Universities of Leeds, Sheffield and York

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1 promoting acce to White Roe reearch paper Univeritie o Leed, Sheield and York Thi i an author produced verion o a paper publihed in Cement and Concrete Compoite. White Roe Reearch Online URL or thi paper: Publihed paper Theodorakopoulo, D.D. and Swamy, R.N. (2008) A deign model or punching hear o FRP-reinorced lab-column connection. Cement and Concrete Compoite, 30 (6). pp White Roe Reearch Online eprint@whiteroe.ac.uk

2 A DESIGN MODEL FOR PUNCHING SHEAR OF FRP - REINFORCED SLAB - COLUMN CONNECTIONS D.D. Theodorakopoulo, * Proeor Department o Civil Engineering Univerity o Patra Patra GR 26500, Greece d.d.theod@upatra.gr R.N. Swamy, Proeor Department o Mechanical Engineering Univerity o Sheield Mappin Street Sheield S1 3JD, England * Correponding author Biography: Dimitrio D. Theodorakopoulo i a Proeor and the Director o Laboratory o Tranportation Work in the Department o Civil Engineering, Univerity o Patra, Greece. He i engaged in reearch in variou apect o concrete tructure, pavement material and deign. He alo received the ACI Deign Practice Award in 2005 award with Pro. R.N. Swamy. Honorary Lie Member and ACI Fellow Em Pro Narayan Swamy i at the Structural Integrity Reearch Intitute and the Center or Cement and Concrete, Univerity o Sheield, England. Hi reearch interet are in contruction material and concrete tructure. He ha received many Award or hi reearch, and he i the Founder and pat Editor o the Journal Cement and Concrete Compoite.

3 ABSTRACT The overall aim o thi paper i to develop a uniied deign method or the punching hear reitance o lab column connection irrepective o the type o internal reinorcement. In the irt part o the paper a deign model or the punching hear reitance o concrete lab-column connection reinorced with ibre reinorced polymer (FRP) i propoed. Thi deign model i baed on the author theoretical analyi or uch lab, which conider the phyical behavior o the connection under load. The eect o the inherent linear brittle repone, the lower elatic modulu and the dierent bond eature, a compared to teel, o the FRP reinorcement are all accounted or in the preent tudy. The propoed model doe not incorporate any itting actor to match the theory to the trend o the available FRP lab tet reult. The excellent agreement between the predicted and publihed tet reult hould give conidence to engineer and deigner in uing FRP a a ound tructural reinorcement or lab-column connection. It i then hown that the propoed deign model or FRP lab and the previou model o the author or teel reinorced lab are both identical in nature and tructure, thu contituting a uniied approach to deign or punching hear in lab. On the bai o the uniied model comparion and correlation between an FRP lab and a reerence teel reinorced lab, conirmed by the available tet reult, are preented. The uniied model alo enable the development o a more rational and reliable equivalent teel reinorcement ratio which can be applied to exiting code equation or teel reinorced lab to etimate the punching reitance o FRPreinorced lab. Keyword: Punching trength, Deign, Slab-column connection, Fibre-reinorced polymer (FRP) 2

4 INTRODUCTION The ue o FRP reinorcement in practice, epecially where the corroion o teel bar i a concern, i very much hampered by the abence o reliable deign method to determine the ultimate trength o tructural element, epecially lat lab and bridge deck, made with FRP-reinorced concrete. For example, although a ew deign method exit to predict the ultimate punching hear trength o lab-column connection reinorced with internal FRP reinorcement, mot o thee recommendation are either empirically baed to it the available tet data [1] or contitute a reinement o variou code prediction or teel-reinorced lab on account o the lower elatic modulu o FRP bar [1-6]. However, the applicability o the above mentioned modiied code prediction to FRP- reinorced lab i quetionable becaue o the dierence that exit between FRP and traditional teel reinorcement. FRP compared with teel, ha a brittle linear elatic repone, a hown in Fig. 1a, but more importantly, it ha many dierent bond eature. Punching hear tet reult reported by variou invetigator [1, 4-5, 7-10] relect thee dierence, and demontrate that they aect the ultimate punching load o an FRP lab. In a recent contribution, Theodorakopoulo and Swamy [11] have propoed a imple analytical model to predict the ultimate punching hear trength o FRPreinorced lab-column connection. The model i baed on the phyical behavior o the connection under load, and determine the depth o the compreion zone to account or the FRP elatic modulu, tenile trength and bond characteritic. The determination o the depth o the compreion zone i uually a major obtacle to any atiactory theory or the ultimate trength in hear. 3

5 The overall objective o thi paper i to preent a imple and reliable deign method, accounting or determining the hear capacity o FRP-reinorced lab-column connection at ultimate load. The uniquene o the propoed model lie on the way it i developed, and it i hown that thi model i identical in nature and tructure to that ued or the deign o teel reinorced concrete lab ailing in punching hear. Thi act oer engineer an uniied deign approach or the deign o thee tructural member, irrepective o whether the internal reinorcement i made o teel or FRP. Baed on the uniied deign model comparion and correlation between the punching hear trength o an FRP lab and a reerence teel lab are preented. In addition, a rational and reliable equivalent teel reinorcement ratio to etimate the punching hear trength o an FRP lab rom exiting code proviion or a teel reinorced lab i derived. MODIFIED CODE EXPRESSIONS FOR FRP SLABS To evaluate the punching hear capacity o FRP-reinorced lab, reearcher have modiied the code equation or teel-reinorced lab o ACI [12] and BS [13], given below, to account or the lower elatic modulu o FRP reinorcement. In what ollow, all quantitie are in metric unit, i.e., trength in N, tree in MPa, elatic moduli in GPa and dimenion in mm. According to ACI [12] the punching hear reitance o an interior quare column teel-reinorced lat lab, in the abence o lexural reinorcement a an inluential parameter, i given a 4

