Correlation of tensile and flexural responses of strain softening and strain hardening cement composites

Transcription

1 Availale online at Cement & Conete Composites 3 () Correlation of tensile an flexural responses of strain softening an strain harening cement composites Chote Soranakom, Barzin Moasher * Department of Civil an Environmental Engineering, Arizona State University, Tempe, AZ 57, Unite States Receive 3 May 7; receive in revise form January ; accepte 3 January Availale online Feruary Astract Close form equations for generating moment curvature response of a rectangular eam of fier reinforce conete are presente. These equations can e use in conjunction with ack localization rules to preict flexural response of a eam uner four point ening test. Parametric stuies simulate the ehavior of two classes of fier reinforce conete: strain softening an strain harening materials. The simulation reveale that the irect use of uniaxial tension an compression responses uner-preicte the flexural response for strain softening material while a goo preiction for strain harening material was otaine. The importance of strain softening range on the flexural response is iscusse using non-imensional post-peak parameters. Results imply that the rittleness an size effect are more pronounce in the flexural response of rittle materials, while more accurate preictions are otaine with uctile materials. It is also emonstrate that correlations of tensile an flexural results can e estalishe using normalize uniaxial tension an compression moels with a single scaling factor. Ó Elsevier Lt. All rights reserve. Keywors: Fier reinforce conete; Uniaxial tension test; Bening test; Flexural ehavior; Moment curvature response. Introuction Reinforcement of cementitious materials with short ranomly istriute fiers has een successfully practice for more than years [,]. Due to the low fracture toughness of cement-ase materials, tensile acks occur easily ue to applie stress, restraint, or environmental conitions. The interfacial on evelope etween the fiers an matrix utilizes the strength an stiffness of the fiers in reinforcing the rittle matrix. Once the matrix acks, loa still transfers aoss the ack faces through the riging fiers. As the loa on the composite is inease, the process of fier pullout affects loa carrying capacity an further contriutes to energy issipation. It has also een known when using a high volume fraction of fiers with a high specific surface area, the ack riging potential an the strength * Corresponing author. Tel.: + 95 ; fax: aresses: chote@asu.eu (C. Soranakom), arzin@asu.eu (B. Moasher). of the composite are inease [3 5]. An inease in volume fraction of the fier shows an enhancement of tensile strength in a variety of isete fier systems, incluing Steel Fier Reinforce Conete (SFRC) [ ], Glass Fier Reinforce Conete (GFRC) [9], Slurry Infiltrate Mat Conete (SIMCON) [,], Ductal [] an Engineere Cementitious Composite (ECC) [3 5]. Similar ehaviors are also oserve in continuous fier systems such as Ferrocement (FRC) [] an Textile Reinforce Cements (TRCs) [7,], which show improve tension capacity an uctility. In the continuous systems, the inease in strength is associate with istriute acking mechanisms an strain harening ehavior [9]. The non-linear ehavior of fier reinforce conete is est characterize y close loop controlle tests conucte in tension or flexure. Fig. a presents tensile stress strain response as compare to equivalently elastic flexural stress vs. eflection of Alkali Resistant (AR) glass faric reinforce composite material []. It is itically important to oserve that the general shapes of these 95-95/$ - see front matter Ó Elsevier Lt. All rights reserve. oi:./j.cemconcomp...7

2 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () 5 77 Nomenclature eam with C a parameter for normalize moment in Tale c a parameter to efine localize ack zone D a parameter for neutral axis epth ratio in Tale eam epth E tensile moulus E c compressive moulus E ci initial compressive moulus E post-ack tensile moulus F force component f stress at vertex in stress iagram h height of each zone in stress iagram I moment of inertia k neutral axis epth ratio L clear span M moment at first acking M fail moment at failure M i normalize moment M/M M low moment at the lowest point efore continue the n ascening curve M max maximum moment in moment curvature iagram M max,m max local maximum moment an in moment curvature iagram P total loa applie to four point ening eam specimen S spacing in four point ening test y moment arm from neutral axis to center of each force component a tu e c e e ctop e cu e cy e t e tot e trn e tu c g k k cu r c r r cst r cy r t x n normalize transition strain normalize tensile strain at ottom fier normalize ultimate tensile strain compressive strain first acking tensile strain compressive strain at the top fier ultimate compressive strain compressive yiel strain tensile strain tensile strain at ottom fier strain at transition point ultimate tensile strain normalize