(2 ( θ + π / 2)) in(2 ( θ + π / 2)) I p p xy p xy (31) where: I x, I y are the moment of inertia of the reinforement with repet to the axe x and y of the onrete etion. I xy i the produt of inertia of the reinforement with repet to the axe x and y of the onrete etion. The alulation of the prinipal axe of inertia of the etion and it entre of gravity are performed with repet to the onrete etion alone for impliity without onidering the ontribution of the reinforement bar. A more rigorou alulation ould be done with repet to the homogenized etion. Equation 25 take into aount the interation between both axe of urvature. Thu, a it wa oberved from the numerial imulation, if the olumn i not braed and the relative eentriity (η) tend to zero, the ritial axial load of the olumn (N r ) i about the weak axi, and onequently the flexural tiffne of the member EI orrepond to the weak axi. Beide, if the olumn i ubjeted to biaxial bending with zero axial load, the relative eentriity (η) i infinite, and in thi ae, the flexural tiffne orrepond to an intermediate value between the trong axi and the weak axi. Thi tiffne will be equal to the weak axi if the olumn bend with repet to thi axi, and equal to the trong axi if the member bend with repet to the trong axi. For other loading ondition, the tiffne of the olumn will orrepond with an intermediate value between both axe of urvature. 15

17 Conequently, the interpolation funtion δ depend on the relative eentriity (η) and on the relative biaxial bending angle (β d ). The interpolation funtion δ wa obtained from leat quare fit of the numerially imulated data: η δ = o 2 β (32) d η + 2 where β d i the relative biaxial bending angle. It i poitive in ounterlokwie ene. M dv iv β d = atan (33) M du iu M du, M dv are the firt order bending moment with repet to the prinipal axe of the etion u and v repetively (Figure 5.b): M M du dv = M dx = M oθ + M dx p p dy inθ + M inθ dy p oθ p (34) Equation 25 repreent the behaviour of an unbraed olumn ubjeted to an axial load and both ingle and double urvature. Suh a funtion take into aount the interation between both flexural axe. On the other hand, if the olumn i braed and i ubjeted to ingle urvature bending with an axial load, the equivalent moment of inertia (I e ) orreponding to it flexure axi (I u or I v ) will be eleted. 4. VERIFICATION OF THE PROPOSED METHOD Beaue of the implifiation that were adopted, it beome neeary to analye the auray obtained uing the propoed equation of the tiffne EI with repet both to the numerial imulation and to experimental reult from the literature Verifiation with the numerial reult. 16

18 The auray of the propoed equation EI in thi paper an be evaluated uing the ratio of the firt order bending moment obtained with numerial imulation (M 1 ) NS and the propoed method (M 1 ) method (Figure 2). However, thi proedure i not appropriate for ae ubjeted to the ritial axial load, where thi ratio tend to infinite. To overome thee inadequaie, the ratio ξ NS i eleted a referene to evaluate the auray. where: R NS = ( N ) R NS ξ NS = (35) Rmethod 2 ( N ) + ( M ) ( M ) ) 2 i u R method = ( N ) NS 1 NS u, máx NS 2 ( N ) + ( M ) ( M ) ) 2 i u NS 1 method u, máx NS (N u ) NS ritial axial load of the etion in imple ompreion (M u,max ) NS bending maximum apaity of the ro etion obtained from the numerial imulation (Figure 2) Table 6 and Table 7 how the auray with repet to the numerial imulation in term of the type of load, ro-etion and urvature. The ame ae ued to infer the propoed EI equation (Table 5) were ued for the verifiation. It an be een that the average for all the experiment i 1.09 with a variation oeffiient of The ame auray i oberved for the different type of load, ro-etion and urvature. It an be notied that the propoed method adjut aurately to the numerial reult Verifiation with experimental reult. To evaluate the auray (ξ) with repet to the experimental reult, the following trength ratio wa adopted: where: N tet N N method maximum experimental axial load tet ξ = (36) 17

