Uniaxial Modelling of Behavior of the Concrete in Fast Dynamics: Approach to Seismic Behavior

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Engneerng, 0, 3, 56-568 do:0.436/eng.0.36066 Publhed Onlne June 0 (http://www.scrp.

1 Engneerng, 0, 3, do:0.436/eng Publhed Onlne June 0 ( Unaxal Modellng of Behavor of the Concrete n Fat Dynamc: Approach to Semc Behavor Abdelhak Baraka, Mohammed Benal Benmanour, Yazd Abdelazz, Fakhreddne Djedd Unverty of Bechar, Bechar, Algera Unverty of Oran, Oran, Algera E-mal: numelab@yahoo.fr Receved December, 00; reved May 0, 0; accepted June 3, 0 Abtract The advantage of the computer enable u to mulate of complex tructure ubjected to dynamc loadng. To come up to the necety to know the real behavor of uch materal, we explot thee advantage bang on expermental data avalable n the lterature. Snce the repone of the materal depend on tre velocty, o t eental to provde a computatonal code wth dynamc behavor. To perform uch mulaton, we have elaborated a behavor law governed by loadng velocty effect on concrete and ther atttude cyclc non elatc, for an approach of emc behavor. Th paper how the procee we have followed to formulate th vcou damage law whoe am behavor predcton for concrete under dynamc tree. Then, the model valdated wth expermental reult and mulaton of ome avalable tet on Hopknon bar. Keyword: Concrete, Vcou-Damage, Stran Rate. Introducton Under rapd and dynamc mpact, concrete how a mechancal behavor, partcularly entve to rate loadng [-5]. Thee condton leave the model mplfed by tatc computaton, even balanced by afety coeffcent, unrealtc. Hence dynamc behavor analy of concrete tructure requre the ue of model and law that take nto account a certan number of eental phenomena relatve to uch tuaton. Among the avalable approache n lterature, we have adopted a phenomenologcal thermodynamc approach to decrbe the damaged non lnear hyteretc behavor of concrete, an approach ued by Labordere [6]. Th approach ha been developed to mulate the behavor of concrete tructure ubjected to monotonou or cyclc alternated loadng qua-tatc mode [6,7]; that to ay at a tran rate lower than S-. Wthn the frame of dynamc behavor mulaton of concrete ubjected eentally to emc tree, then under other type of mpact (blat, percuon ), we wll adapt th model to thee type of tmulaton. But th cannot be poble unle we ntroduce vcou effect nduced by the hgh loadng rate on materal.. Couplng of the Model to Vcoty Wth reference to Dubé work [8], who had adopted for uch parng Perzyna vcoty model [9], many reearcher expreed concrete vcoty through parameter gven n term of rate tran varaton [0-4]. The ntroducton of velocty effect n our damage model, wa done through a non dmenonal parameter whoe varaton lnked frt to dynamc ncreae factor (non dmenonal) whch telf expreed n term of tran rate. Th vcoty parameter noted η, wll be pared wth adjutng parameter noted a, hence the threhold functon become: η a D K ( D ) Y o + D b () where: Y o : ntal ll of damage. a,b : potve real parameter to be determned by adjutng. In cae of dynamc behavor, damage varable calculated by mean of the followng equaton: Copyrght 0 ScRe.