compressive strain normalize post-ack moulus normalize compressive strain normalize ultimate compressive strain compressive stress acking tensile strength constant tensile stress at the en of tension moel compressive yiel stress tensile stress normalize compressive yiel strain unloaing factor Susipts t, t, t3 tension zone,, 3 c, c compression zone,,,, 3, 3 stage,.,., 3., 3. accoring to the value of curves are similar as they represent initial linear portions followe y a range with a reuce stiffness that is ue to istriute acking. There is however a funamental ifference in the magnitues of stress from the tests an the associate eformations. The two main parameters characterizing the tensile response are the first acking tensile strength or Ben Over Point (BOP) an the Ultimate Tensile Strength (UTS). In the flexural loaing case, the first acking is referre to as the Limit of Proportionality (LOP) an ultimate strength as the Moulus of Rupture (MOR). The fact that the MOR value may e several times higher than the ultimate tensile strength UTS can e attriute to several parameters, incluing the nature of ata reuction, size effect, an also nature of the loaing. This isepancy has een well known in the fiel []. This phenomenon is est shown y comparing the cumulative proaility istriution functions for the four strength parameters. The istriution of BOP an UTS in tension followe y LOP an MOR in flexure are shown in Fig. using the results from thin section faric cement composites []. Note that there are funamental over-preictions of first acking an ultimate strength in flexure as compare to the tension case y as much as 3%. These inicate that use of flexural ata as funamental material properties in the esign of cement composites may e misleaing an unconservative. Close form relationships that explain such ifferences etween the tensile an flexural strengths have een recently propose y the authors for oth strain harening [,3] an strain softening type composites []. The present formulation comines an extens the availale moeling techniques using a unifie approach. In orer to correlate tensile an flexural ata for various materials, a unifie close form solution for generating moment curvature response of homogenize materials is presente for various fier reinforce composite systems. Since the solution is erive explicitly, iterative proceures require for hanling material non-linearity are not necessary; hence this metho is attractive for use in an inverse analysis algorithm to ack-calculate material parameters from convenient flexural tests. Furthermore, the moment curvature response generate y close form solution can e use as an input section property for a eam element in non-linear finite element analysis to preict flexural

3 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () Tensile Strain (mm/mm) Equivalent Flexural Stress (MPa) MOR LOP UTS Flexure Tension BOP 3 5 Flexural Deflection (mm) 5 5 Tensile Stress (MPa) Failure Proaility, p.... BOP UTS LOP MOR Tensile or Flexural Strength (MPa) Fig.. Relationship etween uniaxial tension test an flexural test of faric reinforce cement: (a) comparison of experimental responses an () comparison of cumulative proaility istriutions for LOP, MOR, BOP, an UTS []. ehavior of more complex structures. Finally, the close form solutions of the propose equations can e susequently use in a simplifie esign proceure for cement composites.. Derivation of close form solutions for moment curvature iagram Fig. presents a constitutive moel for homogenize strain softening an harening fier reinforce conete. As shown in Fig. a, the linear portion of an elastic perfectly plastic compressive stress strain response terminates at yiel point (e cy, r cy ) an remains constant at compressive yiel stress r cy until the ultimate compressive strain e cu. The tension moel in Fig. is esie y a trilinear response with an elastic range efine y E, an then post-acking moulus E. By setting E to either a negative or a positive value, the same moel can e use to simulate strain softening or strain harening materials. The thir region in the tensile response is a constant stress range efine with stress r cst in the post-ack region. The constant stress level l can e set to any value at the transition strain, resulting in a continuous or iscontinuous stress response. Two strain measures are use to efine the first acking an transition strains (e,e trn ). The tensile response terminates at the ultimate tensile strain level of e tu. The stress strain relationship for compression an tension can e expresse as >< E c e c e c e cy r c ðe c Þ¼ E c e cy e cy < e c e cu ðþ >: e c > e cu Ee t e t e >< Ee þ E ðe t e Þ e < e t e trn r t ðe t Þ¼ ðþ lee e trn < e t e tu >: e t > e tu where r c, r t, e c an e t, are compressive an tensile stresses an strains, respectively. In orer to erive the close form c cy= E t = E cst= E, E = E, < E = E c E = E, - < E cst= E, = cy= cu cu c trn= tu= tu t Fig.. Material moels for homogenize fier reinforce conete: (a) compression moel an () tension moel.