19 N method maximum axial load uing the propoed method The propoed equation (equation 17) wa ompared with the ame 613 experimental tet from the literature ([3], [20]-[42]) that were ued to validate the numerial model (etion 2). If ξ (equation 36) ha a value greater than one, the propoed method i on the afe ide. Table 2 how the range of variation of the parameter tudied in the experimental reult. To alulate the ultimate bending moment of the ro-etion, the parabolaretangle diagram for onrete under ompreion defined in the EC-2 (2004) ode [2] wa applied (laue from Euro Code 2 (2004) [2]). The onrete trength (f ) in eah experimental tet wa taken a the value of the mean ompreive trength in order to alulate the elati onrete modulu E (equation 7). Table 8 lit the author that performed the experimental tet, a well a the auray degree ξ of the propoed method both for hort-term and utained load (average ratio, variation oeffiient, perentile 5% and 95%). The evaluation of the method independently of the type of load and type of ro-etion ha alo been inluded in thi table. It an be een that an average ratio for hort-term load of 1.10 with a variation oeffiient of 0.14 wa obtained. For utained load, an average ratio of 1.11 with a variation oeffiient of 0.12 wa obtained. Finally, for all the experiment, an average ratio of 1.10 with a orreponding oeffiient of variation of 0.15 wa obtained. Table 8 how that the auray i lightly better for retangular etion than for non-retangular etion. Figure 6 how the ratio ditribution ξ and it trend line in term of the mot important parameter. The auray degree i analyzed with the ame referene variable that the eleted for the omparion with the numerial reult. 18

20 For all the graph, the trend line i plaed in a poition of ξ that i lightly higher than one, the reult lying on the afe ide. Generally, the trend line eem to be dereaing, apart from the yielding tre of the teel (f y ) and the relative firt order eentriity (η), where the trend line eem to be inreaing. Conequently, the propoed method detet the variation of uh variable properly. Finally, a omparion between the reult from propoed EI equation and the method propoed by the ACI-318(08) [1] and Euro Code-2) [2] wa arried out in onnetion with the experimental reult from the literature. Table 1 how the E.I equation ued in both deign ode The method from the ACI-318(08) ode [1] ugget the ue of the magnifier method for the deign of unbraed olumn. In order to take into aount the eond order effet, the following magnifiation fator i propoed: δ n Cm = Nd 1 φ N r (37) where: - C m i the oeffiient for alulating the equivalent uniform bending moment. It i equal to one for the ae of olumn ubjeted to an equal bending moment at both end auing ymmetri ingle urvature bending. - φ i the trength redution fator. It i et to a value of one to perform thi omparative analyi. - N r = π 2 EI/l 2 p where EI i the flexural tiffne of the olumn. The EI i alulated with equation from the ACI-318 (08) [1]. The following expreion i ued to alulate the Young modulu of onrete: ( f in MPa). E = 4700 f 19

21 The firt order bending moment i magnified in eah diretion independently for the ae of biaxial bending. The method that wa propoed from the EC-2 ode [2] ugget alo the magnifiation fator in order to take into aount the eond order effet (Setion EC-2 (2004) ode [2]): The effetive elati modulu of the onrete etion i obtained from: E { γ ( + ϕ )} d,eff Em / E 1 = (38) eff γ E partial afety fator (equal to 1.2). For thi omparative tudy, it ha a fixed value of 1. E m onrete eant elati modulu: E ( f 10) 0. 3 = (f m m en MPa) (39) f m mean value of onrete ylinder ompreive trength. In thi analyi, it i equal to the trength of onrete (f ) for eah experiment. ϕ eff equivalent reep oeffiient: ϕeff = κ ϕ (40) ϕ κ reep oeffiient ratio between the quai-permanent and the total load If the olumn i ubjeted to axial load and biaxial bending, then equation 1 i applied. To ompute the ultimate bending moment of the etion in the ACI-318(08) ode [1] the equivalent retangular onrete tre ditribution wa ued. While the parabolaretangle diagram for onrete under ompreion wa ued for the ode EC-2 (2004) ode [2]. 20

