IDEALIZED STRESS-STRAIN RELATIONSHIP IN TENSION OF REINFORCE CONCRETE MEMBER FOR FINITE ELEMENT MODEL BASED ON HANSWILLE S THEORY

Transcription

1 VOLUME 2, O. 2, EDISI XXIX JULI 24 IDEALIZED STRESS-STRAI RELATIOSHIP I TESIO OF REIFORCE COCRETE MEMBER FOR FIITE ELEMET MODEL BASED O HASWILLE S THEORY Hardi Wibowo ABSTRACT Untuk penganaliaan kontrol retak (crack control) pada model truktur beton bertulang (reinforced concrete/rc) menggunakan oftware finite element eperti DIAA diperlukan pemodelan hubungan tegangan-regangan (tre -train relationhip). Data model hubungan tegangan-regangan ini dalam rangka mengakomodai efek rekatan (bond -lip effect) antara tulangan (reinforcement) dengan beton (concrete) yang melingkupinya. Hubungan tegangan - regangan yang diperlukan adalah nilai rata-rata tegangan-regangan pada tulangan dan beton. Data hubungan tegangan-regangan rata-rata ini bia diturunkan dengan menggunakan teori Hanwille. Pada tulian ini akan diuraikan mengenai teori Hanwille untuk menentukan hubungan tegangan-regangan terebut dan diberikan atu contoh perhitungan dan curva hubungan tegangan-regangan rata-rata dari ebuah batang beton bertulang. GEERAL To model reinforced concrete member for finite element (FE) analyi, reinforcement teel bar were modeled a embedded element. In thi element, the bar element do not have independent degree of freedom. Intead, the tiffne of the bar element were uperpoed on that of mother concrete element. In FE model, perfect bonding between concrete and embedded reinforcement i aumed. Bond-lip effect between reinforcement and urrounding concrete can be taken into account by uing an average tre-train e tr g e vera a t a t e I b + relationhip of reinforced concrete including tenion tiffening effect. The average tretrain relation derived from the bond-lip differential equation propoed by Hanwille will be explained here. Fig. below how the chematic figure for thi tre-train relationhip. The tate I correpond to perfect bonding, while the tate II to perfect cracking. The average tre of a RC member i expreed in term of average teel tre and average concrete tre c, c t a t e I I A : U n c r a c k e d t a t e B : I n t i a l c r a c k i n g t a t e C : S t a b i l i z e d c r a c k i n g t a t e n D : P o t y i e l d i n g t a t e I i, r = I I C D A v e r a g e t r a i n (a) Average tre-train curve Staf Pengajar Juruan Teknik Sipil Fakulta Teknik Univerita Diponegoro MEDIA KOMUIKASI TEKIK SIPIL 45

2 Idealized Stre-Strain Relationhip in Tenion of Reinforced Concrete Member For Finite Element Model Baed on Hanwille Theory y e tr e el St E S t e e l t r a i n (b) tre-train curve for teel f ct E c C o n c r e t e t r a i n c (c) tre-train curve for concrete Fig. Schematic figure for Stre-train curve of RC member in tenion. dv c... () V V where for a uniaxial tre tate, L L c cdv, dv... (2) L L Where : V At L i the total volume of the RC member with the cro-ectional area At A c A and length L. A : reinforcement ratio A c A =Cro-ectional area of reinforcement A c =Cro-ectional area of concrete Under the aumption of perfect bonding ued in the preent meared crack FE analyi, the average concrete train equal the average teel train. However, the average concrete train c include contribution of crack opening in concrete. We can eparate the average train into an intact part and a cracking part, a L du du c L L * L cm w / L u...(3) where cm denote the average train over the intact part L * ; w u u u i the crack width, and the ummation i taken over all crack in L. In ummary, ince normal tree are zero in the cracking part, we have c cm, m,...(4) c m cm w / L 46 MEDIA KOMUIKASI TEKIK SIPIL