6 ACI Vc = 0.33 ' c b o d (1) where c i the peciied cylinder compreive trength o concrete, b o i the perimeter at the critical ection located at 0.5 d away rom the column ace and d i the average eective lab lexural depth. In BS [13], or teel reinorced lab, V c i calculated a V BS c = 0.79(100ρ ) 1/3 ( cu /25) 1/3 (400/d) 1/4 b p d (2) where cu i the cube concrete compreive trength, b p i the rectangular, regardle o the column hape, critical perimeter located at 1.5d away rom the column ace and ρ i the teel reinorcement ratio. Baed on thee equation the ollowing modiication have been propoed or FRP lab. El-Ghandour, Pilakouta and Waldron [2] modiied the ACI code equation by introducing the term (E /E ) 1/3, where E and E are the modulu o elaticity o FRP and teel, repectively. Thu, Eq. (1) or FRP reinorced lab, become, V 3 = 0.33 (E / E ) 1/ b d (3) ACI c,el c o The Intitution o Structural Engineer [3] recommended the ue o an equivalent area o teel in the BS 8110 equation, Eq. (2), by multiplying the actual area o the FRP reinorcement, ρ, by the modular ratio ρ = ρ E / E (4a) 5

7 which implie that, given the tructure o Eq. (2), the Code equation i again multiplied by the term (E /E ) 1/3. El-Ghandour et.al. [4] propoed a new modiication o Eq. (2), baed on their experimental work o FRP lat lab. According to thi approach the equivalent area o teel i obtained a in Eq. (4a), multiplied by a train correction actor, a hown below ρ = ρ (E /E )(0.0045/ y ) (4b) where i the propoed train limit or FRP reinorcement and y i the teel yield train. Thu, Eq. (2) or FRP lab become, 1/3 1/3 1/4 [ (E /E )(0.0045/ )] ( /25) (400/d) b d BS Vc, EL = ρ y cu p (4) Matthy and Taerwe [5] propoed the ollowing equation, or two-way lab reinorced with FRP bar or grid, a a modiication o BS 8110 equation V BS,MT c = / 4 [ 100ρ (E / E )] 1/ 1/ (1/ d) b d cm p (5) where cm i the mean cylinder concrete compreive trength at 28 day. Furthermore, Opina, Alexander and Cheng [1] propoed an empirical equation, baed on Eq. (5), given by 6

8 Vc,0AC = 2.77( ρ c) 1/ 3 (E / E ) 1/ 2 b p d (6) It can be een that in Eq. (6) the eect o modular ratio E /E i taken a the quare root intead o the cube root, in order to produce better reult, wherea the cale eect on the punching o lab with FRP reinorcement i omitted, ince thi eect wa reported not to be evident baed on the available FRP tet reult [1]. A comprehenive review on the reliability o mot o the above mentioned predictive equation o tet reult or FRP-reinorced lab can be ound in Opina et al [1]. They reported that among the punching hear trength etimator conidered, the modiied expreion (Eq.(5)) o the BS equation i clearly uperior with an average tet-to predicted trength ratio o the available tet reult o 1.17 and a tandard deviation o However, thi equation igniicantly underetimate the trength o the ix lab reinorced with Carbon FRP o erie C1, C2 and C3 teted by Matthy and Taerwe [5], with the average tet-to-predicted ratio being More recently, Opina [6] propoed the ollowing equation or predicting the punching capacity o two-way lab reinorced with either teel or FRP bar V = N b k d (7a) c, O c o where N i a contant equal to 5/6 (or c in MPa, b o in mm and d in mm). The term kd i the depth o the neutral axi auming elatic, cracked condition, where 1/2 2 E E E k = ρ 2 ρ ρ E + (7b) c Ec Ec 7

9 where E = E and ρ = ρ or teel lab, E = E and ρ=ρ E /E or FRP lab, and E c i the modulu o elaticity o concrete. It can be een that Eq. (7), in eence, contitute a modiication o the ACI 318 [12] equation through the introduction o the actor k, which repreent the eect o the lab reinorcement ratio, teel or FRP. Neverthele, Eq (7) i till a conervative predictor when applied to available FRP lab tet reult [6]. ANALYTICAL MODEL FOR FRP SLABS According to the theory o Theodorakopoulo and Swamy [11] the ultimate punching hear trength o FRP-reinorced concrete lab, accounting or the cale eect, i given a V u = ct cot θ ξ b p (X) (8) where (X) 2X (X ) X + (X ) = (8a) X = 0.25d (8b) In the equation above, 2 / 3 ct = 0.27 cu and cu are the tenile and cube compreive trength o concrete, repectively; θ i the mean angle o the ailure urace taken a 30, ξ =(100/d) 1/6 i the cale eect actor, and b p i the critical perimeter o BS 8110 deined in Eq. (2). Furthermore, X, which i independent o the material propertie, and (X ) are the neutral axi depth or critical hear ection and critical lexural 8

10 ection, repectively. (X) i taken a the harmonic mean o X and (X ), and repreent the combined neutral axi depth o the lab a explained in Re [14, 15]. Equation 8(a) or (X) expree, in eect, that the governing ailure load under punching hear i due to the complex moment hear interaction where punching i conidered a a orm o combined hearing and plitting, occurring without cruhing, but under complex three dimenional tree [14]. PROPOSED DESIGN EQUATION FOR FRP REINFORCED SLABS For the purpoe o evaluating the deign punching hear trength o FRP-reinorced lab-column connection, the calculation o the neutral axi depth o the lexural ection, (X ), at ailure, can ollow a procedure imilar to that propoed or the teelreinorced lab in Theodorakopoulo and Swamy [14]. Thu, adopting, or the ake o implicity, the rectangular concrete tre block aociated with ACI 318 (Fig. 1b) where the term 0.80 cu repreent the cylinder compreive trength o concrete and uing the equilibrium equation, one obtain 1 ρ ρ E (X ) = d = (0.25d) (9) cu cu where = / E and are the actual train and tre o FRP reinorcement, repectively. To evaluate the FRP train in Eq. (9) the analyi due to Theodorakopoulo and Swamy [11] i employed. Thi procedure aume that, becaue o the bond lip 9