22 Table 9 and Table 10 how a omparion between the reult from the propoed method and the method from the ACI-318 (08) ode [1]and the EC-2 ode [2], with repet to the experimental reult. In general, the propoed method ahieve an average ratio that i loer to one, the lowet variation oeffiient, and it preent an eential improvement for utained load, mainly with regard to biaxial bending. It i important to oberve that the method propoed by the ACI-318(08) ode [1] appear to be more onervative. Regarding the reult obtained for non-retangular etion, the propoed method preent a better auray degree than the deign ode, that i, lower variation oeffiient, higher 5% perentile, lower 95% perentile and an average value loe to If the 5 % perentile of the propoed method i ompared with the deign ode, it an be oberved that it i higher for hort-term load and lower for utained load. The auray degree of the ode i different for hort-term load from for utained load, being more onervative for the lat utained load (mainly in biaxial bending). However, with the propoed method the value of the 5% perentile i 0.9 for almot all type of urvature. 5. EXAMPLE In order to illutrate the pratial appliation of the propoed method, the longitudinal reinforement of an unbraed olumn i alulated. The olumn ha a bukling length of 5 meter and it i ubjeted to ontant fore along the length of the element orreponding to the ultimate limit tate for the permanent or variable tate. Thee are N d = 1000 kn, M dx = 24 kn.m and M dy = 40 kn.m with repet to the entre of gravity of the gro etion. The ro-etion i preented in Figure 7. The mehanial propertie of the material are f k = 30 MPa and f yk = 500 MPa. The reep 21

23 oeffiient (ϕ) i equal to 2 and the ratio between the quai-permanent and the total axial load (N g /N tot ) i equal to 0.6. The ize of the reinforement i obtained by following the tep explained in etion 1 and 3 uing the bai hypothei from the EC-2 (2004) ode [2] to ompute the ultimate bending moment. Initially, the following parameter are omputed: f f ϕ d yd eff = = = f f k yk / γ = 30 /1.5 = 20 MPa / γ = 500/1.15 = MPa ( N N ) ϕ = = 1. 2 g tot The flexural tiffne of the olumn EI i obtained uing equation 17, for whih the following omputation mut be performed: -Moment of inertia (I x, I y ) and produt of inertia (I xy ) of the gro etion in m 4 with repet of it entre of gravity: I I x xy = I y = = Angle of the prinipal trong axi of inertia (θ p ) with repet to the x-axi. It i poitive in the ounter-lokwie ene (eq. 26). I x = I y θ = π 4 rad. p - Prinipal moment of inertia of the gro etion (I u, I v ) in m 4 with repet of it entre of gravity: (eq. 29): I = 0.002; u I v = Radiu of gyration of the onrete etion with repet to the prinipal axe of inertia in m (eq. 30): 22

24 i u = v ; i = Moment of inertia of the reinforement bar (I x, I y ) in m 4 with repet to the entre of gravity of the gro etion. For the teel ditribution preented in Figure 7, uh a moment of inertia an be expreed in term of the total area of reinforement (A ) in m 2 : I x = I y = A - Moment of inertia of the reinforement bar (I u, I v ) in m 4 with repet to the prinipal axe of inertia of the onrete etion (u,v) in term of the total area of reinforement (A ) in m 2 (eq. 31): I u = A ; I = v A - Deign bending moment with repet to the prinipal axe of inertia of the onrete etion (eq. 34): M = kn. m; M = kn m du dv. - Critial radiu of gyration of the onrete etion (eq. 22): Mdv 0 i = iu = m - Mehanial lenderne of the olumn (eq. 23): λ m = l p i = Firt order relative eentriity (η) (eq. 19): η = M d e0 4 i = M d = N 4 i M 2 dx d + M 2 dy = = kn. m - Relative biaxial bending moment (β d ) (eq. 33): 23