3 VOLUME 2, O. 2, EDISI XXIX JULI 24 where the overbar denote averaged quantitie over the total region, while the ubcript m tand for averaged quantitie over the intact part. In the preent meared FE analyi, we obtain the averaged quantitie over the total region a output. Since the elatic perfectly platic model i aumed for tre-train relation of reinforcement teel a hown in Fig.(b), the tre-train relation for concrete a hown in Fig.(c) wa derived from the average tre-train relation from Hanwille. The derived tre-train relation of concrete i modeled a a multilinear curve in FE analyi. COSITUTIVE MODEL AD BOD-SLIP DIFFERETIAL EQUATIO Conider a differential length d x of reinforcement embedded in concrete a hown in Fig.2, then the force due to bond between teel reinforcement and the urrounding concrete mut be ame a the change of axial force on teel or the concrete cro-ection. c+d c c v + d c v Fig.2 Stree in differential element of RC member. Conidering equilibrium of force in the longitudinal direction -d c (x)a c = d (x)a = v (x)u d x... (5) Where : v = bond tre c = Stre in concrete = Stre in concrete U = Perimeter of cro-ection of reinforcing bar =d d = Diameter of teel bar x = Longitudinal coordinate of the member Dividing Eq.5 by A d x d ( x) U 4 ( ) v x v ( x)...(6) d x A d Auming that the cro-ection remain contant then the lip v or relative diplacement between teel and concrete i equal to the difference of train between teel and concrete. From Fig.2 v = - c -...(7) dv or c...(8) MEDIA KOMUIKASI TEKIK SIPIL 47

4 Idealized Stre-Strain Relationhip in Tenion of Reinforced Concrete Member For Finite Element Model Baed on Hanwille Theory where : v = relative diplacement between teel and concrete c = diplacement in concrete = diplacement in teel = diplacement due to hrinkage = Shrinkage train c = Strain in concrete = Strain in teel In Eq.8, and c can be replaced by teel tre and concrete tre repectively and E n=modular ratio= E c E = modulu of elaticity of teel E c = modulu of elaticity of concrete dv E ( x) n c ( x)... (9) Hanwille ued the following contitutive model defined in exponential form for bond tre and relative diplacement or lip v (x) = Af cw v (x)... () where A and are contant and are obtained empirically, and f cw =compreive trength of cube of concrete. Uing the value of v (x), we get d 2 v 4Af cw n v ( x) 2 d E... () dv ow the olution of v= and, will give the length of lip region v(x) : 2 dv 2 2 ( n ) Af cw x E d...(2) The boundary condition decribed above mean that at a location where there i no lip between teel reinforcement and the concrete i taken a the origin of coordinate ytem. FIRST CRACK In a RC member ubjected to axial force, when the tre attain the tenile trength of concrete, the firt crack appear and then the tre of concrete and teel at the cracking region change and a relative diplacement (lip v) between teel and concrete i produced. The tre condition a hown in Fig.3 i produced. The length of the region where crack produce relative diplacement i denoted here by L ER and i called introduction length or tranmiion length.,r f c t, r I c x ) ( x ), r,r x L ER Fig.3 Stre of concrete and teel reinforcement after firt crack in a RC member ubjected to axial tenile force 48 MEDIA KOMUIKASI TEKIK SIPIL

5 VOLUME 2, O. 2, EDISI XXIX JULI 24 To decribe the tre c (x) and (x) of concrete and teel in the region of length L ER, thee quantitie are repreented a a function of lip. The coordinate ytem choen i hown in Fig.3. At the origin of thi coordinate ytem, no lip i produced between teel and concrete, i.e. at x =, dv From Eq.9, for x=, we have dv E () n () c... (3) (x) at x=l ER i,r and (x) at x= i,i. When at x=l ER, (x) i,r then c ()=f ct. Equating the total force at the crack and at the ection x=,r A = A c c + A or,r = c + At x =,,r = c () + () And then we find f ct n, r E... (4), r E, r E, r n... (5) n, r E dv, r... (6) n IITIAL CRACKIG STATE In a reinforced concrete member ubjected to axial tenile force, when the tre attain the tenile trength of concrete, the firt crack appear and the relative lip between teel reinforcement and urrounding concrete i produced. In thi method for crack width evaluation, the ame contitutive relation between the bond tre v and the lip v wa ued a ued by Hanwille theory, given in Eq.. In thi equation, i not a non-dimenional parameter but dimenional one. If the unit of length i cm, then Hanwille reported that A=.58 and =.3 are tandard value for a deformed bar. After firt cracking of concrete, further increae of the axial force increae the number of crack and accordingly deformation due to cracking, and thu pacing between adjacent crack reduce. Hence, depending upon the pacing between the crack, two cracking tate can be defined, one i initial cracking tate and the econd one i tabilized cracking tate. In the initial cracking tate a hown in Fig.4(a), the tranmiion length of two adjacent crack do not overlap each other and there i a tate I region between two adjacent crack. Since there i no relative lip in the tate-i region, there i no interaction between two adjacent crack, i.e. the opening of one crack doe not affect the width of around crack. Further increae of axial force caue the generation of more crack till the pacing between crack become o mall o that the tranmiion length of crack overlap each other. Thi i called tabilized cracking tate a hown in Fig.4(b). In thi tate, the opening of one crack affect the width of around crack. Crack width expreion in both cracking tate i given by w=2v at the crack poition. The crack width w R for the firt crack i derived a given by Eq.8 for, I i with, r and, r are given in Eq.4 f and ct, r repectively. MEDIA KOMUIKASI TEKIK SIPIL 49

6 Idealized Stre-Strain Relationhip in Tenion of Reinforced Concrete Member For Finite Element Model Baed on Hanwille Theory concrete reinforcing teel, c II m I II f ct c L ER tatei L ER a=l ER ( (a) Initial cracking tage, c I I m cm c a=l ER ( (b)stabilized cracking tage Fig.4: Stre ditribution in cracked teel reinforced concrete member. i =Average train at the boundary between initial cracking tate and tabilized cracking tate. i i obtained from the average teel train,m in the initial cracking tate. We have, m, II, I... (7) where i defined a II E,... (8) r E, and,ii,ii,i At the boundary of initial and tabilized cracking tate, = thu,ii =,r and,ii =,r =,II -,I 5 MEDIA KOMUIKASI TEKIK SIPIL

7 VOLUME 2, O. 2, EDISI XXIX JULI 24 i a non-dimenional factor defining crack pacing in term of tranmiion length. Hanwille ha conidered the poibility of tre drop due to further cracking in the initial cracking tate and thu he ha alo propoed expreion for crack width other than firt crack, thu giving crack width le than firt crack. A long a initial cracking tate prevail, the crack width expreion for firt crack give the maximum crack width. Thi procedure i intended to evaluate the maximum crack width. Thu, in our propoed method, we aume a contant tre tate in the initial cracking tate a hown in Fig.. STABILIZED CRACKIG STAGE In the tabilized cracking tage, we have: II E r E... (9) and =a/l ER i the non dimenional crack pacing. Hanwille propoe the following expreion a a mean value of m on the bai of the tatitical experimental data. 2 ( * * m max /. )... (2). where 2 II E 2 r E 2 II E 2 r E... (2) When evaluating the crack width, it i needed to know II from FE analyi. II i the tree in teel in tate II a defined in Fig. 4(b). The relationhip between II and the average tree in FE analyi can be derived from the equilibrium condition. c =Axial force of RC lab At uncracked location, the force will be hared by concrete and teel reinforcement. c At... (22) c Ac c A At crack, the force i taken by only reinforcing teel, and the reulting tree on the reinforcing bar i II. c II... (23) A Putting the value of c from Eq.2.4 into Eq.2.4 we get the relation A c II c... (24) A Or II c... (25) Thu the Eq.24 can be ued for getting the value of II from c and. The c and are obtained from out put of FEM analyi. The average train derived from the bond-lip differential i given by Hanwille, a follow: 2 II II m m E II m 2... (26) From thi average train expreion, we can pecify the average tre-train of a RC member, which i ued in the FE analyi. A EXAMPLE Finally, we have concluion that tretrain relationhip to model RC member for FEM ha three tage i.e.: un-cracked tage, initial cracking tage, and tabilized cracking tage (ee fig.a). In the u n-cracked tage (line -a), range of train i I and f average tre i ct n. In the MEDIA KOMUIKASI TEKIK SIPIL 5

8 Idealized Stre-Strain Relationhip in Tenion of Reinforced Concrete Member For Finite Element Model Baed on Hanwille Theory initial cracking tate (line a -b) range of average train i I i where i i average train at the boundary between initial cracking tate and tabilized cracking tate. In thi point = and =2, then. fct i n 2 and average E,r tre i. And in the Stabilized cracking tate range of average train i i = m and equation below are hold: 2 II II m m E II m 2 In thi condition C therefore, II and S, II, where, r, II Yield trength of reinforcement. Following i an example of calculation uing data a below: Modulu of elaticity of teel E = 2 /mm 2, Compreive trength of concrete = cy = 4 /mm 2, Tenile trength of concrete f ct = 2.69 /mm 2, Reinforcement ratio =.9, Modular ratio n o = 7, Bond lip contant A =.29 (in mm), Bond lip contant =.3, Shrinkage train of concrete o =. The calculation reult can be preented a a table and figure a hown below: Ave. train Ave.RC tre (Mpa) Ave.reinf. tre (Mpa) ote,,, Point,9 2,9925,3529 Point A,32 2,9925,27677 Point B,35 2,99872,3926,42 3,864,65887,48 3,37356,873,53 3,5699 2,746,58 3,7484 2,26533 Stabilized,62 3, ,4579 cracking,67 4,2325 2,6475 tate,72 4,367 2,83583,77 4,4989 3,236,82 4,6855 3,275,86 4, ,3974,9 5,635 3,58372,96 5, ,76975, 5,4359 3,95558, 5, ,9746,2 5,55 4,2987 Table.: Calculation reult 52 MEDIA KOMUIKASI TEKIK SIPIL

9 VOLUME 2, O. 2, EDISI XXIX JULI 24 Average Stre (MPa) a b 3 2 RC Member Reinforcement Average Strain () (a). Stre-train relationhip for RC member and reinforcement 3 Stre (MPa) Strain () (b). Stre-train relationhip for concrete Stre (MPa) Strain ( ) (c). Multi-linear tenion oftening curve for DIAA input data Fig. 5. Reult of calculation of tre-train relationhip and Multi-linear tenion oftening curve MEDIA KOMUIKASI TEKIK SIPIL 53

10 Idealized Stre-Strain Relationhip in Tenion of Reinforced Concrete Member For Finite Element Model Baed on Hanwille Theory REFERECES AASHTO, 998, LRFD Bridge Specification, SI unit, Second edition, American Aociation of State Highway and Tranportation Official, 444 orth Capitol Street,.W., Suite 249 Wahington, D.C. 2 Clark, A. P., 956, Cracking in reinforced concrete flexural member, ACI Journal, Proceeding, Vol.52, o.8. Crifield, M. A. and Will, J.,989, Analyi of R/C panel uing different concrete model, Journal of Engineering Mechanic, ASCE, Vol. 5, o. 3 Hanwille, G., 986, Zur Rißbreitenbechränkung bei Verbundträgern, Technich- Weienchaftliche Mitteilungen, Intitut für Kontruktiven Ingenieurbau Ruhr- Univerität Bochum, Mittelilung r Holmberg, A. and Lindgren, S., 97, Crack pacing and crack width due to normal force or bending moment, Document D2, 97, ational Swedih building reearch. JSCE, 998, Committee of Concrete, Standard pecification for concrete tructure [Deign], (in Japanee). azaqpur, A. G. and ofal, M., 99, Analytical modelling of non-linear behaviour of compoite bridge, Journal of Structural Engineering, ASCE, Vol. 6, o MEDIA KOMUIKASI TEKIK SIPIL