2 56 A. BARAKA ET AL. Y Y 0 D Y η a + Y 0 3. Etablhng of the Unaxal Model for the Concrete Ung otropy hypothe to decrbe concrete behavor n monoaxal model, wll mplfy the equaton form, from a tonoral wrtng to a calar wrtng, hence the deformaton wll be wrtten a follow: σ β fd ε + (3) E D E D b b ( ) ( ) 0 0 The tre gven by the followng formula: ( ) 0 () σ εe D β fd (4) Hence the development law that we adopt for damage parameter D, the one gven by (). The Rate of volumnal refund of energy expreed by the followng equaton: ( ) ( D ) σ : σ + γ ( σ : σ trσ ) + β f Trσ I Y E Th rate take the followng unaxal form: Y 0 σ + β σ E0 ( D ) We expre Y n term of deformaton, by ubttutng (4) n (6): fβ Y ( E0ε + fβ) E0 D 4. Determnaton of the Parameter In th computatonal model [6,7,5], t neceary to determne the followng parameter: E 0 ; f ; β : repectvely, the modulu of elatcty n t orgn; compreve trength or tenle trength n tatc behavor and the parameter of non-elatcty determned n tatc; all gven tartng from the expermental curve. a, b : non dmenonal parameter characterzng the damage varable determned by adjutment on the curve of expermental behavor n tatc. Y 0 : the ntal threhold of damage: In compreon, t ha been condered more convenent to take nto account that the damage begn once f (5) (6) (7) exceedng concrete elatc tre n qua tatc tate [7], that to ay σ 0.98E ε (8) e Takng σe from par (8) and puttng D c 0 n the Equaton (6), we have: Y 0c 0 σe + βc fc σe E The ntal threhold of damage n tenon condered once t attan 90% of the tenle trength tre of concrete ft [4]. By ubttutng σ e 0.90 ft and D t 0 n the Equaton (6), we get: Y 0t f ( βt) t E 0 0 (9) (0) η : non dmenonal materal parameter of vcoty, calculated n term of dynamc ncreae factor (DIF) n compreon or tenon. 5. The Parameter of Vcou Work Hardenng η The parameter η nfluence the vcou repone of concrete behavor n compreon or tenon. Governed by the deformaton velocty varaton, t gve the form of the curve after work hardenng. The expreon of vcou work hardenng value drectly n term of tran rate ε reman poble, but complcated. Th why, we expre vcoty parameter n term of dynamc ncreae factor (DIF) noted R, whch t elf etmated from tran rate and retance charactertc of the concrete n tatc n CEB-FIP formula [6]. Bang on the expermental curve of Gary [7], Bhoff [] and thoe of Cotovo [8] n compreon and on expermental work of dynamc tenon of Brara [4] and Toutlemonde [9,0], alo by the ue of analytcal relaton of Ngo [] to verfy the fttng of deformaton ncreae n term of tenle tre ncreae; we have concluded that η evolvement can be decrbed by an equaton of the form: n η m R + ( ln ( R )) + ( m) () where, m and n are determned from two behavor chart obtaned n dynamc under dfferent tran rate bt reolvng a non lnear ytem of two equaton wth two unknown. 6. Dynamc Increae Factor (DIF) R f : Dynamc retance obtaned to ε ( compreon Copyrght 0 ScRe.

3 A. BARAKA ET AL. 563 or tenon). f : tatc retance obtaned to ε ( compreon or tenon). R: Dynamc ncreae factor (DIF). 6.. In Compreon We ue CEB-FIP formula [6] to ae dynamc ncreae factor value n compreon: 06α ε ε 30S ε R 3 ε γ ε > 30S ε ε : Stran rate, at a range of to 300 S. ε : Statc tran rate, at a range of logγ 6.56 α. α ; fc0 0 MPa. f c 5+ 9 fc0 6.. In Tenon () The aement of dynamc ncreae factor value n tenon performed through CEB-FIP formula [6]: 06α ε ε 30S ε R 3 ε γ ε > 30S ε ε : Stran rate, at a range to 300 S. ε : Statc tran rate, at a range log β 7.δ.33 δ ; fc0 0 MPa. f c fc0 7. Evoluton of the Deformaton and Rupture Crteron (3) In damage mechanc, the parameter D uually the prvleged ndcator of the rupture once t reache the value of. Yet, a the model for unaxal behavor of a fragle materal uch a concrete, we can adopt a rupture crteron by deformaton. The deformaton ll of dynamc falure ε d r, aeed n term of falure ll by tatc deformaton ε r combned wth a tran rate ε. Th wa done under everal type of formulaton, ether drectly n term of tran rate a one uggeted by J. Mazar [], or n term of dynamc ncreae factor a hown n Ngo and Mend []. Concernng our model, we ugget the decrpton of the amplfcaton of deformaton ndrectly n term of tran rate ung the followng relaton: d n ε ε R (4) r r m d Wth: εr εr, f R. The parameter m' and n' are determned by adjutng from two behavor curve (two chart are uffcent) obtaned under dynamc tet. 8. Smulaton of the BRARA Tenle Tet by Chppng In 997; the tenle tet by chppng were realzed n the unverty of Metz on ample concrete MB50. Hgh mpact peed (up to ε 30 S ) that permtted to have the chppng of the concrete pecmen, were reached by Hopknon Bar [4,3]. Hopknon Bar aemblng of th tet baed on the ejecton of a projectle that ht an mpact bar at a hgh peed, producng wave preadng through the mpact bar and the ample, whch generate a pullng on the free de of the ample. The ample are degned n cylndrcal form, whch 40 mm n dameter and ha length of 0 mm. They are mpacted by a metallc cylnder (entry bar) that ha the ame ecton wth one meter n length (Fgure ). Thee ample n MB50, concrete have the followng charactertc (Table ): Under two dfferent mpact velocte, the experment we have mulated are dvded nto two notaton: BE6 tet, the lowet wth a tran rate of 35.9 (S ); and BE tet whoe tran rate about two tme BE BE6, BE Tet The BE6 the lowet tet, t mpact velocty equal 7. Projectle Entry bar Gauge φ 40 mm 0 mm 0 mm φ 40 mm Specmen Fgure. Expermental aemblng of tenle tet on Hopknon bar. Table. Mechancal charactertc of the concrete ued n Brara tet. f c (MPa) f t (MPa) E 0 (MPa) Denty ρ (Kg/m 3 ) Copyrght 0 ScRe.