25 M tan -1 dv i v β rad d = =. M du i u - Effetive tiffne fator α (eq. 18) for η < 0.2: α = ( λ 0.25 ϕ m ) ( η 0.2 ) + = ( ) ( eff ( fd ) </ ) + ( ) = Redution fator of the tiffne of the reinforement ξϕ (eq. 24): ξ ϕ = m 1.9 ϕ eff exp( λ 25) = exp( ) = </ - Interpolation oeffiient (eq.32): η = o 2 β = η + 2 δ d - Equivalent moment of inertia inm 4 (e.25): GroSetion Reinforement I I e e = I = I x x δ + I δ + I y y (1 δ ) = (1 δ ) = A - Seant onrete elati modulu (E d ) for deign The EC-2 (2004) ode [2] adopt a value of 1.2 for the afety fator of the onrete elati modulu (γ E ). Moreover, if the real value of the mean onrete ompreive trength i unknown (f m ), it i omputed uing the following equation: f m = f k + 8 (in MPa). 0.3 ( f 10) 1.2 = MPa Ed = E γ E = m 81 - Elati modulu of the longitudinal reinforement: E = MPa - Flexural tiffne of the olumn EI in kn m 2 (eq.17): EI E I E I d e e = α + = ϕeff 1+ ξ ϕ A 24

26 The ritial axial load (N r ) in kn i equal to (eq.6): N r 2 π EI = = A l 2 p The magnifiation fator i equal to (eq.5): δ n = 1 1 ( N N ) d r A = A 1.0 The magnified bending moment in kn.m i equal to (eq.2): * A M d = δ n M d = </ M d = kn. m A To determine the required longitudinal reinforement, the deign fore (N d, M d *) and the ultimate fore of the etion (N u, M u ) are mathed and a non-linear ytem of two equation and two unknown (A, x) i thu obtained. Thi ytem of equation an be olved by uing the well-known Regula Fali method. Figure 8 how the variation of M * d and M u in term of A for the given axial load N d. The interetion between both urve determine the required area of reinforement A to be equal to m 2, whih i equal to 12 rebar with a diameter φ=16 mm (24.12 m 2 ). M * d N = N ( A,x ) ( A ) = M ( A,x ) d u u 6. CONCLUSIONS Thi paper propoe a new equation to obtain the effetive tiffne EI of lender reinfored onrete olumn both for verifiation and deign ubjeted to ombined axial load and biaxial bending that i valid for hort-time and utained load, and for both normal and high trength onrete. The method i only valid for olumn with equal effetive bukling length in the two prinipal bending plane. 25

27 The new equation extend the propoed EI equation in the Biaxial bending moment magnifier method by Bonet et al [6], whih wa valid only for retangular etion to etion with any hape of the ro-etion. Furthermore, a new EI equation under uniaxial bending and axial load valid for any type of ro-etion i propoed. The propoed formulation for biaxial bending i an extenion of the general flexural tiffne equation EI for uniaxial bending obtained by alulating the equivalent moment of inertia of the gro etion and the reinforing bar. Suh formulation inlude the exiting interation between both flexural axe and the ae of the axial load and ingle urvature. The effet of braed truture i taken into aount in the behaviour of the olumn ubjeted to an axial load and uniaxial bending with repet to the trong axi. The method wa ompared with 613 experimental tet and it proved to be reaonably aurate for pratial engineering deign appliation. A notieable improvement in the predition auray of olumn trength wa ahieved uing the new flexural equation of EI when ompared with the urrent equation of the ACI-318 (08) ode [1] and the EC-2 (2004) ode [2]. It i important to highlight that thi improvement i more relevant for utained load and biaxial bending. For the ae of ingle bending urvature and utained load, the average and variation oeffiient are 15% and 75% lower than the Euro Code 2 [2] repetively. For biaxial bending and utained load the average obtained with the propoed method i 20% lower than the Euro Code 2 [2] being till onervative, while the variation oeffiient are imilar. Otherwie, the ACI-318 [1] i more onervative and ha higher attering tan the Euro Code 2 and the propoed method. 26