11 ailure that occur at the inal tage o ailure o teted lat lab, the actual FRP train i a raction o the FRP train *, calculated on the aumption o perect bond and train compatibility, i.e., = k * (10a) or u * = k with k = 0.55 (10b) u Thu, the introduction o the coeicient k in Eq. (10) relect the bond characteritic o the FRP reinorcement wherea the aigned value o 0.55 ha been baed on inormation reported by Opina et al [1] rom tet on lat lab reinorced with gla ibre polymer reinorcement. Furthermore, baed on equilibrium o orce in the lexural ection, it ha been hown [11] that the FRP train *, normalized with repect to ultimate FRP train u, can be related to the normalized FRP reinorcement ratio ρ /ρ b by ρ ρ b = cu cu cu cu cu u ( )/( ) = * * * * cu + E cu + u E u cu/ u + / u / (11a) u / 1 Thu, olving Eq. (11a) with repect to * / u and making ue o Eq. (10b) one receive u 2 cu/u + (cu/u ) + 4(1 cu/u )/(ρ /ρb) = (11b) 2 * + 10

12 u 2 cu / u + ( cu/ u ) + 4(1 cu/ u )/(ρ /ρ b ) = k (11c) 2 + In the above, cu i the peciied value o the concrete compreive train at ultimate and ρ b i the FRP reinorcement ratio at balanced condition, that i, an FRP ratio where concrete cruhing and FRP rupture occur imultaneouly. The ρ b ratio, the value o which depend on the ultimate FRP train u conidered, i calculated on the aumption o perect bond between FRP and concrete, and train compatibility condition. The eectivene o uing the normalized ratio * / u, / u and ρ /ρ b in Eq. (11b-c) ha been explained in Theodorakopoulo and Swamy [11] and the main concluion drawn are ummarized a ollow. 1. The variation o * / u veru ρ /ρ b i almot independent or a wide range o u (0.0105, and ) and cu ( and ) value conidered, or the whole range o ρ /ρ b > 1. For ρ /ρ b 1, under the aumption o perect bond, FRP ailure govern and, thereore * / u = Similarly, the variation o / u veru ρ /ρ b (ollowing the lip behavior between FRP and concrete) i again almot independent or the wide range o u and cu value mentioned above. Thi in imple word mean that, wherea the value o ratio / u are dierent or variou value o u ued, the value o the actual FRP train i maintained nearly contant. Thi concluion i o great importance and it will be ued in the development o the propoed deign model or FRP lab in the next ection. 3. Under the condition o bond-lip, Eq. (11c), the new ratio ρ /ρ b that deine the limit o the lexure mode o an FRP-reinorced concrete lab i approximately 11

13 equal to 0.33, intead o 1.00 or the cae o perect bond, or any value o u conidered (ρ /ρ b range rom 0.34 to 0.32 or u = ). 4. The above mentioned value o ρ /ρ b = 0.33 depend only on the elected value o k = 0.55 and increae with increaing value o k i.e., i FRP reinorcement with better bond eature i ued. FRP deign equation or punching hear The oregoing conideration provide uicient background to allow the ormulation o the FRP deign model, thu relecting the tructural behavior o the lat lab ytem and enuring generality without any lo in accuracy. Thu, the value adopted in the preent tudy or the concrete compreive train at ultimate and or a reerence value o the ultimate FRP train (any value could be ued) are given a cu = , ud = (=3 cu ) and ud = E ud (12) Fig. 2, baed on thee value, how the relationhip between * / ud or / ud and the ratio ρ /ρ b. It can be een that, a mentioned previouly, or ρ /ρ b > 0.33 concrete cruhing in the lexural ection o a lat lab govern, wherea or ρ /ρ b 0.33 FRP rupture govern. By introducing the parameter α and λ deined a ρ ρ α = E ud ud = (13a) 0.145cu 0.145cu and λ = or ud * = k (13b) λ = ud ud 12

14 the combined neutral axi depth (X) Eq. (8a), on account o Eq. (9) and (13), i expreed a (X) 2α λ = (0.25d) (14) 1+ α λ The coeicient λ in Eq. (13b) indicate the tre or train at which the FRP reinorcement work at ailure tage. It i obviou that λ i alway le than unity or lab with ρ /ρ b > 0.33, which mean < ud. It i to be pointed out that, even though, the calculated value o α and λ depend on = 0. ud 0105, their product α λ = ρ E / cu i independent o any value o ud that could be ued. Thi mean that the value o ud i not an inluential parameter o the combined neutral axi depth in Eq. (14) and, thereore, o the deign punching trength derived below. From the lat expreion it can be een that i α λ = 1.00, then (X) = 0.25d, which implie, with the aid o Eq. (8b), that or thi particular FRP lab the depth o the neutral axi or both the hear and lexural ection are equal to 0.25d. Such a lab i deined a an FRP control lab. In addition, one can eaily conclude rom Eq. (14) that the combined neutral axi depth o an FRP lab decreae, not in a proportional way, with decreaing value o α λ. Evaluation o coeicient λ On ubtituting Eq. (13a) into Eq. (11a) and making ue o Eq. (12), one obtain that the ratio ρ /ρ b i equal to α, i.e., 13

15 ρ ρ b = α (or cu = and ud = ) (15) and, thereore, in what ollow, all comment mentioned previouly or ρ /ρ b, are alo valid or α. The unknown a yet value o λ in Eq. (13a) can be calculated or deign purpoe on account o Eq. (11c) (12) and (15), a ollow ( α ) k λ < 1 or α > 0.33 (16) = = / u 6 d From the above, it i apparent that the adopted value o = 0. ud 0105 in Eq. (12) i ully documented by the implicity o the relationhip between α ρ / ρ b and in Eq. (15) and λ and α in Eq. (16). In the light o the above conideration, Eq. (8) can take the orm V ud / 3 6 α λ = (100 / d) 1/ cu bp d or α > 0.33 (17) 2 1+ α λ which, in conjunction with expreion (13a) and (16), i the deign prediction equation or the ultimate punching trength o FRP-reinorced concrete lab-column connection. Equation (16) and (17), due to Eq. (15), are obviouly valid or α > 0.33 ince i) or α 0.33 FRP rupture govern and ii) the propoed FRP deign model i intentionally retricted to the cae where the punching hear capacity i le than the hear orce at the lexural capacity o a lab. However, the application o Eq. (17) to 14