4 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () 5 77 solutions for moment curvature response in non-imensional forms, the material parameters shown in Fig. a an are efine as a comination of two intrinsic material parameters: the first acking tensile strain e an tensile moulus E in aition to seven normalize parameters with respect to E an e as shown in Eqs. (3) (5). x ¼ e cy e ; a ¼ e trn e ; tu ¼ e tu e ; k cu ¼ e cu e ð3þ r c ðkþ Ee r t ðþ Ee >< ck k x ¼ cx x < k k cu >: k cu < k >< þ gð Þ < a ¼ l a < tu >: tu < ðþ c ¼ E c E ; g ¼ E E l ¼ r cst ð5þ Ee The normalize tensile strain at the ottom fier an compressive strain at the top fier k are efine as: ¼ e tot ; k ¼ e ctop ðþ e e They are linearly relate through the normalize neutral axis parameter, k. ke k ¼ e k ðþ or k ¼ k k ð7þ Sustitution of all normalize parameters efine in Eqs. (3) () into Eqs. () an () results in the following normalize stress strain moels: In the erivation of moment curvature iagram for a rectangular oss section with a with an epth, the Kirchhoff hypothesis of plane section remaining plane for flexural loaing is applie. By assuming linear strain istriution aoss the epth an ignoring shear eformation, the stress strain relationships in Fig. a an are use to otain the stress istriution aoss the oss section as shown in Fig. 3 at three stages of impose tensile strain: <, < a an a < tu. For stage an 3 there are two possile scenarios: the compressive strain at top fier is either elastic ( k x) or plastic (x k k cu ). These cases will e treate in susequent sections. Normalize heights of compression an tension zones with respect to eam epth an the normalize magnitues of stress at the vertices with respect to the first acking stress Ee are presente in Tales an, respectively. The area an centroi of stress in each zone represent the force components an lines of action. Their normalize values with ctop= f c h c k y c F c h t y t F t tot= f t ctop= h c k f c y c F c ctop= h c h c k f c F c F c y c y c h t h t tot= f t f t y t F t y t F t h t h t tot= f t f t y t F t y t F t ctop= h c k f c y c F c ctop= h c f c F c h c k F c y c y c h t y t f t y t F t y t3 h t f t F trn t 3 h t3 f t3 3 F t3 tot= h t y t f t y t F t y t3 h trn t f t F t 3 h t3 f t3 3 F t3 tot= Fig. 3. Stress strain iagram at ifferent stages of normalize tensile strain at the ottom fier (): (a) an k x; (.) < a an k x; (.) < a an x < k k cu ; (c.) a < tu an k x; (c.) a < tu an x < k k cu.

5 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () Tale Normalize height of compression an tension zones for each stage of normalize tensile strain at ottom fier () Normalize height hc hc ht ht ht3 Stage ( ) an ( k x) Stage. ( < a) an ( k x) k xð kþ k k xð kþ k k ð kþð Þ Stage. ( < a) an (x < k k cu ) k ð kþð Þ Stage 3. ( > a) an ( k x) Stage 3. ( > a) an (x < k k cu ) k xð kþ k xð kþ k ð kþða Þ ð kþð aþ k ð kþða Þ ð kþð aþ Tale Normalize stress at vertices in the stress iagram for each stage of normalize tensile strain at ottom fier () Normalize stress fc Ee fc Ee ft Ee ft Ee ft3 Ee Stage ( ) an ( k x) Stage. ( < a) an ( k x) Stage. ( < a) an (x < k k cu ) Stage 3. ( > a) an ( k x) xc xc ck k ck k xc ck k Stage 3. ( > a) an (x < k k cu ) + g( ) + g( ) + g(a ) + g(a ) l l xc Tale 3 Normalize force component for each stage of normalize tensile strain at ottom fier () Normalize force component F c Ee F c Ee Stage ( ) an ( k x) ck ð kþ Stage. ( < a) an ( k x) ck ð kþ Stage. ( < a) an (x < k k cu ) Stage 3. ( > a) an ( k x) Stage 3. ( > a) an (x < k k cu ) xc ðk þ xk xþ xc ðk þ xk xþ x c ð kþ ck ð kþ x c ð kþ F t Ee F t Ee F t3 Ee ð kþ ð kþ ð kþð Þðg gþþ ð kþ ð kþð Þðg gþþ ð kþ ð kþða Þðga gþþ ð kþð aþl ð kþ ð kþða Þðga gþþ ð kþð aþl Tale Normalize moment arm of force component for each stage of normalize tensile strain at ottom fier () Normalize moment arm y c Stage ( ) an ( k x) Stage. ( < a) an ( k x) kþxð kþ y c y t y t 3 k 3 k xð kþ 3 3 ð kþ 3 ð kþ g g gþ3þ3 3ðg gþþ ð kþ y t3 Stage. ( < a) an (x < k k cu ) Stage 3. ( > a) an ( k x) Stage 3. ( > a) an (x < k k cu ) kþxð kþ 3 k xð kþ 3 ð kþ 3 ga ga gþ3aþ3 3ðga gþþ ð kþ ðaþþ ð kþ respect to acking tensile force Ee an eam epth are presente in Tales 3 an, respectively. Tale 5 shows the steps in etermination of net section force, moment, an curvature at each stage of applie tensile strain,. The net force is otaine as the ifference etween the tension an compression forces, equate to zero for internal equilirium, an solve for the neutral axis epth ratio k. The expressions for net force in stage an 3 are in the quaratic forms an result in two solutions for k. With a large scale of numerical tests covering a practical range of material parameters, only one solution of k yiels the vali value in the range < k < an it is presente in Tale. The internal moment is otaine y operating on the force components an their istance from