28 The equation propoed in thi paper are more omplex than the propoed by other author or deign ode; however, from the pratial point of view it appliation i very eay with preadheet or mall omputer program. A more eonomial deign i obtained with a higher auray degree than with the atual deign ode. The method i ueful for truture in building ine it preent a high degree of auray for appliation in profeional pratie, uh a heking reinfored onrete etion or in the deign phae. ACKNOWLEDGEMENTS The author wih to expre their inere gratitude to the Spanih Miniterio de Cienia e Innovaión for help provided through projet BIA and BIA and to the European Community with the Feder fund. BIBLIOGRAPHY [1] ACI , "Building Code Requirement for Reinfored Conrete", Amerian Conrete Intitute, Detroit, 2008, pp 471 [2] European Committee for Standardization: Euroode 2: Deign of onrete truture- Part 1: General rule and rule for building, EN Deember 2004 [3] Mavihak, V.; Furlong, R.W.: "Strength and tiffne of reinfored onrete olumn under biaxial bending ", Reearh Report 7-2F, Center for Highway Reearh, Nov 1976 [4] Mirza, S.A.:"Flexural tiffne of retangular reinfored onrete olumn", ACI, Strutural Journal, V.87, Nº4, 1990, pp [5] Weterberg, B., Slender olumn with uniaxial bending, International Fedaration for Strutural Conrete (fib), Tehnial report, bulletin 16 Deign example for 27

29 1996 FIP reommendation Pratial deign of trutural onrete, January 2002, pp [6] Bonet, J. L., Miguel, P. F., Fernandez, M. A., Romero, M. L., Biaxial bending moment magnifier method, Engineering Struture, Volume 26, Iue 13, November 2004, pp [7] Tikka, Timo K., Mirza, S. Ali, Nonlinear EI Equation for Slender Reinfored Conrete Column, ACI Strutural Journal, Volume 102, Iue 6, November 1, 2005, pp [8] Tikka, Timo K., Mirza, S. Ali, Effetive flexural tiffne of lender trutural onrete olumn, Canadian Journal of Civil Engineering, Volume: 35, 2008, pp [9] Ehani, M.R.; Alameddine, F.: "Refined tiffne of lender irular reinfored onrete olumn", ACI, Strutural Journal, V.84, 1987, [10] Sigmon, G.R.; Ahmad, S.H.:"Flexural rigidity of irular reinfored onrete etion", ACI Strutural Journal, V.87, Nº5, 1990, pp [11] Breler, B.:"Deign riteria for reinfored olumn under axial load and biaxial bending", ACI, Journal of the Amerian Conrete, V.57, Nº5, 1960, pp [12] Pallaré. Lui, Bonet, Joé L., Miguel, Pedro F., Fernández, Miguel Á., The influene of the weak axi on the behavior of high trength RC lender olumn ubjeted to biaxial bending, Engineering Struture, Volume 31, Iue 2, February 2009, pp [13] Mari, A.R.: "Nonlinear Geometri, Material And Time Dependent Analyi Of Three Dimenional Reinfored And Pretreed Conrete Frame", Report No. 28

