Predicting the Shear Capacity of Reinforced Concrete Slabs subjected to Concentrated Loads close to Supports with the Modified Bond Model

Transcription

1 Predicting the Shear Capacity of Reinforced Concrete Slab ubjected to Concentrated Load cloe to Support ith the Modified Bond Model Eva O.L. LANTSOGHT Aitant Profeor Univeridad San Francico de Quito Eva Lantoght, born 1985, received her civil engineering degree from Vrije Univeriteit Bruel (008), M.S. from the Georgia Intitute of Technology (009) and PhD from Delft Univerity of Technology (013). She i a reearcher at TU Delft and aitant profeor at USFQ. Summary Cor VAN DER VEEN Aociate Profeor Delft Univerity of Technology C.vanderveen@tudelft.nl Cor van der Veen received hi civil engineering degree from Delft Univerity of Technology in 198 and PhD in At preent he i aociate profeor in the reearch group of concrete tructure. Ane DE BOER Senior Advior Minitry of Infratructure and the Environment ane.de.boer@r.nl Ane de Boer, born 1951, received hi civil engineering and PhD degree from Delft Univerity of Technology and i orking a enior advior at the Dutch Centre for Infratructure. The hear problem i typically tudied by teting mall, heavily reinforced, lender beam ubjected to concentrated load, reulting in a beam hear failure, or by teting lab-column connection, reulting in a punching hear failure. Slab ubjected to concentrated load cloe to upport, a occurring hen truck load are placed on lab bridge, are much le tudied. For thi purpoe, the Bond Model for concentric punching hear a tudied at firt. Then, modification ere made, reulting in the Modified Bond Model. The Modified Bond Model take into account the enhanced capacity reulting from the direct trut that form beteen the load and the upport. Moreover, the Modified Bond Model i able to deal ith moment change beteen the upport and the pan, a occur near continuou upport, and can take into account the reduction in capacity hen the load i placed near to the edge. The reulting Modified Bond Model i compared to the reult of experiment that ere carried out at the Stevin laboratory. A compared to the Eurocode (NEN-EN :005) and the ACI code (ACI ), the Modified Bond Model lead to a better prediction. Keyord: arching action; bond; compreive trut; continuou upport; edge loading; punching hear; hear; lab; torion. 1. Introduction The hear problem i typically addreed by tudying to ell-defined cae: 1) the one-ay hear capacity of beam (beam hear), and ) the to-ay hear capacity of lab (punching hear). Oneay hear in beam i mot often tudied on mall, heavily reinforced, lender beam, teted in four point bending [1, ], reulting in the emi-empirical expreion a given in NEN-EN :005 [3] and ACI [4] for the beam hear capacity. To-ay hear in lab i tudied on lab-column pecimen [5]. Thee experiment form the bai of the emi-empirical punching hear proviion a given in NEN-EN :005 and ACI Beide thee to tandard cae of the hear problem that have been idely tudied over the pat decade [6], other loading cae, often at the interection of beam hear and punching hear, arie in practice. An example i the hear capacity of exiting reinforced concrete olid lab bridge, ubjected to concentrated live load hen located cloe to the upport, reulting in large hear tree at the upport [7]. The available code proviion are not fully uitable to determine the hear capacity of lab ubjected to concentrated load cloe to upport, a problem at the interection beteen one-ay and to-ay hear. To decribe the behaviour of reinforced concrete lab ubjected to concentrated load cloe to the upport, a ne model i propoed. Thi model i a combination of load-bearing quadrant and trip, and i baed on the Bond Model [8, 9]. The reulting Modified Bond Model can be conidered a mechanical model, in hich the concept of a limiting one-ay hear tre i incorporated. Where mot beam hear and punching hear model make a trict ditinction beteen thee to mode of