6 7 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () 5 77 Tale 5 Equilirium of force, moment an curvature for each stage of normalize tensile strain at ottom fier () Stage Tension Compression Force equilirium Internal moment k x F c + F t F c y c + F t y t. < a k x F c + F t + F t F c y c + F t y t + F t y t. < a x< k k cu F c F c + F t + F t F c y c + F c y c + F t y t + F t y t 3. > a k x F c + F t + F t + F t3 F c y c + F t y t + F t y t + F t3 y t3 3. > a x< k k cu F c F c + F t + F t + F t3 F c y c + F c y c + F t y t + F t y t + F t3 y t3 Note that curvature = e c /(k). Tale Neutral axis epth ratio, normalize moment an curvature for each stage of normalize tensile strain at ottom fier () Stage k M / ( k ¼ for c ¼ p ffiffi þ c þc for c < orc > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. k ¼ cþd c þdc D D ¼ gð þ Þþ c M ¼ ½ðc Þk3 þ3k 3kþŠ k M ¼ ðcþcþk3 3Ck þ3ck C k C ¼ g3 þ3g 3 gþ / ¼ ð kþ / ¼ ð kþ. k ¼ D Dþxc M ¼ð3xc þ C Þk C k þ C / ¼ ð kþ D ¼ gð þ Þþþx c pffiffiffiffiffiffiffiffiffiffi 3. k 3 ¼ D3 c D3 D3 c D 3 ¼ gða a þ Þþlð aþþa C ¼ g3 3g þ3 x 3 cþg M 3 ¼ ðc3 cþk3 3 3C3k 3 þ3c3k3 C3 k3 / 3 ¼ ð k3þ C 3 ¼ 3ðl la ga þa Þþga 3 þg 3. k 3 ¼ D3 D3þxc M 3 ¼ðC 3 þ 3xcÞk 3 C 3k 3 þ C 3 / 3 ¼ ð k3þ D 3 ¼ x c þ ga þ ðl ga la þ aþþg C 3 ¼ 3ðl la ga þa Þþga 3 x 3 cþg the neutral axis an the curvature is etermine as the ratio of compressive strain at top fier (e ctop = ke ) to the epth of neutral axis k. The moment M i an curvature / i at each stage i are then normalize with respect to the values at acking M an /, respectively an their close form solutions are presente in Tale. M i ¼ M i M ; M ¼ Ee ð9þ / i ¼ / i / ; / ¼ e ðþ As mentione earlier, the compressive strain at the top fier k in stage or 3 coul e either in elastic ( k x) or plastic (x < k k cu ) range, epening on the applie tensile strain an neutral axis parameter k. The range can e ientifie y assuming k < x [Fig. 3(.) or (c.)] an using the expression k or k 3 in Tale to etermine k from Eq. (7). Ifk < x hols true, the assumption is correct, otherwise k > x an the expression k or k 3 is use instea. Once, the neutral axis parameter k an the applicale case are etermine, the appropriate expressions for moment an curvature in Tale an Eqs. (9) an () are use. 3. Crack localization rules When a flexural specimen is loae eyon the peak strength, the loa eeases an two istinct zones evelop as the eformation localizes in the acking region while the remainer of the specimen unergoes general unloaing. To correlate the stress-ack with relationship into the stress strain approach, localization of major acks is simulate as an average response over the ack spacing region. Results are use as a smeare ack in conjunction with the moment curvature iagram to otain loa eformation ehavior. Fig. a presents the schematic moment curvature iagram with ack localization rules an Fig. shows a four point ening test with localization of smeare ack occurs in the mi-zone; while the zones outsie the acking region unergo unloaing uring softening [5,]. The length of the localize zone is efine as cs representing prouct of a normalize parameter c an loaing point spacing S = L/3, where L is the clear span. For the simulations of fier reinforce composites in this paper, it was assume that acks were uniformly istriute throughout the mi-zone an a value of c =.5 was use. Moment istriution along the length of a eam is otaine y static equilirium an the corresponing curvature is otaine from a moment curvature relationship. As shown y a soli curve in Fig. a, a typical moment curvature iagram is ivie into two portions: an ascening curve from to M max an a escening curve from M max to M fail. For a special case of low-fier volume fraction where an ascening curve from to M max, representing the tensile acking strength is followe y a sharp rop

7 Moment C. Soranakom, B. Moasher / Cement & Conete Composites 3 () non-localize zone (,M j- j- M max localize zone portions, the curvature corresponing to the loa step eyon the M low is etermine y the thir portion (M low to M max ). M M max ( j,m j M max M fail. Algorithm to preict loa eflection response of four point ening test M low loaing unloaing Curvature The loa eflection response of a eam can e otaine y using the moment curvature response, ack localization rules, an moment-area metho as follows. non-localize zone localize P/ zone cs S S/ Axis of Symmetry Fig.. (a) Moment curvature iagram an ack localization rules an () four point ening test. in the post-peak response, the post-peak moment curvature response exhiits two portions: a escening curve from M max to M low an ascening again from M low to M max. In this case, there are two local maxima, which either point coul e the gloal maximum. To preict loa eflection response, static equilirium is use an an array of loa steps is erive from a series of isete ata points along a moment curvature iagram. For each loa step, the moment an corresponing curvature istriution along the eam are calculate. While the specimen is loae from to M max (or M max ), the ascening portion of the iagram is use. Beyon the maximum loa, as the specimen unergoes softening, the