30 USB/SESM-84/12, Departament of Civil Engineering, Univerity of California, Berkley, California, USA, June 1984 [14] Bonet, J. L.; Romero, M. L.; Miguel, P. F.; Fernández, M. A.: A fat tre integration algorithm for reinfored onrete etion with axial load and biaxial bending Computer & Struture, Volume 82, Iue 2-3, January 2004, pp [15] Comité Euro-internaional du beton: "CEB-FIB Model Code 1990" C.E.B. Bulletin Nº º y 205, 1991 [16] Comité Euro-internaional du beton: "High Performane Conrete. Reommended extenion to the Model Code 90 reearh need", C.E.B.. Bulletin, Nº 228, 1995 [17] Comité Euro-internaional du beton: "Manual of Buling and Intability", C.E.B. Bulletin Nº 123, 1978, pp 135 [18] Comité Euro-internaional du beton: "Bukling and Intability - Progre Report", C.E.B.. Bulletin, Nº 155, 1983 [19] Bonet Senah, J.L.:"Método implifiado de álulo de oporte ebelto de hormigón armado de eión retangular ometido a ompreión y flexión biaxial", Tei Dotoral, Dpto. Ingeniería de la Contruión y Proyeto de Ingeniería Civil, Univeridad Politénia de Valenia, Julio 2001 [20] Galano, L.; Vignoli, A. Strength and Dutility of HSC and SCC Slender Column Subjeted to Short-Term Eentri Load, ACI Strutural Journal, Volume 105, Iue 3, May 1, 2008, pp [21] Pallaré, L., Bonet, J.L., Miguel, P.F., Fernández Prada, M.A.: Experimental reearh on high trength onrete lender olumn ubjeted to ompreion and 29

31 biaxial bending fore, Engineering Struture, Volume 30, Iue 7, July 2008, pp [22] Germain, O., Epion, B., Slender high-trength RC olumn under eentri ompreion, Magazine of onrete reearh, Vol. 57, No. 6, Agoto 2005, pp [23] Sarker P. K., Adolphu S., Patteron S., Rangan B.V., High-trength onrete olumn under biaxial bending, Speial Publiation Reent Advane in onrete tehnology, año 2000, pp [24] Kim, J.K..; Lee, S.S.: "The behaviour of reinfored onrete olumn ubjeted to axial fore and biaxial bending", Engineering Struture, V.23, 2000, pp [25] Claeon C., Gylltoft K., Slender onrete olumn ubjeted to utained and hort-term eentri loading, ACI Strutural Journal, Vol. 97, No. 1, Enero- Febrero 2000, pp [26] Claeon C., Gylltoft K., Slender high-trength onrete olumn ubjeted to eentri loading, Journal of Strutural Engineering, Vol. 124, No. 3, Marzo 1998, pp [27] Foter J.F, Attard M. M., Experimental tet on eentrially loaded high-trength onrete olumn, ACI Strutural Journal, Vol. 94, No. 3, Mayo-Junio 1997, pp [28] Lloyd N.A., Rangan B. V., Studie on high-trength onrete olumn under eentri ompreion, ACI Strutural Journal, Vol. 93, No. 6, Noviembre- Diiembre 1996, pp

32 [29] Taylor, Andrew W.; Rowell, Randall B. and Breen, John E.: Behavior of Thin- Walled Conrete Box Pier ACI Strutural Journal, Volume 92, Iue 3, May 1, 1995, pp [30] Kim, J.K., Yang, J.K.: "Bukling Behaviour of Slender High-Strength Conrete Column", Engineering Struture, V.17,Nº 1, 1995, pp [31] Hu C.T.T., Hu L.S.M, Tao W.H., Biaxially loaded lender high-trength reinfored onrete olumn with and without teel fibre, Magazine of onrete reearh, Vol. 47, No. 173, Deember 1995, pp [32] Tao, W.H., Hu C.-T. T.: "Behavior of quare and L-haped lender reinfored onrete olumn under ombined biaxial bending and axial ompreion", Magazine of Conrete Reearh, V 46, Nº 169, De 1994, pp [33] Wang G.G., Hu C.T., Complete biaxial load-deformation behavior of RC olumn, Journal of Strutural engineering, Vol. 118, No. 9, September 1992, pp [34] Hu, Cheng-Tzu Thoma: Channel-Shaped Reinfored Conrete Compreion Member Under Biaxial Bending, ACI Strutural Journal, Volume 84, Iue 3, May 1, 1987, pp [35] Iwai, S., Minami, K., Wakabayahi, M., Stability of lender reinfored onrete olumn ubjeted to biaxially eentri load, Bulletin of the diater prevention reearh intitute, Vol. 36, No. 321, pp [144]. [36] Hu, Cheng-Tzu Thoma: Biaxially Loaded L-Shaped Reinfored Conrete Column, ASCE Journal of Strutural Engineering, Vol. 111, No. 12, Deember 1985, pp