16 tet lab with α 0.33, and reported to have ailed by a mixed ailure mode, that i, lexure punching or punching lexure might be jutiied. In uch a cae, the value o λ = / ud i obviouly calculated on the bai o = u, irrepective o whether the peciied FRP train u i le or greater than the reerence FRP train ud = , ince in a real tet lab the FRP reinorcement experience train up to the tenile train at ultimate, u. Veriication o tet reult and dicuion The propoed deign equation ha been applied to predict the punching hear capacity o 28 FRP-reinorced concrete lab reported in the literature. The geometry o the teted lab, the material propertie, the analyi and the reult are hown in Table 1. It can been een that the lab analyzed cover many variable that inluence punching hear behaviour, uch a, ize o loaded area, eective depth o lab, concrete trength, FRP reinorcement ratio and, very importantly, dierent type o FRP reinorcement with varied manuacturing procee, elatic modulu and ultimate tenile trength. For the propoed deign model the predicted-to tet punching hear trength ratio i with a tandard deviation o The latter i much le than 0.150, which i generally acceptable rom a tructural point o view. Thu, the deign model appear to be equally reliable and conitent a the author propoed theoretical analyi [11], and compare avourably to exiting deign model or FRP lab [1, 5-6]. It hould be pointed out that the propoed deign model, baed on the momenthear interaction, relect the phyical behaviour o an FRP-reinorced concrete labcolumn connection under load. It i derived entirely rom baic engineering principle and conider the ailure mechanim o FRP-reinorced lab, and in particular, it 15

17 incorporate, through the coeicient k ued or the calculation o the actual FRP train and the coeicient λ, the bond-lip behaviour between FRP bar and concrete. The latter play a dominant role in the ailure proce o all FRP-reinorced concrete tet lab [1,5,10], and thereore, there i a need or continuing reearch on the quality o bond between each type o FRP reinorcement and concrete to conirm the value o k on a broader bai. For example, reerring back to Table 1, it can be oberved that the propoed deign model underetimate coniderably the punching ailure load or erie C1, C2 and C3 in Matthy and Taerwe [5] tet, with the average o predicted-to-tet trength ratio or thee ix lab (rom No7 to No 12) being Thi underetimation may be explained in term o the poibly better bond characteritic o the particular type o reinorcement ued in erie C (Carbon NEFMAC) than thoe on which the value o k = 0.55 wa baed (Gla FRP) [11]. Thu, applying the propoed deign model to lab o erie C uing an increaed value o k by 30%, i.e., k = 1.30 x 0.55 = 0.715, it can be ound (not hown here) that a much better agreement between predicted-to tet trength or thee ix lab i obtained with the new average o ratio o Thi indicate an average increae in punching trength o only 18% (0.945/0.800 = 1.18), and thi i due to the act that, wherea the contribution o the critical lexural ection to ultimate punching reitance increae proportionally with increaing value o k (Eq (9) and (10b)), the contribution o the critical hear ection remain contant, (X =0.25d). The propoed deign model doe not incorporate any empirical actor to match the prediction to available FRP lab tet reult. A a reult, the propoed deign equation (17) i not ubject to any limitation a ar a the material propertie and reinorcement ratio are concerned. Indeed, although the available tet reult are limited, it i oberved rom Table 1 that the deign prediction are cloe to tet 16

18 trength even or lab H1 made with high trength concrete ( cu = MPa) and lab H2/H2 reinorced with a large amount o FRP reinorcement (ρ = 3.78 %). It i alo worth tating, baed on the development o the model, that any dierence between the ultimate trength and/or the elatic modulu o the FRP reinorcement peciied by the manuacturer and tet propertie o FRP i not a concern or the deign prediction o the punching ailure load, except or lab with low lexural reinorcement, a explained below. Thu, it i o interet to mention the deign prediction or FRP tet lab reinorced with value o α equal to or le than According to the propoed FRP deign model uch lab mut ail in lexure. In act, mot o thee lab in Table 1, uch a H1, SG1, SC1 and SG2 have been reported to have ailed either by a mixed (lexure-punching) or by a bond-lip mode. Normally, the comparion between the model prediction and tet load or thee lab hould not be included in Table 1, ince the model preented here i intentionally retricted or the cae where punching hear capacity i le than the hear orce, V lex, at the ultimate lexural capacity o the lab. However, becaue o the mode o ailure o the above mentioned lab, the hear capacitie o the lab can be conidered only lightly above the tet ailure load, and thereore a comparion between deign and tet reult can be made. Thi i jutiied by the V ud /V t ratio being 0.872, 0.993, and or tet lab H1, SG1, SC1 and SG2, repectively. It hould be noted that or thee lab, becaue o their mode o ailure, the value o λ (column 11, Table 1) ha been calculated, a explained in a previou ection, on the bai o Eq. (13b) and or = u. Finally, it i worth noting that or all tet lab hown in Table 1, the calculated value o α λ (column 12) are le than 1.00, a a conequence o the bond-lip behavior o FRP reinorcement. Thi indicate that the neutral axi depth o their lexural ection, on account o Eq (8e) 17

19 and (14), i le than that o the FRP control lab, i.e., le than 0.25d, thu veriying the act that lexural crack height in FRP reinorced member are expected to be larger than thoe in teel reinorced member. In what ollow, the author deign equation or teel-reinorced concrete lab [14] i briely preented and the two model are compared. In addition, a new equation o the equivalent teel ratio i propoed on the bai o equal deign prediction or two lab identical in all repect but the type o reinorcement. DESIGN MODEL FOR STEEL SLABS For the two-way normal concrete lab reinorced with teel bar, the ollowing deign equation ha been propoed or the ultimate deign trength, V ud (Theodorakopoulo and Swamy) [14]. By deining, α ρ y = cu λ = and (18) y with α or 0.20 α α or 0.50 α 1.00 λ = (19) α or 1.00 α α or 2.50 α