2 failure, the Bond Model conider the hear-carrying behaviour a an action of to-ay quadrant and one-ay trip. A uch, it i the mot uitable model for the conidered cae that i a combination of one-ay hear and to-ay hear.. Bond Model for concentric punching hear direction of reinforcement Column l trip Radial trip The Bond Model for concentric punching hear [8, 9] i a mechanical model for lab-column connection that explain the load tranfer beteen plate and column by combining radial arching action and the concept of a critical hear tre, a ued for beam hear. Shear, V, (moment gradient) reult here the magnitude of the force T or lever arm z varie along the length of the member. A uch, hear i carried by a combination of beam action and arching action: dtz ( ) dt ( ) dz ( ) V z T (1) dx dx dx For lab, arching action, expreed by the radial compreion trut, i the dominant mechanim in the radial direction. It i aumed that the load i ditributed in the radial direction from the column by arching. remote end: zero hear In the Bond Model, arching i repreented by four trip branching out from the column, parallel to the Fig. 1: Layout of trip reinforcement, Fig. 1. Thee trip eparate the column from the lab quadrant. The length of the trip, l trip, i determined from the column to a remote end, a poition of zero hear. The trip are loaded in hear on their ide face only and are decribed a cantilever beam a hon in Fig.. Thee cantilever have negative and poitive moment capacitie of M neg and M po that can be combined into M, the total flexural capacity of the trip. At the ide of the column, the axial load P AS,1 i acting. The length l i the loaded length of the trip, and the uniformly ditributed load. The loading term i an etimate of the hear that can be delivered by the adjacent quadrant of the lab to one ide face of the trip. For a trip ith to ide face, the total uniformly ditributed load on the trip i. Uing force and moment equilibrium of the cantilever trip (Fig. ) reult in the folloing expreion for the total flexural capacity M and the concentrated load P AS,1 : l M () ide of the column or load remote end PAS,1 l (3) l trip Solving Eq. () for the unknon l loaded length l and ubtituting thi into Eq. (3) reult in: PAS,1 M (4) M To find the maximum column axial M po neg load P AS, the capacity of all four trip can be ummed: P AS,1 M po PAS 8 M (5) 0 0 The maximum value of the loading term can be found baed on the equivalence beteen the maximum value of beam action hear and a M neg M limiting nominal one-ay hear tre a precribed by the code. Uing the Fig. : Equilibrium of radial trip one-ay hear capacity from ACI 318-

3 11, empirically defined a the inclined cracking load [10], a found to lead to the bet reult [9]. When the maximum value of the loading term i limited by beam action hear, it i expreed a follo: ACI 0,1667d f (6) ck In Eq. (6), ACI i given in [kn/m] ith f ck in [MPa] and d in [mm]. 3. Development of Modified Bond Model 3.1 Concentrated load cloe to the upport continuou upport F pre,3 F pre, F pre,1 M po,x 0 c = l load load 0 y x To apply the Bond Model to the cae of lab ubjected to concentrated load cloe to upport, it i neceary to take into account direct load tranfer. For one-ay lab, the different propertie in the pan direction and in the tranvere direction need to be taken into account. The four cantilevering trip branching out from the load can be tudied together to ketch the aumed moment ditribution, Fig. 3. In Fig. 3, the geometry and layout from the lab hear experiment carried out at Delft Univerity of Technology [11] are ued. For the x-direction trip beteen the concentrated load and the upport, direct load tranfer beteen the load and the upport i taken into account. Regan decribed the punching capacity of lab under concentrated load cloe to upport by conidering the 4 ide of the punching perimeter eparately [1]. To take into account the beneficial influence of direct load tranfer, the capacity of the ide of the punching perimeter at the upport a enhanced ith a factor d x / a v, in hich d x i the effective depth to the reinforcement in the x-direction and a v i the face-to-face ditance beteen the load and the upport. Similarly, it i propoed to increae the capacity of the trip beteen the load and the upport by enhancing the capacity ith d x / a v for 0,5d x < a v < d x and 4 for a v 0,5d x. 3. Load cloe to the continuou upport l c = b load M neg,x imple upport l M neg,y Fig. 3: Sketch of application of Bond Model to lab under concentrated load cloe to upport. M up a M pan Fig. 4: Illutration of moment at a continuou upport A firt extenion of the Bond Model deal ith load applied cloe to the continuou upport, in hich the poitive moment reinforcement can increae the total flexural capacity. A the negative and poitive moment capacitie are not activated in the ame croection of the trip and becaue yielding of the compreion reinforcement i not aumed in the model, the folloing expreion i propoed to take into account the effect of the poitive moment reinforcement: M = M neg + λ moment M po. The factor λ moment range from 0 (for imply upported edge) to 1 (for fully retrained cae) and equal: λ moment = M up / M pan 1 (Fig. 4). When the concentrated load i placed cloe to a continuou upport, to quadrant experience the change in moment from hogging moment M up to agging moment M pan. A a reult, the combined effect of the top and bottom reinforcement hould be taken into account on the three trip that border thee to quadrant: the to y-direction trip a ell a the x-direction trip beteen the load and the upport.