curvature istriution epens on the localize or non-localize zones an its prior strain history (unacke or acke). For an unacke section, the curvature unloas elastically. If the section has een loae eyon M, the unloaing curvature of acke sections follows a quasilinear recovery path expresse as: / j ¼ / j n ðm j M j Þ ðþ EI where / j an M j represent the previous moment curvature state an / j, an M j are the current state. E an I represent the elastic moulus an the moment of inertia of unacke section. The unloaing factor n is etween an ; n = inicates no curvature recovery while n = is unloaing elastically with initial stiffness EI. An unloaing factor n = was use in the present stuy uner the assumption that acks o not close when material softens in isplacement control. For a section in the localize zone, the unloaing curvature is etermine from the escening portion of the moment curvature iagram (M max to M fail ) or (M max to M low ). For a special case of low-fier content that the moment curvature iagram is ivie into three P/ () For a given oss section an material properties, the normalize tensile strain at the ottom fier is inementally impose to generate the moment curvature response using Eqs. (9), (), an the expressions given in Tale. For each value of in stage an 3, the conition for compressive stress k < x or k > x is verifie in avance of moment curvature calculation. () Since a moment curvature iagram etermines the maximum loa allowe on a eam section, the isete moments along the iagram are use to calculate the applie loa vector P =M/S as shown in Fig.. (3) The eam is segmente into finite sections. For a given loa step, use static equilirium to calculate moment istriution along the eam an use moment curvature relationship with ack localization rules to ientify the curvature. () The eflection at mi-span is calculate y numerical moment-area metho of isete curvature etween the support an mi-span. This proceure is applie at each loa step to until a complete loa eflection response is otaine. A simplifie proceure for irect calculation of the eflection is presente in an earlier work []. 5. Parametric stuy of material parameters Two sets of parametric stuies were conucte to aress the ehavior of strain softening an strain harening materials. The flexural strength an uctility for each material parameter stuie were expresse as the normalize moment curvature response, which is inepenent of section size an first acking tensile strength. Fig. 5 presents the parametric stuy of a typical strain softening material with material parameters specifie as a compressive to tensile strength ratio cx =, normalize ultimate compressive strain k cu = 3 an ultimate tensile strain tu = 5. For each case of stuy, all parameters were hel constant to the typical values while the parameters sujecte to stuy were varie. In orer to avoi a iscontinuous tensile response at the transition strain a for strain softening materials, the post-peak moulus g is etermine y Eq. ()

8 7 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () 5 77 α=5, η=-. α=, η=-.7 α=5, η=-.7 α=., η= Strain (mm/mm) E = GPa ε =. μ =.33 β tu = 5 γ = ω = λ cu = Stress (MPa) μ=.99,η=-. μ=.7,η=-.37 μ=.33,η=-.7 μ=.,η= Strain (mm/mm) E = GPa ε =. α = β tu = 5 γ = ω = λ cu = Stress (MPa) γ=,ω=.5 γ=3,ω=3.33 γ=,ω=5 γ=,ω= -.. Strain (mm/mm) E = GPa ε =. η = -.7 α = μ =.33 β tu = 5 λ cu = Stress (MPa) Normalize Moment, M' Normalize Moment, M' Normalize Moment, M' 3 α=5 α= α=5 α=. 3 Normalize Curvature, φ' 3 μ=.99 μ=.7 μ=.33 μ=. 3 Normalize Curvature, φ' 3 γ=,ω=.5 γ=3,ω=3.33 γ=,ω=5 γ=,ω= 3 Normalize Curvature, φ' Fig. 5. Parametric stuy of a typical strain softening material: the effect of parameters a, l an c & x to normalize moment curvature iagram. ð lþ g ¼ ða Þ ðþ Fig. 5a epicts the compression an tension moel with the transition strain a varie from. to 5. Fig. 5 reveals that an inease in the transition strain a, ineases oth flexural strength an uctility. Fig. 5c epicts the material moel with the resiual tensile strength l varie from. to.99, simulating a range of rittle conete to elastic perfectly plastic response of high volume fraction fier reinforce conete. Fig. 5 shows that moment curvature iagram is quite sensitive to the variations in parameter l as it significantly affects the peak an post-peak response. The flexural response changes from a rittle to uctile material as l changes from. to.99. In orer to stuy the effect of compressive stiffness, the range of parameters c an x were use together to represent the inease in relative compressive to tensile stiffness from. < c/x < (/ to /.5) at a fixe compressive to tensile strength ratio (cx = ) as shown in Fig. 5e. Fig. 5f reveals that the changes in the relative stiffness slightly affect the peak moment from.7 to. an marginally ineases the stiffness of the moment curvature response. It is also conclue that the normalize compressive moulus c an