20 then V ud 1 2/3 1/6 2α λ = cu (100/d) bp d (20) 2 1+ α λ In the above, and y are the teel tre and the teel yield tre, repectively. Thereore, the coeicient λ in Eq. (19) indicate the eectivene o the teel tre, i.e., the tre at which the tenion teel work (either greater or le then y ) at the ultimate tage o punching. Detail o the calculation o λ can be ound in Re. [14]. It i again worth noting that the deign equation (20) or teel-reinorced lab, baed on the author theoretical analyi or punching hear o teel-reinorced lab [15], doe not employ any actor etimated empirically rom tet data. Furthermore, a in the cae o FRP lab, or the particular teel lab or which α λ = 1, it i implied that both the neutral axi depth o the hear and lexural ection are equal to 0.25d, and thereore, uch a lab i deined a a teel characteritic or control lab. It ha been hown that Eq. (20) predict the teel-reinorced lab tet reult in a better way than Deign Code with a maller tandard deviation [16]. Furthermore, Opina et al (teel lab SR-1) [1] and Matthy and Taerwe (teel lab R1, R1, R2 and R3) [5], cat thee teel reerence lab or comparion purpoe to their FRPreinorced lab. Applying Eq. (20) (not hown here) to the above mentioned teel lab one can ind predicted-to-tet trength ratio o or lab SR-1 and (on the mean) or lab R1, R1, R2 and R3. It i to be pointed out that thee ratio are o comparable magnitude to thoe (on the mean) o the correponding FRP-reinorced lab o thee reearcher. 19

21 A UNIFIED MODEL FOR PUNCHING SHEAR A comparion o the deign expreion (16)-(17) and (19)-(20) or FRP- reinorced and teel reinorced lab, repectively, how that the two model are identical in nature and tructure. Both model include all the key parameter that play an important role on punching hear behavior, uch a, ize eect, ize o the column area, lab eective depth, reinorcement ratio and concrete trength. It i obviou that they dier only in the value o αλ ince the parameter α λ expree the dierent engineering propertie and bond characteritic o the FRP reinorcement, a compared to parameter α λ or teel reinorcement. Alo, the term 2αλ/(1+ αλ) in both equation expree the interaction o the two critical ection conidered in developing the propoed equation, namely, hear and lexural. A a reult o thi moment-hear interaction, it can eaily be een rom the two deign equation that the inluence o the teel or FRP ratio and concrete trength on punching hear trength are not iolated and ingle contributor, a aumed in code equation. Finally, in addition to the above conideration it appear that the deign equation (17) and (20) retain the tructure and implicity o variou code equation or teel lab or modiied equation or FRP lab and, thereore, they are eay to apply by reearcher and deign engineer. Thu, a a concluion, it can be aid that a imple and reliable uniied deign model or punching hear trength o lab-column connection, baed on ound engineering principle, i poible and applicable to all lab irrepective o whether the internal reinorcement i made o teel or FRP. Baed on the uniied model, the punching trength o an FRP lab and a reerence teel lab are eay to compare and 20

22 correlate, and a new equation o the equivalent teel ratio i propoed in the next ection. Comparion between FRP and teel lab (experimental evidence) Matthy and Taerwe [5], in their ytematic FRP reinorced lab tet, given in Table 1, have compared the trength reult with thoe obtained by the teel reinorced reerence lab R1/R1, with c = 150/230 mm, d = 90mm, cu = 41.9 MPa, ρ = 0.58%, y = 500 MPa, E = 200 GPa, which imply α = 0.48 and λ = 1.24, and ailure load o 240/255 kn, repectively. The comparion o tet trength wa baed on the ollowing general characteritic: the lexural trength which i proportional to ρ u or ρ y, the equivalent reinorcement ratio ρ = ρ E / E given by Eq. (4a) and the lexural tine o the ection expreed by ρ E d 2 or ρ E d 2 or FRP and teel lab, repectively. Their general concluion are briely ummarized here or the ake o completene. FRP-reinorced concrete lab, uch a o erie C1 and CS deigned with a imilar lexural trength a reerence lab o erie R1, have igniicantly lower punching trength. Comparing lab with imilar eective depth and dierent type o lexural reinorcement, the obtained ailure load are roughly imilar or equal equivalent reinorcement ratio ρ E / E, uch a lab o erie R1, C2 and H2 or o erie C1 and CS. FRP-reinorced concrete lab deigned with a imilar lexural tine a teel reerence lab R1/R1 have imilar or higher punching trength or erie C2/C2 and C3/C3 and lightly lower punching trength or lab H2 and H3. 21

23 Comparing FRP lab with imilar lexural tine but with dierent eective depth and reinorcement ratio, uch a C2/C2 and C3/C3, the eect o increaing the lab depth on the punching reitance (comparing lab C1/C3) eem to be more pronounced than the eect o increaing the reinorcement ratio (comparing lab C1/C2). Comparing lab H2 with the teel reerence lab R1 o imilar lexural tine, it i concluded that to obtain imilar punching reitance the FRPreinorced lab hould have an FRP ratio that i uiciently higher than teel ratio. Baed on the above conideration, it can be aid that all three characteritic, namely, lexural trength, equivalent reinorcement ratio and lexural tine o the ection ued by Matthy and Taerwe [5] contitute a good bai or comparion purpoe between FRP and teel reinorced lat lab. However, none o thee characteritic account or parameter that inluence, a indicated by experimental evidence, the tructural behavior o a connection, uch a moment-hear interaction, lipping o the FRP reinorcement at ailure tage and level o the concrete trength value. Theoretical comparion between FRP and teel lab The comparion and correlation between FRP and teel reinorced concrete lab o thi ection have been obtained on the bai o the uniied deign model preented previouly. The cae o lab identical in all repect except the type and percentage o reinorcement and ailing in punching hear i examined. One can argue, baed on 22