4 564 A. BARAKA ET AL. m, generatng a tran rate that equal ε 35.9 S. Under thee condton the pecmen ha hown a ngle falure durng tme perod of S at a dtance of 65.8 mm from the free edge. Rupture dynamc tre etmated at σ td 9. MPa. The BE tet ha an mpact velocty that equal 5 m, generatng a tran rate equal ε 70.4 S. After a tme perod exceedng gnfcantly S, two fure appear at the dtance of 4 mm and 69 mm from the free edge. Rupture dynamc tre eteemed at σ td 33.5 MPa around 69 mm. 8.. Model Adjutng The determnaton of the dfferent parameter eental to the calculaton, and acheved by adjutng on the tatc curve, ha gven the followng value a t 3.55 and b t.00. The tatc behavor chart made by our model ha got the form hown n fgure (Fgure ). After adjutng to dynamc behavor data, vcou parameter took the followng value m t and n t Model repone to dfferent tran rate hown on the fgure (Fgure 3) Numercal Smulaton of the Tet Fnte element grd we ued, ha a fber of mm alongde the ample (0 fber). Such dcretzaton enable u to follow the development of mechancal charactertc wth tme n ther locaton wth an accuracy of about mm. Concernng the boundary condton of the pecmen, the blockng up happen on the mpacted de; however the free de bear the appled expermental velocty. of the damage. The parameter D evolve to be mportant once mpact tme reache the value S, but the ultmate value of reached only at S, at the 6t fber at a dtance of 59 mm from the free edge (Fgure 5). Th value ndcate the poton of the mcro fure that caue the rupture, at a gap of 6 to 7 mm of expermental falure poton (65.8 mm from the free edge). The tme of rupture n the mulaton appearance dentcal to the expermental one ( S). At frt, tre dffuon through the cylnder happen a f t wa under tatc loadng, but after a certan perod of tme, tre development evolve wth tme n each fber eparately from the other, the mulated rupture tre σ tdr equal 9.09 MPa (Fgure 6), knowng that the expermental value σ tdr 9. MPa. Fnally, we turn attenton on nput force n the mulaton. Th force obtan the maxmum value of N, n a tme perod of t S, allow the comparon between expermental and mulated value (Table ) BE Tet Smulaton Under an mpact velocty V 0 5 m and gven tran rate that equal ε 70.4 S, BE tet mulaton ha hown that D ultmate value of reached uccevely and almot multaneou n two locaton, ndcat BE6 Tet Smulaton Th mulaton provded u wth a very rapd movement Fgure 3. Model Repone under dfferent tran rate obtaned wth the parameter m t and n t Fgure. Statc behavor curve of be ample made by our model wth value of a t 3.55 and b t.00. Fgure 4. Sample dcretzaton n mm fber. Copyrght 0 ScRe.