9 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () compressive yiel strain x have a marginal effect on the preicte moment curvature response as long as the compressive strength is aout one orer of magnitue higher than the first acking tensile strength. Fig. presents the parametric stuy of a typical strain harening material, with a compressive to tensile acking strength ratio of cx =, normalize ultimate compressive strain k cu = 3 an ultimate tensile strain tu = 5 similar to the previous case stuy. The post-peak response was ignore for this case y setting l to a very low value of.. Fig. a shows the compression an tension moel of a typical strain harening material with varying a =. 5. Fig. shows that the inease in a irectly ineases the normalize moment an curvature. Fig. c an emonstrate that ineasing the post-ack moulus g also significantly ineases the moment curvature iagram. Similar to the strain softening materials, the inease in relative compressive to tensile stiffness at constant compressive to tensile strength ratio cx (as shown in Fig. e) has a sutle effect to the moment curvature response as shown in Fig. f. These results inicate that the most significant parameters affecting the moment capacity are the transition strain a an the stiffness in the post-acking tensile range g. α = 5 α = α = 5 α =. E = GPa ε =. η = Strain (mm/mm) β tu = 5 μ =. γ = ω = λ cu = 3 η=.99 η=.7 η=.33 η=. E = GPa ε =. a = - - Stress (MPa) -.. Strain (mm/mm) β tu = 5 μ =. γ = ω = λ cu = 3 γ=,ω=.5 γ=3,ω =3.33 γ=,ω=5 γ=,ω= - - Stress (MPa) E = GPa ε =. -. η =.33 a =. Strain (mm/mm) - β tu = 5 μ =. λ cu = 3 - Stress (MPa) Normalize Moment, M' Normalize Moment, M' Normalize Moment, M' α=5 α= α=5 α=. 3 Normalize Curvature, φ' η=.99 η=.7 η=.33 η=. 3 Normalize Curvature, φ' γ=,ω=.5 γ=3,ω=3.33 γ=,ω=5 γ=,ω= 3 Normalize Curvature, φ' Fig.. Parametric stuy of a typical strain harening material: the effect of parameters a, g an c an x to normalize moment curvature iagram.

10 7 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () Preiction of loa eformation response The algorithm to simulate loa eflection response of a eam uner four point ening test was use as a preictive tool to stuy three classes of materials: SFRC with.5% an.% volume fraction representing strain softening eflection softening an eflection harening materials, respectively, an ECC with.% volume fraction representing strain harening material... Simulation of Steel Fier Reinforce Conete (SFRC) Two mixes of SFRC (H an H) that use hook-en fiers at volume fraction levels of.5% an.% were selecte from the literature [7,] to emonstrate the algorithm to preict loa eflection responses. Tensile og one specimens of 7 mm in net imensions with an enlarge with of mm at the ens were use. The flexural four point ening specimens were mm with a clear span of 75 mm. The average material properties were: compressive strength fc ¼ 3 MPa, initial compressive moulus E ci =.5 GPa an initial tensile moulus E = 5. GPa. The first acking tensile strain e for mix H an H are. an. miostrains, respectively. As shown in the parametric stuies, the material parameters for compression moel have a marginal effect to the preicte flexural response as long as the compressive strength is as much nine times greater than the tensile strength []. Therefore, typical values for compression parameters can e estimate without severely affecting the results. The compressive yiel stress f cy was assume to e.5fc an compressive moulus E c was estimate to e.5e ci. The normalize compressive moulus c was then otaine y E c /E an the normalize compressive yiel strain y x = f cy /(E c e ). The range of ultimate compressive strain e cu etween.35 an. was suggeste y several researchers [7,]. The value of. was selecte in this stuy an the corresponing normalize value was calculate y k cu = e cu /e. The material parameters for tension moel were etermine y fitting the moel to the uniaxial tension test result as shown y the soli line in Fig. 7a an c. All other parameters use in the simulation of flexural ening of mix H an H are provie in the figures. For eflection-softening material represente y mix H, the constant post-peak stress level l =., post-ack stiffness g =.7, post ack stiffness ultimate strain capacity a =., total tensile strain level tu = 3, compression to tension stiffness ratio c =.95 an compression Tensile Stress (MPa) Moifie moel (.ε ) Fitte moel Experiment = mm, = mm, L = 75 mm, E = 5. GPa, ε =. μstr, η = -.7, α =., μ =., β tu = 3, γ =.95, ω =., λ cu = 3. Equivalent Flexural Stress (MPa) Moifie moel (.ε ) Fitte moel Experiment Tensile Strain (mm/mm) Deflection (mm) Tensile Stress (MPa) = mm, = mm, L = 75 mm, E = 5. GPa, ε =. μstr, η = -., α = 3., μ =., β tu = 9, γ =.95, ω =.3, λ cu = 3. Moifie moel (.ε ) Fitte moel Experiment.5..5 Tensile Strain (mm/mm) Equivalent Flexural Stress (MPa) Moifie moel (.ε ) Fitte moel Experiment Deflection (mm) Fig. 7. Simulation of a Steel Fier Reinforce Conete (SFRC): (a an ) tension moel an flexural response of mix H (V f =.5%); (c an ) tension moel an flexural response of mix H (V f =.%).