24 the development o the uniied deign model, that the expreion o α and α are the mot repreentative parameter o the problem. Indeed, Eq. (13a) or α and Eq. (18) or α contain the quantitie ρ u and ρ y (or lexural trength), ρ E and ρ E (or equivalent reinorcement ratio) and, in addition, the concrete trength cu. Furthermore, coeicient λ and λ account or the bond between concrete and FRP (lip behavior) and teel reinorcement (perect bond), repectively. Figure 3 how the variation o the ultimate punching hear trength, Eq. (17) and (20), veru α and α or FRP and teel reinorced lab, repectively. The trength are normalized with repect to trength o the FRP or teel control lab (1/2) /3 cu ξ b p d, and thereore the obtained variation are valid or any level o the concrete trength value. It can be een that both variation are imilar, a ar a the pattern i concerned, and increae monotonically or the whole range o α or α value, approaching almot a horizontal line at high value o thee parameter. Thi coniguration ha our implication: two are eay to undertand, and the other two are more obcure. 1. FRP and teel lab with α = α. According to the curve in Fig. 3, FRP reinorced concrete lab deigned with the ame lexural trength (ρ ud ) a the reerence teel reinorced lab (ρ y ), which implie α = α, hould have igniicantly lower punching trength. Thi concluion o the theory i due i) to the lower elatic modulu o the FRP reinorcement, a compared to teel and ii) to the bond-lip behavior o the FRP reinorcement (k = 0.55). I a higher value i aigned in k, ay 0.715, to relect the ue o FRP reinorcement with better bond characteritic, a Carbon NEFMAC in lab o erie C1, C2 and C3 [5], the predicted punching trength 23

25 increae (not hown in Fig. 3), till remaining lower than the predicted trength o the reerence teel-reinorced lab. Thi i ully jutiied by the tet reult o Matthy and Taerwe [5], a mentioned beore, by comparing the lab o erie C1 and CS with the reernce teel lab R1. 2. FRP and teel lab with equal punching reitance. To obtain equal punching reitance between an FRP lab and a reerence teel lab, one can ollow the arrow hown in Fig. 3. It i clear that the FRP lab hould have an α value that i uiciently higher than α or reaon analogou to thoe o point 1. An example o thi are lab H2 (α = 2.49) and R1 (α = 0.48) with comparable magnitude o ailure load, being 231 kn and 240 kn, repectively [5]. Thi apect will be explained and dicued in detail in the next ection. 3. Eect o increaing reinorcement ratio. Given that α and α are proportional to ρ and ρ, repectively, it can be een rom Fig. 3 that the punching reitance o a connection increae with increaing reinorcement ratio, teel or FRP. It i alo oberved that or a given increaed reinorcement ratio either or FRP or teel lab, it eect on punching trength depend on the rank o the initial value o α (ρ ) or α (ρ ) conidered. For example, and reerring to teel lab, it can be ound that the percentage increae in punching hear reitance rom doubling the teel ratio ρ i 42% and 27% or initial value α o 0.35 and 1.00, repectively. In addition, thi concluion, given that dierent initial value o α may reult rom a change o cu, implie that the level o the concrete trength play a igniicant role on the eect o increaing 24

26 reinorcement ratio. Finally, it hould be noted, or the ake o comparion, that according to BS 8110 [13] deign equation, Eq. (2), the percentage increae in punching trength when the teel reinorcement ratio i doubled, i contant and equal to 26% ( 3 2 =1.26), i.e., independent o the teel ratio the initial lab i reinorced with and the level o the concrete trength. 4. Eectivene o FRP a compared to teel reinorcement. A cloe inpection o two variation o punching reitance in Fig. 3 reveal that the increaed predicted punching capacity baed on equal initial value o α and α (α = α ) and aociated with the ame increae in lexural reinorcement, i greater or teel than FRP lab. Thi reult can be attributed to both the lower elatic modulu (E < E ) and the bond-lip behavior o the FRP reinorcement, taken into account in developing the propoed FRP deign model. Tet reult rom both FRP and teel reinorced lat lab ully jutiy the above mentioned concluion. For example, reerring to lab GFR-1 and GFR-2 in Table 1, one can ind that the tet load increaed by 20% (260/217 = 1.20) when the FRP reinorcement ratio doubled rom 0.73% (initial value o α = 0.49) to 1.46%. On the other hand, Bae [17] reported that the percentage increae in punching reitance rom nearly doubling the teel ratio rom 0.73% (initial value o α = 0.52) to 1.63% wa 30%, wherea Tol [18] reported the ame average increae 30% when the teel ratio increaed rom 0.35% (initial value o α = 0.47) to 0.80%. Thee tet reult rom teel lab with comparable value o initial α but dierent initial value o ρ veriy, in eence, the role o concrete trength, mentioned in point 3 above. 25

27 A more in-depth dicuion o the eect o lexural reinorcement, reinorcement grade (i.e., yield trength, ultimate tenile trength o FRP bar), bond-lip behavior o the FRP reinorcement, concentration o reinorcement under the column and concrete trength (normal weight and high trength concrete) on punching trength cannot be accommodated within the length peciication o thi paper, and will thereore orm the ubject matter o another paper. Equivalent teel ratio The equivalent teel reinorcement ratio required to reine the variou code prediction or teel-reinorced lab, when the ultimate deign punching hear trength o an FRP-reinorced lab i needed, can eaily be etimated on the bai o the above mentioned uniied model, a ollow. By equating the deign prediction rom Eq. (17) and Eq. (20), one obtain α λ (21) λ = α λ or α / α = λ / Thu, Eq.(21), with the aid o Eq. (13a) and (18), ater rearranging the term, yield E λ ud ρ = ρ with ud = (22) E y λ Equation (22) indicate that, according to the author propoed deign expreion, the equivalent area o teel reinorcement can be obtained a in Eq. (4b) (it i noted a dierent value in the train limit o the FRP reinorcement between Eq (22) and (4b)) multiplied urther by a tre correction actor, expreed by λ /λ. 26