4 4. Study of edge effect 4.1 Extreme cae When load are placed cloe to the edge of the lab, the edge effect play a role. Before tudying the cae of concentrated load cloe to the free edge of the lab, to extreme cae are conidered: the cae here the load i in the middle of the idth and no edge effect i preent and the cae here the load i placed right at the edge and only 3 trip can be ued. When no edge effect i preent, the load and trip are a hon in Fig. 5a. All trip are loaded ith and the capacity of each trip i P AS,1 a given by Eq. (4). The econd cae i the cae in hich the load i placed right at the edge, a hon in Fig. 5b. For thi cae, only 3 trip can be ued and only quadrant reult. A a reult, the trip in the y-direction are loaded ith and the trip in the x-direction i loaded ith. The cae of loading ith give a capacity P AS,1 from Eq. (4). For the trip in the y-direction a load of i placed over a loaded length of l a ketched in Fig. 6. Horizontal equilibrium for the trip at the edge give: Pedge l (7) Moment equilibrium give: 4. Decribing the edge effect for the loaded length l M (8) o that: l M (9) Subtituting the value for l into the expreion of P edge from Eq. (7) give the capacity of the y- direction trip a: P edge M(10) To decribe the edge effect, the loaded length of the x-direction trip beteen the load and the edge i tudied. When all trip are loaded ith, a hon in Fig. 5a, the loaded length ill be: l (a) Fig. 5: Strip and load for: (a) load in middle of lab, or (b) load at the edge M neg P edge l Fig. 6: Loading on lab trip in y-direction equal hen load i placed at the edge of the idth M (b) (11) M po For load cloe to the edge, the ditance beteen the edge and the face of the load, l edge (Fig. 7a) i expreed a (ith b r the ditance beteen the centre of the load and the edge and l load the length of the load): lload ledge br (1) If l from Eq. (11) i larger than l edge from Eq. (1), the model ould be auming a loaded length of the trip that i longer than hat i phyically poible. Therefore, for thoe cae the edge effect need to be taken into account: the loaded length l need to be limited to the edge length l edge. The capacity of the trip beteen the edge and the load then become:

5 P l (13) if the edge effect i preent, and edge edge ACI, y P l (14) if no edge effect i preent. y ACI, y α α (a) M neg (b) Fig. 7: Auming a uniform influence of the edge effect: (a) trip and quadrant, (b) reulting loading on trip 4.3 Influence of the torional moment (a) (c) l edge P edge l α x y (b) (d) The previou conideration did not take into account the fact that not the full load ill be tranferred to the trip beteen the load and the edge hen the edge effect i preent. Let no aume that only a fraction α ith α < 1 can be carried off in the quadrant intead of. The value of α can be expreed a: ledge for l edge l. (15) l Thi ituation i ketched in Fig. 7a. A a reult, a load of α intead of act on the y-direction trip beteen the load and the upport, a hon in Fig. 7b. The reulting capacity of thi trip i then: P l. (16) edge ACI, y Fig. 8: Bending moment (a) in S1T1, (b) in S4T1, torional moment M po Note that Eq. (16) i valid for the cae in hich an edge effect i preent a ell a the cae for hich no edge effect i preent through the ue of the factor α. Torion a neglected in the original Bond Model, a the influence of the torional moment i mall hen load are placed on infinitely large lab. Hoever, hen the load i placed cloe to the edge of a lab, torional ditre influence the capacity. A a reult of the influence of torion, a maller capacity than predicted by the (Modified) Bond Model i found. (c) in S1T1, (d) in S4T1. The magnitude and influence of the torional moment i tudied baed on experiment S1T1 and S4T1, a carried out in the Stevin Laboratory of Delft Univerity of Technology [11]. In S1T1, a ingle concentrated load i placed on a reinforced concrete lab in the middle of the idth. In S4T1, the load i placed cloe to the edge of the lab. The concrete compreive trength of lab S1 a f c,mea = 35,8MPa and for S4, f c,mea = 50,5MPa. Both lab have a longitudinal reinforcement ratio of ρ l = 0,996%, yet a different tranvere reinforcement ratio: ρ t = 0,13% for S1 and ρ t = 0,18% for S4. All other material and geometric propertie are the ame for the to experiment. Failure occur in S1T1 for a concentrated load of 954kN, hich correpond to a hear force at the upport of 799kN. In S4T1, failure occur for a concentrated load of 1160kN, correponding to a hear force at the upport of 964kN.