11 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () to tension strength ratio of x =. were use. The soli curve in Fig. 7 shows the preicte flexural response of eflection softening material (mix H) from the simulation process. It is shown that use of uniaxial tension ata unerpreicts the flexural response for this class of material. This is attriute to ifferences in the stress istriution profiles of the two test methos. In the tension test, the entire volume of the specimen is a potential zone for ack initiation. Comparatively, in the flexural test, only a small fraction of the tension region is sujecte to an equivalent ultimate tensile stress. To quantify the ifferences etween the equivalent tensile strengths of tension an flexure, the authors propose a single scaling parameter. By using a multiplier that is applie to the first acking tensile strain e, associate strains an stresses inease y the same scale, resulting in a uniform inease in material strength. An inverse analysis y trial an error was conucte to ientify the appropriate scaling parameter for this case an it foun that the material moel shoul e scale up uniformly y %. The ash lines in Fig. 7a an show the moifie strength an the new preiction, which provies a etter match to the experimental results. If tensile an flexural responses of several samples are utilize, the inverse analysis proceure can e use to estalish the statistical relationship etween the tensile an flexural responses as have een one for other rittle materials [9]. For eflection-harening materials represente y mix H, the soli curve of tension moel that fits to the uniaxial tension test ata are shown in Fig. 7c. The constant post-peak stress level l =., post-ack stiffness g =., transition strain a = 3., ultimate tensile strain level tu = 9, compression to tension stiffness ratio c =.95 an compressive yiel strain to first acking tensile strain ratio of x =.3 were use. The preicte flexural response is shown as a soli line in Fig. 7. The algorithm slightly unerestimates the flexural response. The inverse analysis foun that the strength of the uniaxial moels shoul e inease y % for a reasonale preiction of flexural results as shown y the ashe lines in Fig. 7c an. Note that the correlation of experimental an simulate responses in the eflection harening range is quite reasonale; the isepancy etween the fitte an moifie tension moels is much lower than that of strain softening materials. This inicates that for the eflection harening materials, the moel preictions are quite reasonale... Simulation of Engineere Cementitious Composite (ECC) An ECC mix that use polyethylene fiers at volume fraction levels of.% from the literature [,5] was selecte to emonstrate the aility of the algorithm to preict loa eflection response for strain harening material. The flexural specimens for the four point ening test were mm with a clear span of 3. mm. The average material properties were: compressive strength f c ¼ MPa. The initial compressive moulus E ci =.35 GPa was otaine y ack-calculation of the initial flexural loa eflection response. The initial tensile moulus E =.75 GPa an the first acking tensile strain e = miostrains were otaine irectly from the uniaxial tensile test results. The constant post-peak stress level l =.9, post-ack stiffness g =.9, post-ack strain capacity a = 95., ultimate tensile strain level tu = 7, compression to tension stiffness ratio c =.9 an compressive yiel strain to first acking tensile strain ratio of x =. were use. The compressive yiel stress f cy was assume to e.fc an compressive moulus E c was estimate equal to E ci. The ultimate compressive strain e cu was assume to e.. The material parameters for tension moel were etermine y fitting the moel to the uniaxial tension test result as shown y the soli line in Fig. a. All parameters use in the simulation are provie in the same figure. The soli curve in Fig. shows the preicte flexural response otaine y the simulation process. The preiction for the strain harening material uring the pre- an post-ack stages agree well with the experimental results; note that the formation of the istriute ack system can Tensile Stress (MPa) = 7. mm, =. mm, L = 3. mm, E =.75 GPa, ε = μstr, η =.9, α = 95., μ =.9, β tu = 7, γ =.9, ω =., λ cu =. Fitte moel Experiment.... Tensile Strain (mm/mm) Equivalent Flexural Stress (MPa) Fitte moel Experiment Deflection (mm) Fig.. Simulation of Engineere Cementitious Composite (ECC): (a) tension moel an () flexural response.