28 The unknown a yet tre actor λ / λ, which i equal to α / α,can be determined with the aid o Eq. (16), (19) and (21), i the value o α i known. Figure 4 how the variation o both α and α / α with repect to. It i oberved that, or equal deign prediction o an FRP lab and a reerence teel lab, the value α o α increae, a expected, with increaing α, although in a much leer degree. A a conequence o thi, the ratio α / α decreae rom 0.70 or =0.33 to 0.20 α or α =4.55 at a rather contant rate o decay or high value o α. On the other hand, low value o α produce high value o the ratio α / α = λ / λ and thi can be explained in term o the yielding behavior o the reulting low value o the equivalent teel ratio. Figure 4 alo how that, or the whole range o α, the ratio α / α = λ / λ i lower than unity due to the bond-lip behavior o the FRP reinorcement. Furthermore, equation (4a) and (4b) indicate that or given material propertie ρ,, E, E and, the equivalent teel ratio, taken a a percentage o cu y the FRP ratio ( ρ ρ / ), i contant. However, in the light o the above dicuion, it i apparent that, due to Eq. (22), the ratio ρ / ρ decreae with increaing value o, α which implie that even dierent concrete trength provide dierent equivalent teel ratio. Finally, it can be eaily ound that to obtain value o comparable magnitude or the equivalent teel ratio rom Eq. (4a) (22) or rom Eq. (4b) (22) the value o α hould be around o 4.5 (lab H2/H2 ) and 1.0 (lab C3/C3 and H3/H3 ), repectively. Thee reult are alo hown in Table 2, where the equivalent teel reinorcement ratio according to Eq. (4a), (4b) and (22), or tet lab in Matthy and Taerwe [5], are given or comparion purpoe. One can ee that the reult o Eq. (22) compare avorably to thoe o Eq. (4a-b) or lab R1 (reerence teel lab), C2 27

29 and H2, deigned, a mentioned previouly, with imilar eective depth and or roughly equal equivalent reinorcement ratio, and or which the obtained ailure load are o comparable magnitude. It i alo o interet to note that or lab CS/CS with a large value o E = 148 GPa, the equivalent teel reinorcement ratio according to Eq. (22) [0.28% = 0.19% (148/200) x (0.0105/0.0031) x (0.30/0.50)] i higher than the initial FRP ratio, ρ = 0.19%, or reaon analogou to thoe explained above. Taking the above a a whole, it can be aid that the calculation o the equivalent teel ratio on the bai o the uniied model, i more reliable than thoe baed on Eq. (4a) and (4b). In addition to FRP ratio (ρ ) and modular ratio (E /E ), parameter that igniicantly inluence punching trength, uch a reinorcement ultimate tree ( y, u, through y and ud ), bond eature o the lexural FRP reinorcement (k, through the value o λ ) and concrete trength ( cu, through the value o α and α ) are all accounted or. It i obviou that the ue o the propoed equivalent teel ratio o Eq. (22), to reine the variou code prediction or teel reinorced lab, will provide a reliable etimator or the punching hear trength o FRP reinorced lab, only i the code expreion ued i an accurate predictor or the punching trength o the o-called reerence teel lab. CONCLUSIONS The main concluion derived rom thi tudy may be ummarized a ollow: 1. A deign equation, Eq. (17), i developed to predict the ultimate punching hear trength o FRP-reinorced concrete lab. The approach i baed on the author theoretical analyi or uch lab, which conider the tructural behavior o the connection under load. 28

30 2. The propoed deign model account or the mechanical propertie o the FRP reinorcement, which are uiciently dierent rom thoe o teel, uch a, elatic modulu, ultimate tenile trength and, mainly, the bond characteritic. It, alo, incorporate no empirical actor to match the theory to the trend o the available FRP lab tet reult. A a reult, no limit are placed a ar a the material propertie are concerned. 3. The propoed predictive equation retain the tructure and implicity o the modiied code expreion or FRP lab. In addition, the contribution o the FRP reinorcement ratio and concrete trength on the punching hear trength are both incorporated in a combined way, thu relecting the dependence o the punching ailure load on thee interacting variable. 4. The prediction o the propoed deign equation are in excellent agreement with the available experimental ailure load o FRP tet lab, reported by variou invetigator. Alo, the propoed model compare avorably to exiting deign model or FRP lab, reported in the literature. 5. The propoed deign model or FRP lab and the previou model o the author or teel reinorced lab are both identical in nature and tructure, and include all the key parameter that igniicantly inluence punching hear behavior. Thu, the two model contitute a uniied model to deign or punching hear, irrepective o whether the internal reinorcement i made o teel or FRP. 6. With the aid o the uniied model a new equation o the equivalent teel ratio i propoed on account o a tre actor or teel and FRP. In addition, the uniied model accommodate the comparion and correlation between teel and FRP lab, veriied by experimental reult, in a reliable and conitent way. 29

31 7. Given the agreement between predicted and tet reult, it i concluded that the propoed uniied model provide a convenient and reliable ramework or the punching trength deign o lab reinorced with any type o reinorcement, teel or FRP. 30

32 NOTATION: b o = critical perimeter or punching hear-capacity evaluation, ACI b p = critical perimeter or punching-hear capacity evaluation (BS 8110, preent tudy) d = eective lexural depth o lab E = elatic modulu o teel E = elatic modulu o FRP ' c = peciied cylinder concrete trength cm = mean cylinder compreive concrete trength ct = tenile concrete trength cu = concrete cube trength, cu = c /0.80 = FRP tre u = ultimate tenile trength o FRP ud = ultimate tenile trength o FRP or deign purpoe, equal to E x = teel tre y = teel yield tre k = actor or the neutral axi depth k 1 = maximum concrete tre block parameter k = reduction actor or FRP reinorcement train N = coeicient V c = nominal hear reitance o a lat lab (code proviion) V lex = hear orce at ultimate lexural capacity (FRP lab) V t = ultimate tet punching trength (FRP lab) 31