6 To tudy the influence of the torional moment, linear finite element model are ued. The reulting principal bending moment for S1T1 are hon in Fig. 8a and for S4T1 in Fig. 8b. The principal bending moment are determined a: 1 m1 mx my ( mx my) 4mxy (17) The torional moment for S1T1 and S4T1 are hon in Fig. 8c and in Fig. 8d, repectively. The torque moment are determined a: m t max ( mx my) 4mxy (18) Table 1: Reult of bending and torional moment at location of load Experiment m 1 m tmax m tmax /m 1 S1T1 315 knm/m 0 knm/m 6% S4T1 41,57 knm/m 181,35 knm/m 44% The reulting bending and torional moment at the location of the load, around hich the cruciform hape reulting from the trip of the Modified Bond Model i applied, are given in Table 1. Table 1 ho that for the experiment ith the load at the edge, the influence of the torional moment i ignificantly larger than for the experiment ith the load in the middle of the idth. x y β α α + β). The capacity of the x-direction trip i then: dl Pup (1 ) M, xaci, x (19) a v β l edge P (1 ) M (0) Fig. 9: Propoed deign method for taking into account the influence of the torional moment. x neg, x ACI, x To take the influence of the torional moment into account in the Modified Bond Model, a implified method i propoed hich doe not require the ue of finite element model. Thi method i ketched in Fig. 9. A dicued in the previou ection, the influence of the edge effect can be taken into account by uing the reduction factor α. For the method propoed in Fig. 9, the edge effect i orking on the y- direction trip beteen the load and the edge. The capacity of the trip i then P edge a given in Eq. (16). The influence of torion i taken into account by uing the factor β 1. Becaue the influence of torion i related to the edge effect, thi influence i aumed to act only in the quadrant beteen the load and the edge. Moreover, it i aumed that thi influence act only in the y-direction, the eaker direction in hich a loer amount of reinforcement i provided. In other ord, due to torion, it i aumed that the capacity of the x- direction trip, hich carry the larger part of the load in one-ay lab, i reduced. A a reult, the x-direction trip are loaded ith (1 In Eq. (19) and (0) the folloing ymbol are ued: P up = the capacity of the trip beteen the load and the upport P x = the capacity of the x-direction trip from the load toard the pan of the lab d l = effective depth to the longitudinal reinforcement a v = face-to-face ditance beteen the load and the upport M,x = M neg,x + λ moment M po,x = the moment capacity, a explained in 3. M neg,x = the hogging moment capacity of the x-direction reinforcement M po,x = the agging moment capacity of the x-direction reinforcement ' ACI, x 0,1667dl fc ith f c in [MPa].

7 The value of the factor β, hich take torion into account, can be expreed baed on the reulting bending moment m 1 and the reulting torional moment m tmax. Since the goal of the Modified Bond Model i to provide a deign method hich can be ued ithout the need of finite element program, a implification i to ue β = 0 for loading cae here the maximum bending moment and maximum torional moment in a lab coincide, uch a the cae of load cloe to the edge, and to ue β = 1 hen the maximum bending moment coincide ith a mall torional moment, uch a for loading in the middle of the lab idth. P exp (kn) 5. Comparion to experimental reult (a) V exp (kn) (b) V exp,ec (kn) (c) P MBM (kn) V ACI (kn) V R,c (kn) Fig. 10: Comparion beteen experiment and predicted value according to: (a) Modified Bond Model, (b) hear capacity from ACI , (c) hear capacity from EN :005 To tudy the performance of the Modified Bond Model, all reult of the experiment on lab S1 to S10, teted at Delft Univerity of Technology [11] are compared to the reult obtained ith the Modified Bond Model. The maximum concentrated load in the experiment, P exp, i compared to the maximum capacity obtained ith the Modified Bond Model, P MBM. Mean value are ued for the material propertie. The evaluation hoed an average ratio P exp /P MBM = 1,19 ith a tandard deviation of 0,13 and a coefficient of variation of 11%. Conidering that the problem under tudy i a hear problem ith a large number of parameter that have been varied in the experiment, the tatitical reult are excellent. The reult are hon graphically in Fig. 10a. Next, thee experimental reult are compared to the hear capacitie according to ACI [4] and NEN-EN :005 [3], to evaluate the performance of the Modified Bond Model a compared to exiting code proviion. Note that the code proviion predict a maximum ectional hear force, hile the Modified Bond Model predict a maximum concentrated load. The value of the experimental ectional hear force V exp compared to the hear capacity according to ACI , V ACI, are hon in Fig. 10b. The reduced ectional hear force V exp,ec i compared to the hear capacity according to NEN-EN :005, V R,c, in Fig. 10c. The reduced ectional hear force V exp,ec take into account a reduction of the contribution of the load cloe to the upport (0,5d x a v d x ) to the hear force by β = a v / d x a precribed by NEN-EN : (6). The ratio of V exp /V ACI ha an average value of,67, ith a tandard deviation of 1,00 and a coefficient of variation of 37%. For V exp,ec /V R,c the average value i 1,99, ith a tandard deviation of 0,7 and a coefficient of variation of 13%. The tatitical reult ho that the Modified Bond Model give a better prediction of the experiment. The 45 o line in Fig. 10a, b, and c indicate experimental hear capacitie that are exactly a predicted by the method under conideration. When uing ACI and NEN-EN :005 to compare to the