33 [37] Poton, R. W.; Gilliam, T. E.; Yamamoto, Y. and Breen, J. E.: Hollow Conrete Bridge Pier Behavior, ACI Journal Proeeding, Volume 82, Iue 6, November 1, 1985, pp [38] Wu H., Huggin M. W., Size and utained load effet in onrete olumn, Journal of the trutural diviion, Vol. 103, No. ST3, Marzo 1977, pp [39] Drydale, R.G., Huggin, M.W.: "Sutained biaxial load on lender onrete olumn", Journal of the Strutural Diviion, Proeeding A.S.C.E., V.97, Nº5, May-1971, pp [40] Goyal B.B., Jakon N., Slender onrete olumn under utained load, Journal of the trutural diviion, Vol. 97, No. ST11, November 1971, pp [41] Breen J.E., Ferguon P.M., Long antilever olumn ubjet to lateral fore, Journal Proeeding, ACI. Vol. 66. No. 11, November [34] [42] Viet I.M., Eltner R.C., Hognetad E., Sutained load trength of eentrially loaded hort reinfored onrete olumn, Journal of the Amerian Conrete Intitute, Vol. 27, No. 7, Marh 1956, pp

34 z,w 1 y θ x1 θ z1 x,u 1 i y,v 1 θ y1 u 3 θ z2 j z,w 2 y,v 2 θ x2 θ y2 ε 1 2 x x,u 2 Strain Deompoition into thik layer σ Deompoition into quadrilater (a) (b) Figure 1. Finite element model: a) general arrangement, b) Cro etion integration, Bonet et al. [14]. 33

35 ( M u, max ) NS ( M u ) NS λ m = 0 M ( M 1 ) NS ( M 1 ) method Simplified Method Numerial Simulation λ m 0 N i ( N r ) NS N ( N u ) NS Figure 2. Magnifier bending moment method 34

36 λ m =100 Short-term load f = 30 MPa; ω = 0.50 ( α ) NS λ m =35 λ m = η = e 0 /(4 i ) Figure 3. Effetive tiffne fator α 35

37 Short-term load f = 30 MPa; w = 0.50 λ m = Short-term load f = 30 MPa; w = 0.50 λ m = 100 ( α ) NS propoed numerial tet EC-2 (04) ACI-318 (08) Tikka Tika y Mirza and Mirza (08) Weterberg (02) η = e 0 /(4 i ) ( α ) NS propoed numerial tet EC-2 (04) ACI-318 (08) Tikka Tika y Mirza and Mirza (08) Weterberg (02) η = e 0 /(4 i ) Figure 4. Comparion of α with different author and deign ode. 36

38 y M y v Weak axi u Strong axi λ m = 0 Weak axi M v M ty M dy M d M t β * (+) M u Strong axi 0 θ p (+) M dx M tx x λ m ¹ 0 θ p (+) M dx M tx M x (a) Cro-etion of the olumn (b) Interation diagram (M x,m y ) for deign axial load of N d 0 entre of gravity of the onrete gro etion x, y oordinate axe of the etion u, v prinipal axe of inertia of the onrete gro etion Figure 5. Propoed implified method 37

39 Strength Ratio = N tet/nmethod Strength Ratio = N tet/nmethod Strength Ratio = N tet/nmethod f (MPa) Strength Ratio = N tet/nmethod ω = (A f y )/(A f ) λ me = l p /i η = e 0 /(4 i ) Strength Ratio = Ntet/Nmethod 2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0, β d (º) = atan[(m dv i v )/(M du i u )] Strength Ratio = N tet/nmethod ϕ eff = ϕ (N g /N tot ) Trend line Figure 6. Comparion of the propoed method with the experimental reult 38

40 0.20 y 0.20 v u θ p x φ? In meter 0.40 Figure 7. Example. Cro-etion of the olumn 39