33 V u = ultimate theoretical punching trength (FRP lab) V ud = ultimate deign punching trength (FRP lab) V ud = ultimate deign punching trength (teel lab) (X) = combined neutral axi depth (FRP lab) (X ) * = depth o compreion zone o the lexural ection (FRP lab- perect bond) (X ) = depth o compreion zone o the lexural ection (FRP lab bondlip) X = depth o compreion zone o the hear ection (teel lab and FRP lab) α = parameter equal to ρ y /0.145 cu (teel lab) α = parameter equal to ρ ud /0.145 cu (FRP lab) cu = ultimate concrete compreive train * = FRP train (perect bond, train compatibility) = FRP train (bond lip) u = ultimate tenile FRP train ud = ultimate tenile FRP train or deign purpoe, equal to = teel train c = teel train o the characteritic lab equal to y = teel yield train θ = angle o ailure urace λ = parameter equal to / y (teel lab) λ = parameter equal to / ud (FRP lab) ξ = depth correction actor equal to (100/d) 1/6 32

34 ρ = tenion teel reinorcement ratio ρ = FRP reinorcement ratio ρ b = balanced FRP reinorcement ratio (perect bond) 33

35 REFERENCES 1. Opina CE, Alexander SDB, Cheng JJR. Punching o Two-Way Concrete Slab with Fiber-Reinorced Polymer Reinorcing Bar or Grid. ACI Structural Journal 2003; 100(5): El-Ghandour AW, Pilakouta K, Waldron P. New Approach or Punching Shear Capacity Prediction o Fiber Reinorced Polymer Reinorced Concrete Flat Slab. In: Dolan CW, Rizkalla SH, Nanni A, editor. Proceeding o the Fourth International Sympoium on Fiber Reinorced Polymer Reinorcement or Reinorced Concrete Structure, SP-188, American Concrete Intitute, Farmington Hill, Mich., p Intitution o Structural Engineer. Interim Guidance on the Deign o Reinorced Concrete Structure Uing Fibre Compoite Reinorcement. The Intitution o Structural Engineer, London, El-Ghandour AW, Pilakouta K, Waldron P. Punching Shear Behavior and Deign o FRP RC Flat Slab. In: Silwerbrand J, Haanzadeh G, editor. Proceeding o the International Workhop on Punching Shear Capacity o RC Slab, Dedicated to Proeor Sven Kinnunen. Stockholm: TRITA-BKN Bulletin 57, p Matthy S, Taerwe L. Concrete Slab Reinorced with FRP Grid. II: Punching Reitance. ASCE Journal o Compoite or Contruction 2000; 4(3):

36 6. Opina CE. Alternative Model or Concentric Punching Capacity Evaluation o Reinorced Concrete Two-way Slab. ACI Concrete International 2005; 27(9): Ahmad SH, Zia P, Yu TJ, Xie, Y. Punching Shear Tet o Slab Reinorced with 3-D Carbon Fiber Fabric. ACI Concrete International 1993; 16(6): Banthia N, Al-Aaly M, Ma S. Behavior o Concrete Slab Reinorced with Fiber- Reinorced Platic Grid. ASCE Journal o Material in Civil Engineering 1995; 7(4): Zaghloul AE, Razaqpur AG. Punching hear behavior o CFRP reinorced concrete lat plate. In: Bruno D, Spadea G, Swamy RN, editor. Proceeding o the International Conerence on Compoite in Contruction, p El-Ghandour AW, Pilakouta K, Waldron P. Punching Shear Behavior o Fiber Reinorced Polymer Reinorced Flat Slab: Experimental Study. ASCE Journal o Compoite or Contruction 2003; 7(3): Theodorakopoulo DD, Swamy RN. Analytical Model to Predict the Punching Shear Strength o FRP-Reinorced Concrete Flat Slab. ACI Structural Journal 2007; 104(3): ACI Building Code Requirement or Structural Concrete and Commentary. Farmington Hill, Mich.: American Concrete Intitute,

37 13. Britih Standard Intitution. Structural Ue o Concrete, BS8110: Part 1 Code o Practice or Deign and Contruction. London: Britih Standard Intitution, Theodorakopoulo DD, Swamy RN. A Deign Method or Punching Shear Strength o Steel Fiber Reinorced Concrete. American Concrete Intitute, SP- 216, Innovation in Fiber-Reinorced Concrete or Value, 2003; Theodorakopoulo DD. Swamy RN. Ultimate Punching Shear Strength Analyi o Slab-Column Connection. Cement and Concrete Compoite, Special Theme Iue 2002; 24(6): Theodorakopoulo DD, Swamy RN. Deign Equation to Predict the Ultimate Punching Shear Strength o Slab-Column Connection. In: Brandt A, editor. Proceeding o the International Sympoium BMC7. Waraw, Bae GD. Some tet on the Punching Shear Strength o Reinorced Concrete Slab. Technical Report TRA/321. Cement and Concrete Aociation, London Tol P. Plattjockleken inverkan pabetongplattor ha llathet vid genomtanning. Forok med cikulara platter. Bulletin 146. Stockholm, Dept o Structural Mechanic and Engineering, KTH, (in Swedih with ummary in Englih). 36

38 TABLES AND FIGURES Lit o Table Table 1- Table 2- Predicted deign load compared with FRP tet punching trength Equivalent teel reinorcement ratio Lit o Figure Figure 1- Figure 2- Section orce equilibrium Variation o the normalized FRP train / ud (bond-lip) veru ρ /ρ b value: u = ud = Figure 3- Variation o the normalized deign punching trength (teel and FRP lab) veru α or α : k = 0.55 Figure 4- Relationhip between α or α /α value and α value or equal deign prediction (teel and FRP lab): k =

39 Reerence Slab No * c** (mm) Table 1. Predicted deign load compared with FRP tet punching trength d (mm) cu *** (MPa) ρ (%) u (Mpa) E (GPa) u (x10e3) CFRC-SN Ahmad et CFRC-SN al 7 CFRC-SN CFRC-SN Banthia et I al 8 II C C1' C C2' C Matthy C3' and CS Taerwe 5 CS' H H H2' H H3' SG EL- SC Ghandour SG et al 10 SG SC GFR Opina et GFR al 1 NEF Zaghloul & 9 Razaqpur * numbering o lab *** concrete trength at teting time Average ratio V t (kn) α **** λ α xλ V ud (kn) V ud /V t ** column width: quare or diameter **** α =ρ x E x / cu Standard deviation 0.102