8 experimental reult, an increaing conervatim ith increaing hear capacitie i een. The Modified Bond Model, on the other hand, ho more conitent reult, a the cloud of tet reult lie above and parallel to the 45 o line. 6. Summary and concluion A model for the hear capacity of reinforced concrete lab under concentrated load cloe to upport i propoed: the Modified Bond Model. Thi model i baed on the Bond Model for concentric punching hear and take direct load tranfer beteen the load and the upport into account. It i applicable to lab ith different amount of reinforcement in the x- and y-direction and ith a concentrated load cloe to the upport. For continuou upport, the effect of the poitive moment reinforcement i taken into account. For load cloe to the edge, the edge effect and the influence of torion are tudied and a implified method i propoed.the experimental reult indicate that the Modified Bond Model i an improvement a compared to the code method for lab under concentrated load cloe to upport. Acknoledgement The author ih to expre their gratitude and incere appreciation to the Dutch Minitry of Infratructure and the Environment (Rijkatertaat) for financing thi reearch ork and to InfraQuet for coordinating the cooperation beteen Delft Univerity of Technology, Rijkatertaat and the reearch intitute TNO. Reference [1] COLLINS M.R., BENTZ E.C., and SHERWOOD E.G., "Where I Shear Reinforcement Required? Revie of Reearch Reult and Deign Procedure", ACI Structural Journal, Vol. 105, No. 5, 008, pp [] REINECK K.-H., BENTZ E.C., FITIK B., KUCHMA D.A., and BAYRAK O., "ACI- DAfStB Databae of Shear Tet on Slender Reinforced Concrete Beam ithout Stirrup", ACI Structural Journal, Vol. 110, No. 5, 013, pp [3] CEN, Eurocode : Deign of Concrete Structure - Part 1-1 General Rule and Rule for Building. NEN-EN :005, 005, Comité Européen de Normaliation, Bruel, Belgium, p. 9. [4] ACI COMMITTEE 318, Building Code Requirement for Structural Concrete (ACI ) and Commentary, 011, Farmington Hill, MI, American Concrete Intitute, p [5] ASCE-ACI TASK COMMITTEE 46, "Shear-Strength of Reinforced-Concrete Member - Slab", Journal of the Structural Diviion-ASCE, Vol. 100, No. NST8, 1974, pp [6] REGAN P.E., "Reearch on Shear: A Benefit to Humanity or a Wate of Time", The Structural Engineer, Vol. 71, No. 19, 1993, pp [7] LANTSOGHT E.O.L., VAN DER VEEN C., DE BOER A., and WALRAVEN J.C., "Recommendation for the Shear Aement of Reinforced Concrete Slab Bridge from Experiment ", Structural Engineering International, Vol. 3, No. 4, 013, pp [8] ALEXANDER S.D.B., Bond Model for Strength of Slab-Column Joint, Univerity of Alberta, p. 8. [9] ALEXANDER S.D.B. and SIMMONDS S.H., "Bond Model for Concentric Punching Shear", ACI Structural Journal, Vol. 89, No. 3, 199, pp [10] MORROW J. and VIEST I.M., "Shear Strength of Reinforced Concrete Frame Member ithout Web Reinforcement", ACI Journal Proceeding, Vol. 53, No. 3, 1957, pp [11] LANTSOGHT E.O.L., VAN DER VEEN C., and WALRAVEN J.C., "Shear in One-Way Slab under a Concentrated Load Cloe to the Support", ACI Structural Journal, Vol. 110, No., 013, pp [1] REGAN P.E., Shear Reitance of Concrete Slab at Concentrated Load Cloe to Support, 198, Polytechnic of Central London, London, United Kingdom, p. 4.