12 7 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () 5 77 e aequately esie y the smeare pseuo-strain moel. 7. Discussion The three simulations of fier reinforce conete materials, ranging from low to high fier contents an material response changing from softening to harening, inicate that as the post-peak tensile strength of material ineases, the use of uniaxial response to preict flexural response ecomes more accurate. It also implies that the inease of fier contents in the experiments can effectively suppress the flaws from initiating ack that leas to premature failure, especially uner uniaxial tensile conitions. Therefore, the flaw size istriution in material is less sensitive to the constant vs. linear istriution of tensile stress patterns in uniaxial an flexural specimens, respectively.. Conclusions This paper presents close form solutions for generating moment curvature iagrams of a rectangular eam mae of homogenize fier reinforce conete. The algorithm using a moment curvature iagram with ack localization rules an moment-area metho to preict loa eformation of a eam uner the four point ening test was also evelope. The normalize moment curvature iagram otaine y close form solutions were use in parametric stuies of strain softening an strain harening materials. For a typical strain softening material, the most important factors to flexural strength an uctility are the constant resiual tensile strength an the strain at ultimate tensile strength. For a typical strain harening material, oth tensile strain at transition an post-ack moulus are almost equally important to the flexural ehavior of a eam. Strain softening an harening materials share the similarity that the change of relative compressive to tensile stiffness slightly affects the flexural response as long as the compressive to tensile strength ratio is greater than 9. Loa eflection responses of four point ening tests reveale that the use of uniaxial tensile ata tens to uner-preict the flexural response for the material that has a relatively low post-ack tensile strength. This was in part ue to a ifference in stress istriution etween the uniaxial tension an ening tests. The uner-preicte loa eformation response can e correcte y ineasing strength of the uniaxial material moel using a scaling parameter to the first acking strain. With this approach, other associate strains an stresses will susequently e inease y the same factor. With proper scaling parameters, the preicte responses agree well with experimental oservations. It is also oserve that as fier content, or on is inease such that the post-ack tensile strength is improve, the size effect oserve in preicting the response of flexural samples tens to e reuce an the use of uniaxial response to preict flexural response ecomes more accurate. Acknowlegement The authors acknowlege financial support of the National Science Founation, Program 39-3 uner Dr. P. Balaguru. References [] Moasher B, Shah SP. Test parameters for evaluating toughness of glass fier reinforce conete panels. ACI Mater J 99;(5): 5. [] Moasher B, Li CY. Mechanical properties of hyri cement-ase composites. ACI Mater J 99;93(3): 9. [3] Romuali JP, Batson GB. Mechanics of ack arrest in conete. Proc ASCE 93;9(3):7. [] Majumar AJ, Laws V. Composite materials ase on cement matrices. Philos Trans R Soc Lon Ser A: Math Phys Sci 93: 9. [5] Moasher B, Li CY. Effect of interfacial properties on the ack propagation in cementitious composites. Av Cement Base Mater 99;(3 ):93 5. [] Umekawa S, Nakazawa K. On mechanical properties of stainless steel fier an fier-reinforce stainless-sn PB alloy composite. J Jap Inst Metals 97;3(): 7. [7] Lim TY, Paramasivam P, Lee SL. Analytical moel for tensile ehavior of steel fier conete. ACI Mater J 97;(): 9. 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13 C. Soranakom, B. Moasher / Cement & Conete Composites 3 () the eighth international symposium on rittle matrix composites BMC, Warsaw, Polan;. p [] Soranakom C, Moasher B. Close form solutions for flexural response of fier reinforce conete eams. J Eng Mech 7;33():933. [5] Ulfkjaer J, Krenk S, Brincker R. Analytical moel for fictitious ack propagation in conete eams. J Eng Mech 995;():7 5. [] Olesen JF. Fictitious ack propagation in fier-reinforce conete eams. J Eng Mech ;7(3):7. [7] Swamy RN, Al-Ta an SA. Deformation an ultimate strength in flexural of reinforce conete eams mae with steel fier conete. ACI Struct J 9;7(5): [] Hassoun MN, Sahejam K. Plastic hinge in two-span reinforce conete eams containing steel fiers. Proc Can Soc Civil Eng 95:9 39. [9] Noguchi K, Matsua Y, Oishi M. Strength analysis of Yttriastailize tetragonal Zirconia polyystals. J Am Ceram Soc 99;73(9):7 7.