41 M (mkn) M u N d = 1000 kn * M d A (m 2 ) Figure 8. Example. Reinforement ratio alulation 40

42 Table 1. Comparion of different E.I. equation. Author Parameter Effetive flexural tiffne EI Mavihak and Furlong [3] Mirza[4] Weterberg [5] Bonet et al [6] Tikka and Mirza[7] Tikka and Mirza[8] ACI-318 (08) [1] EC-2 ode [2] N /N u 0.2 E I + E I EI = 1.6 ( 1 2 N / N ) α E I + E I 1+ β e/h; l/h; β d EI = α = ( ( l h) 0.3 ( e h) EI = α α E I ϕ e d u + E I or α = ( ( e h) 0 λm ν; λ m ; ϕ eff ; f ; ω ω 100 λ α = 0.08 ν (1 + ω) ν; αϕ = ϕ 1 m e f e eff ω 200 E E EI = α I + I 1 + ϕ 1+ η e/h; l/h; ϕ eff ; f e/h; ρ l; β d for for eff e h < 0.2 e h 0.2 f α p = ; 200 α E I EI = 1+ β α E ( I I EI = 1+ β d l α = ( ϕ h α = α (1.2 e h) 0.1 p η = 1.9 ϕ E I e/h; l/h; ρ l ; β d α = ( e h) β d ν; λ m ; ϕ eff ; f k ; ρ l eff I ) E eff l exp 0.1 h 0 ) ( e h 0.2) + α α = ( ) e h 0 1 ( ) + β e h β = 7 for ρ 2%; β = 8 for ρ > 2% ( I 1 l ( e h) + + β h β = 7 for ρ 2%; β = 8 for ρ > 2% and β = 9 EI = 0.2 EI = K ρ 0.2% K l l d d E I + E I 0.4 E I or EI = 1+ β 1+ β E I + K = 1 K E = k k 1 I 2 l d l + I /(1 + ϕ ); ρ 1% K ef l ) for ompoite = 0 K l p olumn. = 0.3/( ϕ ) fk λm k1 = ; k2 = ν EI = flexural tiffne of ompreion member; E = modulu of elatiity of onrete; E = modulu of elatiity of reinforement; I = moment of inertia of gro onrete etion; I = moment of inertia of reinforement; N = axial load; N u = maximum load apaity; e/h = eentriity ratio; l/h = geometrial lenderne ratio; β d = ratio of the maximum fatored axial utained load to the maximum fatored axial load aoiated with the ame load ombination; ω = mehanial reinforement ratio; ν = relative normal fore; λ m = mehanial lenderne ratio; ϕ eff effetive reep ratio; f = onrete trength; ρ l = geometrial reinforement ratio; f k = harateriti ompreive ylinder trength of onrete at 28 day. ef 41

43 Table 2. Parameter variation in the experimental tet Parameter Range Compreive onrete trength [f. (MPa)] MPa MPa Steel trength [f y (MPa)] MPa 684 MPa Mehanial reinforement ratio [ω] Geometrial reinforement ratio [ρ g ] Type of etion Retangular / Square / L -hape / Box/ C -hape / Ovoid Mehanial Slenderne [λ m ] Ratio between the prinipal radii of gyration [i v /i u ] 1 3 Relative eentriity [η=e 0 /4/i ] Relative axial load [ν] Relative biaxial angle [β d ] 0 º 90 º Creep oeffiient [ϕ] Equivalent reep oeffiient [ϕ eff ] Ratio between the axial load from the permanent fore and the axial load from the total [N g /N tot ]

44 Table 3. Calibration of the numerial model. Analyi in term of the hape of the etion and the type of load. Short-term Load Sutained Load Total Setion N er ξ m V.C P 5 P 95 R NR All R NR All R NR All R =Retangular or Square; NR= Non- Retangular ξ m : Average ratio; V.C.: variation oeffiient; P 5 : Perentile 5%; P 95 :Perentile 95%; 43