5 A. BARAKA ET AL. 565 Fgure 5. Smulated damage dtrbuton at T S. Fgure 6. Smulated tree dtrbuton at rupture at T S. Copyrght 0 ScRe.

566 A. BARAKA ET AL. ng the tartng of two ucceve falure at the tme t 4.94 0 5 S. The frt one appeared n the 5 t fbber, makng a dtance of 69 mm from the free edge.

6 566 A. BARAKA ET AL. ng the tartng of two ucceve falure at the tme t S. The frt one appeared n the 5 t fbber, makng a dtance of 69 mm from the free edge. The econd appeared n the 79 th fber, at a dtance of 4 mm from free edge (Fgure 7). Thee value how a good conformty between mulated and expermental reult. The maxmum value of the mulated rupture tre σ tdr equal 33.5 MPa (Fgure 8). It to be noted that th tre obtaned nce the tartng of the frt falure (5 t fber ) a our computatonal code top automatcally once the damage full flled n a gven fber (D ). The model gve an nput force value of N n a tme perod that equal t S. The table (Table 3) how the reult obtaned by model mulaton and expermental one. 9. Concluon The development of the model ha been acheved by ntroducng vcoty n damage evolvement, a fact whch allowed the regulaton of dynamc problem and the reproducton of velocty effect notced n the experment. In the bac model we have ued, we ntroduced vcou parameter by reproducng concrete behavor wth dynamc tmulaton, accordng to the avalable expermental reult n lterature. In practce, our model contan four parameter to dentfy by adjutng on expermental data, whch ha Table. A comparon between BE6 tet expermental and mulaton value. BE6 Tet Expermental Smulaton Falure Poton (free edge) (mm) to 60 Rupture tre σ tdr (Mpa) Falure tme 0 5 S Input force F max 0 4 N F max correpondng tme 0 5 S.8 to Table 3. A comparon between BE tet expermental reult and ther mulaton. BE Tet Expermental Smulaton Falure Poton (free edge) ( mm) 69 and 4 69 and 4 Rupture tre σ tdr (Mpa) Falure tme 0 5 S Input force F max 0 4 N F max correpondng tme 0 5 S to..63 Fgure 7. Smulated damage dtrbuton at T S of BE ample. Copyrght 0 ScRe.

A. BARAKA ET AL. 567 Fgure 8. Smulated tree dtrbuton n rupture at T 4.94 0 5 S of BE ample.

7 A. BARAKA ET AL. 567 Fgure 8. Smulated tree dtrbuton n rupture at T S of BE ample. been the cae for the majorty of the model uggeted n that way, except the fact that our model ha the advantage of etmatng dynamc ncreae factor trough a frt law elaborated by mean of experment namely (CEB- FIP) formula. Then, a econd adjutment ha been performed for each concrete, n term of t own repone to dynamc loadng, becaue vcoty problem not totally urrounded by (CEB-FIP) formula nce many concrete don t follow th law fathfully n ther dynamc ncreae factor. Fnally, the developed model agreement wth dfferent experment ha proved t performance n the decrpton of concrete dynamc behavor. Beng mplemented n a calculaton code of tructure n fnte element, t guarantee both the follow up of tructure repone n real tme and after the dappearance of acton, that to ay the aement of redual deformaton at the tme of falure. The model practcally ueful for the mulaton of alternated cyclc behavor n rapd dynamc, whch wll allow the reproducton of the atttude of tructure ubjected to emc tree. In a future propect, we hope expandng the model to take nto account devaton effect n an anotropc contnuou medum lke the concrete. Th model can be adapted to other materal whoe behavor are cloe to concrete behavor (fragle behavor),.e.: rock, ceramc, etc. Parng the model wth heat tranfer effect that take place n certan mpact epecally blat. Takng nto account ome mechancal phenomena occurrng when every concrete tructure under acton lke for example platc deformaton and fatgue. 0. Reference [] H. Renhard, Influence of Stre-Rate and Stran-Rate on Mechancal Properte of Concrete, CEB GTC, 4 Impact and Impulve Loadng of Concrete Structure, 985. [] P. Bhoff and S. Perry, Compreve Behavour of Concrete at Hgh Stran Rate, Materal and Structure, Vol. 4, No. 6, 99, pp do:0.007/bf04706 [3] C. Denoual, Probabltc Approach of the Impact Behavor of Slcon Carbde: Applcaton to Sheldng Mean, The, Ecole Normale upereure, Cachan, France, 998. [4] A. Brara, Expermental Study of the Dynamc Tenon of Concrete by Spallng The, Unverty of Metz, Metz, 999. Copyrght 0 ScRe.

8 568 A. BARAKA ET AL. [5] F. Camborde and C. Marott, Study of Dynamc Behavor of Brttle Materal by Dcrete Element, 4 Congre of Mechanc Touloue, France, 999. [6] C. Labordere, Phenomena n Unlateral Materal Damaged Modelng and Applcaton to the Analy of Concrete Structure, The, Unverty of Par VI, France, 99. [7] M. Benmanour, Modelng the Behavor of Cyclc Alternatng BA under Varou Statc Tet of Pole, The, ENPC Ecole, Par, France, 997. [8] J. F. Dubé, Smplfed Model Behavor and Vco-Damaged Concrete Structure, The, Ecole Normale upereure, Cachan,France,994. [9] P. Perzyna, Fundamental Problem n Vcoplatcty, Advance n Appled Mechanc, 966, Vol. 9, pp [0] J. C. Smo, T. Hughe, Computatonal Inelatcty-Interdcplnary Appled Mathematc, Sprnger-Verlag, New York, 998. [] F. Gatungt, Predctng Fracture of Concrete Structure Sought n Fat Dynamc, The, Ecole Normale upereure, Cachan, France, 999. [] A. Wnnck, C. J. Pearce and N. Bcanc, Vcoplatc Hoffman Contency Model for Concrete, Computer & Structure, Vol. 79, No., 00, pp do:0.06/s (00)000-3 [3] C. Labordere, Stratege and Model for Calculaton Concrete Structure, The, Unverty of Pau, Par, France, 003. [4] F. Barp, Impact Behavor of Concrete: Computatonal Approach, Engneerng Fracture Mechanc, Vol. 7, No. 5, 004, pp do:0.06/j.engfracmech [5] A. Baraka, Contrbuton to the Modelng of Cyclc Behavor of Concrete Alternate wth the Damage Mechanc, The, Unverty of Scence and of Technology, Oran, Algera, 00. [6] CEB-FIP, CEB-FIP Model Code 990, Trowbrdge, Wlthre, UK, 993. [7] G. Gary, Tet at Hgh Speed on Concrete, Specfc problem GRECO Publher, Par, 990. [8] D. Cotovo, Numercal Invetgaton of Structural Concrete under Dynamc (Earthquake and Impact) Loadng, The, Unverty of London, London, 004. [9] F. Toutlemonde, Impact Retance of Concrete Structure, the Behavor of the Materal Cotng, The, ENPC Ecole, Par, France, 995. [0] F. Toutlemonde, J. Sercombe, J-M. Torrent and R. Adelne, Development of a Contaner for the Storage of Nuclear Wate: Impact Retance, Edton Hermè, Revue françae de géne cvl, Vol. 3, 999, pp [] T. D. Ngo, P.A. Mend, D. Teo and G. Kuuma, Behavor of Hgh-Strength Concrete Column Subjected to Blat Loadng, Unverty of Melbourne, Melbourne, 00. [] J. Mazar, Recent Development n Search for Model Comprehenve Model Smplfed for Renforced Concrete Structure under Severe Stre, Numercal modelng and Engneerng of Contructon, Par, 005. [3] A. Brara and J. R Klepaczko New Expermental Study of Rupture of Concrete Spallng, Communcaton Congre GEO, France, 997. Copyrght 0 ScRe.

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction

Specification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear